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calculus | The Limit Comparison Test V1 | https://math.stackexchange.com/questions/2089685/the-limit-comparison-test-v1 | <p>I'm currently learning Calculus 2, more specifically I'm learning about sequences and series. I'm not enjoying this section as much as I thought I would, this is because I'm having to learn all these different tests to determine the convergence and being shown no justification as to why it works. I've been shown the... | <p>The limit comparison test is very powerful. One of my favorite applications is this: Determine the convergence/divergence of
$$\sum_{n=1}^\infty \frac 1{n^{1+1/n}}.$$</p>
<p>The heuristic is very simple. For large $n$, we're saying that basically $a_n = cb_n$ (for some positive number $c$). Ignoring small values of... | 200 |
calculus | Can I find the surface area for cone by the surface area formula? | https://math.stackexchange.com/questions/4165567/can-i-find-the-surface-area-for-cone-by-the-surface-area-formula | <p>From section 8.2 in Stewart's calculus, I think I understand the derivation of the surface area formula <span class="math-container">$\int_{a}^{b} 2 \pi y \sqrt{1 +y'} dx$</span>. It's developed from the surface area of frustums, which is developed from the surface area of a cone, which was found by a sector's area.... | 201 | |
calculus | Problem about Conservative Vector | https://math.stackexchange.com/questions/4093830/problem-about-conservative-vector | <p>Let <span class="math-container">$F:\mathbb R^n\to \mathbb R^n$</span> where <span class="math-container">$F(x_1,x_2,...,x_n)=(x_1,x_2,...,x_n)$</span>. show that <span class="math-container">$F$</span> is a conservative vector fields, that is, there is a potential function <span class="math-container">$f$</span> wh... | <p><span class="math-container">$f(x_1,x_2,\ldots,x_n)=\frac{1}{2}(x_1^2+x_2^2+\ldots+x_n^2)$</span></p>
| 202 |
calculus | Proving the Monotonicity of a function? | https://math.stackexchange.com/questions/850467/proving-the-monotonicity-of-a-function | <p>Given a function of numerous variables, say $f(x,y,z)$, what are the usual approaches one can take to prove that $f(x,y,z)$ is monotonically increasing, or decreasing in $x$?</p>
<p>I am aware that one can calculate the functions derivative and attempt to prove that it is positive or negative for any $y$ and any $z... | <p>It depends on how your function is defined. Many times proving directly is the easiest. </p>
<p>i.e. Arbitrarily fix $x_1 < x_2$ and $y_{*}, z_{*}$ in your spaces $X$, $Y$, $Z$ that compose the domain of your function $X\times Y\times Z$. Show that $f(x_{1},y_{*},z_{*})<f(x_{2},y_{*}, z_{*})$. (same as what J... | 203 |
calculus | $\lim_{h \to 0} \int_{x}^{x+h} \ln(t) dt$ | https://math.stackexchange.com/questions/3463830/lim-h-to-0-int-xxh-lnt-dt | <p><span class="math-container">$\lim_{h \to 0} \int_{x}^{x+h} \ln(t) dt$</span></p>
<p>Unless I'm missing something, isn't this just <span class="math-container">$0$</span> due to how the integral is just <span class="math-container">$\int_{x}^{x}=0$</span> </p>
<p>I'm sure I could integrate the inside and then eval... | <p>Yes, you're correct. You can argue it that way, or even if you go as far as integrating first you'll find the same result:</p>
<p><span class="math-container">\begin{eqnarray*}
\lim_{h\to 0} \int_x^{x+h}\ln(t)dt & = & \lim_{h\to 0} \left . t\ln(t) - t \right |_x^{x+h} \\
& = & \lim_{h\to 0}(x+h)\ln... | 204 |
calculus | Functional Gaussian Integral | https://math.stackexchange.com/questions/4169820/functional-gaussian-integral | <p>I am trying to reproduce the result below</p>
<p><span class="math-container">$$\int\mathcal{D}V~e^{-\int dx~[a V^{2}(x)+iV(x)\int_{-\infty}^{\infty}dt~ \bar{\psi}^{a}(x,t)\gamma_{ab}\psi^{b}(x,t)]}= e^{-\frac{1}{4a}\int dx\int\int_{-\infty}^{\infty}dt dt'(\bar{\psi}^{a}(x,t)\psi^{a}(x,t))(\bar{\psi}^{b}(x,t')\psi^{... | 205 | |
calculus | Apply function fractional times | https://math.stackexchange.com/questions/1250978/apply-function-fractional-times | <p>For example, one can apply $\cos x$ to number $a$ one time to get $\cos a$, two times $\cos \cos a$, three times $\cos \cos \cos a$, and so on. Is there a way to define fractional application for $\cos$? Or for any other function? Maybe exists general theory for that?</p>
| <p>Obviously $f(x,n)$ defined as taking $g$ $n$ times of $x$ is a function $(\mathbb R,\mathbb N)\to\mathbb R$. Any extension of $f$ to $(\mathbb R, \mathbb Q)$, could be considered a way to apply $g$ a rational number of times, if you extend it to $\mathbb R^2$, you could consider it a definition of applying $g$ any (... | 206 |
calculus | Fourier Series of $\cos^{n}x$ | https://math.stackexchange.com/questions/627718/fourier-series-of-cosnx | <p>I need help evaluating the integrals in Fourier Series.</p>
<p>For example, for the function <span class="math-container">$\cos^{2}x$</span>, I can evaluate <span class="math-container">$a_0$</span>, <span class="math-container">$a_n$</span>, and <span class="math-container">$b_n$</span>, where <span class="math-con... | <p>Hint:</p>
<p>$\cos^{2}(x)=\frac{1+\cos(2x)}{2}$ </p>
<p>and </p>
<p>$\cos(x)\cos(y)=\frac{1}{2}\big(\cos(x-y)-\cos(x+y)\big)$</p>
| 207 |
calculus | The Fundamental Theorem of Calculus Questions? | https://math.stackexchange.com/questions/4115061/the-fundamental-theorem-of-calculus-questions | <p><strong>Background Information:</strong>
One of the most important ideas that Green discussed in his Essay is the connection between
what happens within a body and the properties of that body’s surface. He realized that, because the
boundary of an object is one dimension lower than the interior, the connection can b... | <p>Well, we know that the boundary of <span class="math-container">$[a,b]$</span> is <span class="math-container">$\{a,b\}$</span>. Then, in some sense,
<span class="math-container">$$\int_a^b f = \int_{\partial[a,b]} F = \int_{\{a,b\}} F.$$</span>
Put another way,
<span class="math-container">$$\int_{[a,b]} \partial f... | 208 |
calculus | What is the $\lim_{h\to0}$ of the average value of $f(x)$ on the interval $[x, x+h]$ | https://math.stackexchange.com/questions/2121801/what-is-the-lim-h-to0-of-the-average-value-of-fx-on-the-interval-x-x | <p>If $f(x)$ is a continuous function on the interval $[x, x+h]$, find $$\lim_{h\to 0} f(x)_{avg}$$ </p>
<p>I suspect I'm using the limit definition of the derivative, and to obtain the average value I've integrated over $[{x, x+ h}]$: $$\frac{\int_x^{x+h}f(x + h) - \int_x^{x+h}f(x )}{(x+h) - x} $$</p>
<p>What is the... | <p>I will note you did the first step wrong. Note that you should have</p>
<p>$$g(x)=\int_a^xf(t)\ dt$$</p>
<p>Differentiating:</p>
<p>$$g'(x)=\lim_{h\to0}\frac{\int_a^{x+h}f(t)\ dt-\int_a^xf(t)\ dt}h=\lim_{h\to0}\frac1h\int_x^{x+h}f(t)\ dt$$</p>
<p>If we suppose the following statement, perhaps as axiom, that $\i... | 209 |
calculus | Meaning of $\int_{a}^{b}dx$ | https://math.stackexchange.com/questions/4164659/meaning-of-int-abdx | <p>I know the meaning of <span class="math-container">$\int_{a}^{b}f(x)dx$</span>, which is <span class="math-container">$F(b)-F(a)$</span>. Geometrically, it gives us the area under the graph from <span class="math-container">$x=a$</span> to <span class="math-container">$x=b$</span></p>
<p>But what does <span class="m... | <p><span class="math-container">$\int_a^bdx$</span> would simply be the same as <span class="math-container">$\int_a^b 1\ dx$</span>, so it would be the area under the curve <span class="math-container">$f(x)=1$</span>, which is indeed equal to the length of the segment from <span class="math-container">$a$</span> to <... | 210 |
calculus | Riemann sum problem. | https://math.stackexchange.com/questions/2985141/riemann-sum-problem | <p>I had a practice midterm that had the following question: </p>
<p><span class="math-container">$A = \lim_{x \to\infty} R_n = \lim_{x \to\infty} (\sum_{i=1}^{n} f(x_i)\triangle x)$</span></p>
<p>Use this definition to find an expression for the area under the graph of <span class="math-container">$f(x) = \frac{log ... | <p>I get <span class="math-container">$\lim_{n\to\infty}\sum_{i=1}^n 2n\cdot \frac{\log(n+2i)-\log n}{(n+2i)^2}$</span>.</p>
<p>This is straight forward using <span class="math-container">$\log\frac ab=\log a-\log b$</span>, plus a little algebra. </p>
| 211 |
calculus | Derivative using Fundamental Theorem of Calculus when integrand has product of two functions? | https://math.stackexchange.com/questions/4174763/derivative-using-fundamental-theorem-of-calculus-when-integrand-has-product-of-t | <p>I want to find the derivative of the following:</p>
<p><span class="math-container">$$exp \left( -\int_{t-\tau(t)}^t \frac{\mu(x)U(x)}{S} \,dx \right)$$</span></p>
<p>I tried to use the Fundamental theorem of calculus of the form:</p>
<p><span class="math-container">$$\frac{d}{dx}\int_0^x t^3 \,dx = f(x)\frac{dx}{dx... | <p>As you say, the <span class="math-container">$\exp$</span> part is straightforward, so let's look at the derivative of
<span class="math-container">$$
Q = \int_{t-\tau(t)}^t \frac{\mu(x)U(x)}{S} \,dx.
$$</span></p>
<p>A standard thing to do here is to write this as a sum of two integrals, splitting at some arbitrary... | 212 |
calculus | Solving for $i$, given $S=\sum_{n=1}^m \frac{A_n}{(1+i)^{t_n}}$ | https://math.stackexchange.com/questions/4174327/solving-for-i-given-s-sum-n-1m-fraca-n1it-n | <p>I'm faced with a problem that is unfortunately beyond my current mathematical skills.</p>
<p>I have an equation that goes like this:</p>
<p><span class="math-container">$$
S=\sum_{n=1}^m \frac{A_n}{(1+i)^{t_n}}
$$</span></p>
<p>My goal is to transform it so that I arrive at formula to calculate <code>i</code>. I wis... | <p>As said in comments, solving for <span class="math-container">$i$</span> the equation <span class="math-container">$$S=\sum_{n=1}^m \frac{A_n}{(1+i)^{t_n}}$$</span> will require numerical methods.</p>
<p>However, since <span class="math-container">$i \ll 1$</span>, we can try to obtain <em>approximations</em>.</p>
<... | 213 |
calculus | Why do we use only the positive root when differentiating an inverse trig function whose inside is linear? | https://math.stackexchange.com/questions/4171429/why-do-we-use-only-the-positive-root-when-differentiating-an-inverse-trig-functi | <p>I'm taking calc 1, and I'm struggling with these types of problems. Example: differentiate <span class="math-container">$y=\sin^{-1}(-4x-1)$</span></p>
<p>I think I understand how to solve these problems, but my answers typically have <span class="math-container">$\pm$</span> roots, like in this example: <span cla... | 214 | |
calculus | decomposing a fraction into partial fractions | https://math.stackexchange.com/questions/686382/decomposing-a-fraction-into-partial-fractions | <p>could someone please help me to decompose the following fraction into partial fractions?</p>
<p>$$\frac{1}{(a-x)(b-x)^{1/2}}$$</p>
<p>where a and b are just constants.</p>
<p>Thanks</p>
| <p>Usually, a partial fraction decomposition is only possible for rational functions. The square root inside the denominator would prevent this kind of decomposition. In</p>
<p>$$\frac{a-b}{(x-a)\sqrt{x-b}}=\sqrt{x-b}\frac{(x-b)-(x-a)}{(x-a)(x-b)}$$</p>
<p>for $x>b$ one can decompose the second factor, but the squ... | 215 |
calculus | Multiplying top and bottom by $ \cos (x) $ to solve integral? | https://math.stackexchange.com/questions/4140640/multiplying-top-and-bottom-by-cos-x-to-solve-integral | <p>Please take a look at this integral. Why is this method not a valid way of solving this integral?</p>
<p><span class="math-container">$\displaystyle \int \frac{1}{\sin (x) \cos(x)} \ dx = \int \frac{\cos (x)}{\sin (x) \cos^2(x)} \ dx = \int \frac{\cos(x)}{\sin (x) (1-\sin^2 (x))} \ dx = \int \frac{1}{u(1-u^2)} \ du... | <p>You have done the partial fractions incorrectly. It should be:</p>
<p><span class="math-container">$$\frac{1}{u(1-u^2)} = \frac{1}{u} - \frac{1}{2(u+1)} \color{red}{-} \frac{1}{2(u-1)}$$</span></p>
| 216 |
calculus | Trig substitution reversion issue $\pm$ | https://math.stackexchange.com/questions/4140799/trig-substitution-reversion-issue-pm | <p>I am working through the 100 integrals video on YouTube and I came across this question. I solved it correctly, but I want some clarification on a step that I made.</p>
<p><span class="math-container">$$\displaystyle\int \frac{e^x\sqrt{e^x-1}}{e^x+3} \ dx$$</span></p>
<p><span class="math-container">$$ u = e^x +3 \... | <p>Briefly, your original integrand is non-negative for all real <span class="math-container">$x \geq 0$</span>, so the antiderivative sought is increasing. Only the <span class="math-container">$+$</span> branch of square root gives an increasing function of <span class="math-container">$x$</span>.</p>
<p>In more deta... | 217 |
calculus | Hyperbolic Trig Proofs/Definitions | https://math.stackexchange.com/questions/1834088/hyperbolic-trig-proofs-definitions | <p>My first post! Hello World!</p>
<p>I was looking back at my notes from Calculus I & II (my how the time has passed!)
I came back across Hyperbolic Trig Functions, sinh, cosh, etc.</p>
<p>I remember being presented the identities, how to use them, derivatives, integrals, etc. I was wondering if anyone could pro... | <p>$\cosh(x)=\frac{e^x+e^{-x}}{2}$ </p>
<p>$\sinh(x)=\frac{e^x-e^{-x}}{2}$</p>
<p>by definition. Most other identities follow from basic calculus. </p>
| 218 |
calculus | How does $e^{-\ln x} = e^{\ln(1/x)}$ | https://math.stackexchange.com/questions/4159565/how-does-e-ln-x-e-ln1-x | <p>I understand the inverse of e^{x} is the natural logarithm. However I don't understand how the following expression is true:</p>
<p><span class="math-container">$e^{-\ln x} = e^{\ln(1/x)}$</span></p>
<p>Any assistance is appreciated.</p>
| <p>One of the properties of logarithms is the following:</p>
<p><span class="math-container">$$\log({x^k}) = k\log{x}$$</span></p>
<p>Therefore when you have <span class="math-container">$-\ln x$</span>, you essentially go backwards:</p>
<p><span class="math-container">$$-\ln x = -1 \times \ln x = \ln(x^{-1}) = \ln \le... | 219 |
calculus | What is meant by $f(x)$ is function of $x$. Or $f(x)$ as a function of $y$? | https://math.stackexchange.com/questions/4180376/what-is-meant-by-fx-is-function-of-x-or-fx-as-a-function-of-y | <p>I am so confused about the terminology and vocabulary here. I tried googling it but couldn't find anything satisfactory. I have a test tomorrow. I would be glad if someone could explain what this conceptually means.</p>
| <p>I'm guessing you're currently in high school so without beating around the bush and/or being pedantic and asking you for definitions (which is futile as it is obvious you have this question <em>because</em> you don't know your definitions in the first place) <em>and</em> since you have an exam coming up very soon, I... | 220 |
calculus | Why is $\lim_{\delta x\to0} \frac{\delta x}{\delta x} = 1$ | https://math.stackexchange.com/questions/3118997/why-is-lim-delta-x-to0-frac-delta-x-delta-x-1 | <p>Why is <span class="math-container">$\lim_{\delta x\to0} \frac{\delta x}{\delta x} = 1$</span>, considering that both are infinitesimally small but may be different from each other?</p>
<p>Also, if so, why can I not replace <span class="math-container">$\frac{\delta f}{\delta x} = \frac{\frac{1}{x + \delta x} - \fr... | <blockquote>
<p>Why is <span class="math-container">$\lim_{\delta x\to0} \frac{\delta x}{\delta x} = 1$</span>, considering that both are infinitesimally small but may be different from each other?</p>
</blockquote>
<p>No, they are never different from each other. <span class="math-container">$\delta x = \delta x$</... | 221 |
calculus | Bound to $\sum_i^n \sqrt{a_i}$ | https://math.stackexchange.com/questions/4180683/bound-to-sum-in-sqrta-i | <p>I am trying to find a bound to this: <span class="math-container">$\sum_i^n \sqrt{a_i}$</span> when <span class="math-container">$a_i$</span> are positive integers.
I think that the following is true, but can't prove it.
<span class="math-container">$$\sum_i \sqrt{a_i} \le (\sum_i a_i)^{3/4}$$</span>
I need a tighte... | 222 | |
calculus | Trap Rule for sin(x) | https://math.stackexchange.com/questions/393619/trap-rule-for-sinx | <blockquote>
<p>Use the trapezoidal rule with $N=6$ to approximate the arc length of the curve $f(x) = \sin(x)$ from $x=0$ to $x=\pi$.</p>
</blockquote>
<p>So I found that $\Delta x = \frac{\pi}{6}$ which means that my interval points are $0,\frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{5\pi}{6}$ and $\pi$.</p... | <p>The formula for step size is given by: </p>
<p>$$\displaystyle h = \frac{b-a}{N} = \frac{\pi - 0}{6} = \frac{\pi}{6}$$</p>
<p>We are also given that $x_0 = 0$.</p>
| 223 |
calculus | force is generally a function of $\mathbf{r}(t)$, $\mathbf{v}(t)$ and $t$ | https://math.stackexchange.com/questions/3001165/force-is-generally-a-function-of-mathbfrt-mathbfvt-and-t | <blockquote>
<p>Force is generally a function of <span class="math-container">$\mathbf{r}(t)$</span>, <span class="math-container">$\mathbf{v}(t)$</span> and <span class="math-container">$t$</span>. <span class="math-container">$$1-)\begin{cases}
\mathbf F: \mathbb R^3\times\mathbb R^3 \times \mathbb R \ \rightarro... | <p>Imagine a world where the sun is moving rapidly in our chosen coordinate system and has a huge current loop around the equator. The earth has a huge negative charge and is moving in the gravitational and magnetic fields of the sun. At any time <span class="math-container">$t$</span> we need <span class="math-cont... | 224 |
calculus | Can L'Hopital's rule be applied only for a part of a function? | https://math.stackexchange.com/questions/2600840/can-lhopitals-rule-be-applied-only-for-a-part-of-a-function | <p>For example, in
$\lim_{x\to 0_+} (x^2 \ln x+bx+c) $
can it be applied only for $x^2\ln x$? (of course not in this form)</p>
| <p>If you have $\lim(f(x)+g(x)+\cdots)$, then you can always compute the limit term-wise, i.e.</p>
<p>$$\lim f(x)+\lim g(x)+\cdots$$</p>
<p>as long as all the single limits exist. So in order to apply l'Hospital to only one of the terms, first use this rule to get</p>
<p>$$\lim (x^2\ln(x)+bx+x)=\lim(x^2\ln(x))+\lim(... | 225 |
calculus | Can't see why one of these functions is conservative, and the other isn't. | https://math.stackexchange.com/questions/4182392/cant-see-why-one-of-these-functions-is-conservative-and-the-other-isnt | <p>I am really confused here: Why one of these functions is conservative, while the other not?</p>
<p><span class="math-container">$F_{1} = \frac{-y \hat i + x \hat j}{x^2+y^2}$</span></p>
<p><span class="math-container">$F_{2} = \frac{x \hat i + y \hat j}{x^2+y^2}$</span></p>
<p>Suppose both these vector functions are... | <p>You are checking whether a vector field is a gradient by seeing if it is curl free. Over a simply-connected domain, being curl free and being a gradient are equivalent. The purpose of this exercise is that these two concepts, equivalent over nice domains, are no longer equivalent on the punctured plane. Just because... | 226 |
calculus | Moment of Inertia around z axis | https://math.stackexchange.com/questions/1617665/moment-of-inertia-around-z-axis | <p>Hello I am having difficulty with the following;</p>
<p>I am wanting to find I, the moment of inertia about the z axis of the region that is bounded by the paraboloid $z=x^{2}+y^{2}$ and the $z=1$ plane, where the density is proportional to the distance from the z axis.</p>
<p>Here is what I have tried:</p>
<p>I ... | <p>Work in cylindrical coordinates $(r,\theta,z)$. The element of volume is $r\,dr\,d\theta\,dz$. The distance of a point to the $z$ axis is just $r$, and the density $\lambda r$. The paraboloid is $r^2=z$ or $r=\sqrt z$.</p>
<p>So the mass is</p>
<p>$$M=\int_{z=0}^1\int_{\theta=0}^{2\pi}\int_{r=0}^{\sqrt z}\lambda r... | 227 |
calculus | First fundamental theorem of calculus where the bounds are not 0 to x. | https://math.stackexchange.com/questions/4099535/first-fundamental-theorem-of-calculus-where-the-bounds-are-not-0-to-x | <p>Suppose <span class="math-container">$F(x) = \int_{3x+8}^{x^{2}+5x+1}\csc^{2}\left(t\right)dt$</span>. How would one find <span class="math-container">$F'(x)$</span> using the first fundamental theorem of calculus? I am aware of how to do this when the bounds are 0 to f(x) through use of chain rule, but I don't know... | <p>Hint: <span class="math-container">$\int_a^bf(x) dx=\int_a^0f(x) dx+\int_0^bf(x) dx=\int_0^bf(x) dx-\int_0^af(x) dx$</span></p>
| 228 |
calculus | The intuitive meaning of integrals | https://math.stackexchange.com/questions/2992196/the-intuitive-meaning-of-integrals | <p>I am an engineering student and i always encounter problems that needs integrals I know that integral is area under the curve , etc.... but till now i could not develop and intuitive meaning for integration. does integration rely only on the idea of area under the curve. do the physics laws that are based on integra... | 229 | |
calculus | Area under Curve Limits | https://math.stackexchange.com/questions/2484422/area-under-curve-limits | <p>If S be the area of the region enclosed by $y=e^{-x^{2}}$, y=0, x=0 and x=1. </p>
<p>Then
(A) $S \ge \frac {1}{e}$ (B) $S \ge 1-\frac {1}{e}$<br>
(C) $S \le \frac {1}{4}(1+\frac{1}{√e})$ (D) $S \le \frac {1}{√2}+\frac{1}{√e}(1-\frac{1}{√2})$ </p>
<p>The correct answer is A,B and D it is multiple choice
I can ... | <p>(A) follows from (B) as $S\ge1-\frac1e>\frac1e$ (this follows from $e>2$)</p>
<p>Don't know about (D) though... </p>
| 230 |
calculus | how to find a function f(n) (continuous on R) such that $(-1)^{f(n)}$ is positive when $n=1, 2, 5, 6, 9, 10....$, and <0 for other natural number? | https://math.stackexchange.com/questions/4087649/how-to-find-a-function-fn-continuous-on-r-such-that-1fn-is-positiv | <p>Further more, can we have a general way to find <span class="math-container">$f(n)$</span> which is negative whenever we design?
(note: we just take <span class="math-container">$n$</span> as natural number)</p>
<p>I think some function with <span class="math-container">$\sin$</span>, <span class="math-container">$\... | <p>To put it another way: you want continuous <span class="math-container">$f:\Bbb R\to\Bbb R$</span> such that <span class="math-container">$f(n)$</span> is an even integer for <span class="math-container">$n=1,2,5,6,9,10,\ldots$</span> and an odd integer for <span class="math-container">$n=3,4,7,8,11,12,\ldots$</span... | 231 |
calculus | Find max vertical distance | https://math.stackexchange.com/questions/164982/find-max-vertical-distance | <p>What is the maximum vertical distance between the line
$y = x + 20$
and the parabola
$y = x^2$ for $−4 ≤ x ≤ 5?$</p>
<p>What steps do I take to solve this? Do I have to use the distance formula and what do I do with the points it gave me?</p>
<p>If anyone could just bounce me in the right direction that would b... | <p>The vertical distance at $x=a$ is the difference in $y$-coordinates at $x=a$, so it’s $|(x+20)-x^2|$. Now $x^2-x-20=(x+4)(x-5)$, so it’s negative between $x=-4$ and $x=5$. Thus, on the interval $[-4,5]$ we have $|(x+20)-x^2|=x+20-x^2$, not $x^2-x-20$.</p>
<p>Now let $f(x)=x+20-x^2$ and find the maximum of $f(x)$ on... | 232 |
calculus | Limit $\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta }{ \theta } } $ | https://math.stackexchange.com/questions/1111672/limit-lim-theta-to-0-frac-cos2-theta-cos-theta-theta | <p>$$\lim _{ \theta \to 0 }{ \frac { cos2\theta -cos\theta }{ \theta } } $$</p>
<p>Steps I took:</p>
<p>$$\lim _{ \theta \rightarrow 0 }{ \frac { 1-2sin^{ 2 }\theta -cos\theta }{ \theta } } =$$</p>
<p>$$\lim _{ \theta \rightarrow 0 }{ \frac { -2sin^{ 2 }\theta }{ \theta } } +\lim _{ \theta \rightarrow 0 }{ ... | <p>your proof is correct. but if you are going to use $\lim_{\theta \to 0}\frac{1-\cos \theta}{\theta} = 0,$ you could have split $\cos(2\theta) - \cos \theta$ as $(1-\cos \theta) -(1 - \cos 2 \theta)$ at the beginning itself.</p>
| 233 |
calculus | Integral Issues. | https://math.stackexchange.com/questions/1156239/integral-issues | <p>$\displaystyle \int \cosh ^2t\,\sinh ^5t \; \textrm{d}t \,$</p>
<p>Can't for the life of me figure this one out. I have tried various substitutions. The pythagorean hyperbolic identity, the double variable identity. Nothing. Could someone give me a push please. </p>
| <p>With some manipulation using $\cosh^2x-\sinh^2x=1\implies \sinh^4t=(\cosh^2t-1)^2$:
$$\cosh ^2t\,\sinh ^5t =\sinh t \cosh^6 t-2 \sinh t \cosh^4 t+\sinh t \cosh^2 t$$
Now try $x=\cosh t,{\rm d}x/{\rm d}t=\sinh t$</p>
| 234 |
calculus | How to prove this in smart way | https://math.stackexchange.com/questions/1161800/how-to-prove-this-in-smart-way | <p>How to prove this in a a smart way?</p>
<blockquote>
<p>If $y= \sin (m \sin^{-1} (x))$, then $(1-x^2)y^{(n+2)}-(2n+1)x{y^{(n+1)}}+(m^2-n^2)y^{(n)}=0$ derivative.</p>
</blockquote>
<p>I have been able to prove it by differentiating it twice and using Leibniz theorem, but thats very long, is there a nice way to p... | <p>$$y'=m(1-x^2)^{-1/2}\cos(m\sin^{-1}(x))$$
$$y''=mx(1-x^2)^{-3/2}\cos(m\sin^{-1}(x))-m^2(1-x^2)^{-1}\sin(m\sin^{-1}(x)),$$
so that $$(1-x^2)y''-xy'+m^2y=0.$$
this establishes the base case of the recurrence.</p>
<p>Now derive</p>
<p>$$(1-x^2)y^{(n+2)}-(2n+1)xy^{(n+1)}+(m^2-n^2)xy^{(n)}=0$$ and get</p>
<p>$$-2xy^{(... | 235 |
calculus | Equation to a level surface | https://math.stackexchange.com/questions/1188100/equation-to-a-level-surface | <p>Could someone please help me with the following question:</p>
<blockquote>
<p>Consider the function $g(x,y,z)=\ln(x^2-y+z^2)$. Find an equation of the level surface of the function through the point $(-1,2,1)$ which does not have $\ln$. Hint: first find $g(-1,2,1).$</p>
</blockquote>
<p>When I sub in the points ... | <p>$$\ln(x^2-y+z^2)=\ln(x_0^2-y_0+z_0^2)$$
can be rewritten
$$y=x^2+z^2+(y_0-x_0^2-z_0^2).$$</p>
<p>It remains the same paraboloid of revolution, with the apex moving along the axis $y$.</p>
| 236 |
calculus | Finding the parametrization for a sphere? | https://math.stackexchange.com/questions/1423532/finding-the-parametrization-for-a-sphere | <p>Find a parametrization for the circle centered around the origin, of radius 3 and contained in the xz-plane.</p>
<p>So from what I gathered you use the formula of sphere $x^2+y^2+z^2= r^2$ to solve this problem. So you know what the radius is 3 yet how does one find xyz just from having the radius?</p>
| <p>A three dimensional surface in a two dimensional plane!? For a constant <span class="math-container">$y$</span> value we can define a small circle and the plane it is contained in.</p>
<p>We can use spherical coordinates. Choose a particular latitude, translate the circle with arbitrary displacements <span class="ma... | 237 |
calculus | Computing $\bigtriangledown^2(1/r)$ | https://math.stackexchange.com/questions/1445477/computing-bigtriangledown21-r | <p>Given that:</p>
<p>$$\vec{r} = x\hat{i}+y\hat{j}+z\hat{k}$$</p>
<p>and $r$ is the magnitude of $\vec{r}$</p>
<p>Then what is:</p>
<p>$$\bigtriangledown^2(1/r)$$</p>
<p><strong>EDIT:</strong>
I know that $\bigtriangledown^2F(x)$ is the divergence of the gradient of $F(x)$ thus my attempt to solve the question wa... | <p>Hint: In Cartesian coordinates</p>
<p>$$\bigtriangledown^2f(x,y,z)=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+\frac{\partial^2f}{\partial z^2}$$and here$$r=|\vec{r}|=\sqrt{x^2+y^2+z^2}$$
$$f=\frac{1}{r}=(x^2+y^2+z^2)^{\frac{-1}{2}}$$
$$\frac{\partial f}{\partial x}=-\frac{x}{(x^2+y^2+z^2)^{\f... | 238 |
calculus | Finding discontinuities points | https://math.stackexchange.com/questions/1536722/finding-discontinuities-points | <blockquote>
<p>find discontinuities points of the function $f(x)=x-\lfloor{x}\rfloor$</p>
</blockquote>
<p>I know that there is no limit $f(x)=\lfloor{x}\rfloor$ when $x\in \mathbb{N}$ Is it sufficient to say that therefore there are discontinuities points when $x\in \mathbb{N}$?</p>
| <p>In every open interval $(n,n+1)$, we have $\lfloor x\rfloor=n$, hence $f(x)=x-\lfloor x\rfloor=x-n$, which is well-known to be a continuous function.</p>
<p>Hence the only discontinuities are at $n$, as $\lim_{x\to n^{-}}f(x)=\lim_{x\to n^{-}}(x-n+1)=1$ while $\lim_{x\to n^{+}}f(x)=\lim_{x\to n^{+}}(x-n)=0$.</p>
| 239 |
calculus | How to plot $f(x)=x^{2/3}$ | https://math.stackexchange.com/questions/1691903/how-to-plot-fx-x2-3 | <p>I'm using Leithold's book to teach calculus. In a exercise Leithold asks how to draw $f(x)=x^{2/3}$. I don't know how to plot this function since I can't use the derivative methods he develop afterwards. Until this page of the book Leithold only covers limits, continuity, tangents and basic derivatives. He didn't ta... | <p>Using continuity, you can find that f is continuous at the origin:</p>
<p>$$
\lim_{x \to 0} f(x) = f(0) = 0 \\
$$</p>
<p>Using limits, you can find what happens at the ends:</p>
<p>$$
\lim_{x \to -\infty} f(x) = +\infty \\
\lim_{x \to +\infty} f(x) = + \infty \\
$$</p>
<p>Using limits, you can find the inclinati... | 240 |
calculus | Calculus problem - Unknown variable in a quadratic | https://math.stackexchange.com/questions/1980858/calculus-problem-unknown-variable-in-a-quadratic | <p>Is there an $a$ such that $\lim_{x \rightarrow -3} \frac{10x^2+ax+a+8}{x^2+x-6}$ exists?</p>
<p>I can't seem to find how to actually solve it other than guessing, and I'm not sure there actually is a solution.</p>
| <p>Hint:</p>
<p>Consider $2$ cases, when the numerator is evaluated to $0$ and when it is not at $x=-3$.</p>
<p>For the case when the numerator is $0$,</p>
<p>You can use L'hopital's rule and evaluate $$\lim_{x \rightarrow -3} \frac{20x+a}{2x+1}$$</p>
<p>Alternatively, $$10x^2+49x+57=(x+3)(10x+19)$$ Since we know t... | 241 |
calculus | $dy/dx$ problems, please help | https://math.stackexchange.com/questions/2056607/dy-dx-problems-please-help | <p>Find $dy/dx$ given $y\cos(xy)=3$.
Also find $dy/dx$ given $y=(2+\sin x)^{\cos x}$</p>
<p>I'm having a hard time solving for $dy/dx$ given $y\cos(xy)= 3$. Because of the $3$, wouldn't the right side of the equation equal $0$? And dividing $0$ by the derivative of the left side to get $dy/dx$ alone also equal $0$?... | <p>For $y\cos(xy)=3$ one must find $\dfrac{dy}{dx}$ by implicit differentiation. For students who find implicit differentiation difficult I recommend first considering both $x$ and $y$ as functions of some third variable such as $t$ and </p>
<p>$(1)$differentiate both sides of the equation with respect to $t$, </p>
<... | 242 |
calculus | How to show $\lim_{n \rightarrow \infty} \frac{[a^{n+1}]}{[a^n]}=a$? | https://math.stackexchange.com/questions/2083127/how-to-show-lim-n-rightarrow-infty-fracan1an-a | <p>How to show that $\lim_{n \rightarrow \infty} \frac{[a^{n+1}]}{[a^n]}=a$, where
$[a]$ = integer part of a?<br>
Here $a>1$. But I suspect it is true for all $a \ne 0$. </p>
| <p>For $|a|>1$,</p>
<p>$$\frac{[a^{n+1}]}{[a^n]}=\frac{a^{n+1}-\{a^{n+1}\}}{a^n-\{a^n\}}=a\frac{1-\dfrac{\{a^{n+1}\}}{a^{n+1}}}{1-\dfrac{\{a^{n}\}}{a^n}}\to a.$$</p>
<p>As the fractional parts are bounded, the numerator and denominator both tend to $1$.</p>
<hr>
<p>This can be extended to $|a|\ge1$ as with $|a|=... | 243 |
calculus | How to solve this complicated integral | https://math.stackexchange.com/questions/2246369/how-to-solve-this-complicated-integral | <p>I am trying to compute the following integral:
$$
I = \int^\infty_1\frac{\operatorname{frac}(x)\cos(a\ln x)}{x^b}\,dx
$$
where $\operatorname{frac}(x) = x - \operatorname{int}(x)$ is the fractional part of $x$, $a > 0$ and $b > 1$.</p>
<p>This is what I got so far.</p>
<p>Let $\operatorname{int}(x) = n$ so ... | <p><strong>Hint</strong>:</p>
<p>$$\int x^c\cos(a\ln x)\,dx=\int e^{(c+1)t}\cos(at)\,dt$$ can be integrated analytically, for example by means of complex numbers.</p>
<p>Indeed,</p>
<p>$$\int e^{bt}\cos(at)=\int\Re(e^{(b+ia)t})=\Re\left(\int e^{(b+ia)t}\right)=\Re\left(\frac{e^{(b+ia)t}}{b+ia}\right),$$ which can be... | 244 |
calculus | A clue to solve this equation | https://math.stackexchange.com/questions/2271578/a-clue-to-solve-this-equation | <p>how to prove that if $f(x,y)=0$ and $g(x,z)=0$ and if $f$ and $g$ are differentiable,then:</p>
<p>$$\dfrac{\partial f}{\partial y}.\dfrac {\partial g}{\partial x}dy=\dfrac{\partial f}{\partial x}.\dfrac {\partial g}{\partial z}dz$$</p>
<p>I think $y$ and $z$ should be dependent, however there is no mentioning for... | <p>We have</p>
<p>$$\dfrac{\partial f}{\partial x}dx+\dfrac{\partial f}{\partial y}dy=0,\\
\dfrac{\partial g}{\partial x}dx+\dfrac{\partial g}{\partial z}dz=0.$$</p>
<p>Then eliminating $dx$,</p>
<p>$$\frac{\dfrac{\partial f}{\partial y}}{\dfrac{\partial f}{\partial x}}dy=\frac{\dfrac{\partial g}{\partial z}}{\dfrac... | 245 |
calculus | Prove that the following expression is always less than x for all values of x and k. | https://math.stackexchange.com/questions/2273552/prove-that-the-following-expression-is-always-less-than-x-for-all-values-of-x-an | <p>Prove that </p>
<p>$$\frac{x^2+kx}{2x+k}$$</p>
<p>is less than x for all values of x and k where x>0, k>0 and k is a constant.</p>
<p>How would I prove this? I have differentiated it with respect to x and noticed that the derivative is always less than 1 for all values of x and k, this means that if the value o... | <p>Multiplying by $2x+k$ positive, $$x^2+kx<x(2x+k)$$</p>
<p>then</p>
<p>$$x^2>0.$$</p>
<hr>
<p>Alternatively,</p>
<p>$$\frac{x^2+kx}{2x+k}-x=-\frac{x^2}{2x+k}<0.$$</p>
| 246 |
calculus | Simple Algebra ,Radicals ,Prime Numbers | https://math.stackexchange.com/questions/2275235/simple-algebra-radicals-prime-numbers | <p>a,b are prime numbers
c∈ℕ</p>
<p>2√a + 7√b = c√3</p>
<p>a²+b²+c²=?</p>
<p>I don't really know how to solve it</p>
| <p>$$2\sqrt{\frac a3}+7\sqrt{\frac b3}\in\mathbb N$$ is only possible if the radicals have rational values (no linear combination of irrationals gives an integer).</p>
<p>Then, only one prime gives the square of a rational when divided by $3$: obviously $3$. From this, $c=9$.</p>
| 247 |
calculus | Quick Questions for Evaluating an Integral | https://math.stackexchange.com/questions/2439131/quick-questions-for-evaluating-an-integral | <p>The calculus shown below is confusing to me. I understand the first step, moving m outside the integral and rewriting in terms of dt, but how does the rest of the evaluation work?</p>
<p>$$\int m \frac{d^2x}{dt^2}dx = m\int\frac{d^2x}{dt^2}\frac{dx}{dt}dt = \frac{m}{2}\int\frac{d}{dt}\left(\frac{dx}{dt}\right)^2dt ... | <p>With $v=\dfrac{dx}{dt}$,</p>
<p>$$\int\frac{d^2x}{dt^2}dx=\int\frac{dv}{dt}dx=\int (dv)\frac{dx}{dt}=\int v\,dv=\frac12v^2.$$</p>
| 248 |
calculus | Nature of roots of a hectic polynomial | https://math.stackexchange.com/questions/2667240/nature-of-roots-of-a-hectic-polynomial | <blockquote>
<p>Let $p(x)$ be a $100$-degree polynomial with $100$ real and distinct roots, say $\alpha_1,\alpha_2,\cdots,\alpha_{100}$, and so $$p(x)=A(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_{100}),$$
where $A\in\mathbb{R}\setminus\{0\}$ and $α_{i}\neq 0$ for all $i\in[1,100]$.
Find nature of roots of the equati... | <p>You have
$$((px)'x)'=(px)''x+(px)'=(p''x+2p')x+(p'x+p)=p''x^2+3p'x+p.$$</p>
<p>The roots of $(px)'$ are the extrema of $px$, which are real and comprised in the $100$ intervals $(\alpha_k,\alpha_{k+1})$, where we define $\alpha_0:=0$.</p>
<p>Then again, the roots of $((px)'x)'$ are real and comprised in the $100$ ... | 249 |
calculus | Solving exponential equation with two variables | https://math.stackexchange.com/questions/2713725/solving-exponential-equation-with-two-variables | <p>Given are two equations:</p>
<p>$$v_1 = v_0 (1 - e^{-\frac{t_1}{\tau}})$$</p>
<p>$$v_2 = v_0 (1 - e^{-\frac{t_2}{\tau}})$$</p>
<p>We know that</p>
<p>$$t_2 > t_1$$
$$v_2 > v_1$$
$$\tau > 0$$
$$v_0 > 0$$
$$\tau, v_0 \in ℝ$$</p>
<p>Given $t_1, v_1, t_2, v_2$, how can we solve for $\tau, v_0$?</p>
| <p>Let $p:=e^{-t_1/\tau}$ so that $e^{-t_2/\tau}=p^\alpha$, where $\alpha$ is known. The equation can be written</p>
<p>$$v_1(1-p^\alpha)=v_2(1-p).$$</p>
<p>$\alpha$ can be an integer $>4$ so that there are certainly cases such that there is no closed-form solution. (In fact, there are only closed-form solutions f... | 250 |
calculus | Error in evaluation of $\displaystyle\lim_{x\to 0} \frac{x\cos x - \ln (1+x)}{x^2}$ | https://math.stackexchange.com/questions/2723256/error-in-evaluation-of-displaystyle-lim-x-to-0-fracx-cos-x-ln-1xx | <p>Evaluate $$\displaystyle\lim_{x\to 0} \frac{x\cos x - \ln (1+x)}{x^2}$$</p>
<p>Here's my method but that results into an error. </p>
<p>\begin{align}
\lim_{x\to 0} \frac{x\cos x - \ln (1+x)}{x^2}
&=\lim_{x\to 0}\frac{\cos x}{x} - \lim_{x\to 0}\left(\frac{1}{x}\right)\lim_{x\to 0}\left(\frac{\ln(1+x)}{x}\right)... | <p>The error lies in these steps:</p>
<p>$$\lim_{x\to 0}\frac{\cos x}{x} - \lim_{x\to 0}\left(\frac{1}{x}\right)\lim_{x\to 0}\left(\frac{\ln(1+x)}{x}\right)\color{red}{=}\frac{\cos x}{x} - \frac{1}{x} \\$$
This is <strong>not correct</strong> because once you split the limit, you need to put the values. Evidently it f... | 251 |
calculus | Easy question: Why is $+ C$ outside the brackets? | https://math.stackexchange.com/questions/2741914/easy-question-why-is-c-outside-the-brackets | <p>$$100(-10te^-0.1t + 10 \int e^{-0.1t}dt) = 100(-10te^-0.1t -100e^{-0.1t})+C$$</p>
<p>Why is the $+C$ outside of the brackets if the integration was done inside? I'm looking at my math book and I'm baffled.</p>
<p>Thanks for the help.</p>
| <p>As far as I can see, you are multiplying the integral by a constant. The $+C$ is just another constant, it can take any value. So it does not matter if you have $+1000C$ or $-0.00001C$, because they are just constants, and you can denote them as $+C$.</p>
| 252 |
calculus | Show that there exists a $x \in \mathbb{R}$ such that | https://math.stackexchange.com/questions/2840854/show-that-there-exists-a-x-in-mathbbr-such-that | <p>No idea where to start on this question. Any help is appreciated:</p>
<blockquote>
<p>$$\text{Show that there exists a $x \in \mathbb{R}$ such that } x^{21}+\frac{200}{1+x^4+\cos^2x}=120$$</p>
</blockquote>
<p>Thank you</p>
| <p><strong>Hint:</strong></p>
<p>Define $f : \mathbb{R} \to \mathbb{R}$ as $f(x) = x^{21}+\frac{200}{1+x^4+\cos^2x}$.</p>
<p>Clearly $f$ is continuous and $\lim_{x\to\pm\infty} f(x) = \pm\infty$.</p>
<p>Hence $f$ is surjective.</p>
| 253 |
calculus | Curve Length Of A Unit Sphere Which Intersect With A Plane | https://math.stackexchange.com/questions/2862222/curve-length-of-a-unit-sphere-which-intersect-with-a-plane | <p>Find the curve length of the intersection between the unit sphere $x^2+y^2+z^2=1$ and the plane $x+y=1$</p>
<p>I have read <a href="https://math.stackexchange.com/questions/2004224/parametrization-of-the-intersection-between-a-sphere-and-a-plane">this</a> and <a href="https://math.stackexchange.com/questions/233924... | <p>Actually you don't need to do any integration.
First of all, you know the curve is a circle, so to find the perimeter, you only need to know the radius.</p>
<p>The idea is that you can find the distance between the plane and the centre of the ball (I will leave it as an exercise). Let me call it $D$. You know that ... | 254 |
calculus | How to prove the following statements about tangent lines to $y=ax^2+bx+c$? | https://math.stackexchange.com/questions/2958859/how-to-prove-the-following-statements-about-tangent-lines-to-y-ax2bxc | <p>Consider the graph of the equation <span class="math-container">$y=ax^2+bx+c$</span>, <span class="math-container">$a≠0$</span>. Prove the following:</p>
<p>a. If <span class="math-container">$a$</span> and <span class="math-container">$c$</span> have the same sign, that is <span class="math-container">$ac > 0$<... | <p>A line through the origin has the equation <span class="math-container">$y=mx$</span>. It tangents the parabola if it makes a "double" intersection with it.</p>
<p>In other words,
<span class="math-container">$$mx=ax^2+bx+c$$</span></p>
<p>must have a double root. This occurs when the discriminant</p>
<p><span cl... | 255 |
calculus | How fast is the area of rectangle increasing? | https://math.stackexchange.com/questions/3077488/how-fast-is-the-area-of-rectangle-increasing | <p>The length of a rectangle is increasing at a rate of 8 cm/s and
its width is increasing at a rate of <span class="math-container">$3$</span> cm/s . When the length is
20 cm and the width is 10 cm, how fast is the area of the rectangle
increasing?</p>
<p>So on internet I found a solution but I didn't do that way and... | <p>One millisecond later, the sides are <span class="math-container">$20.008$</span> and <span class="math-container">$10.003$</span> and the relation <span class="math-container">$l=2w$</span> is no more true.</p>
<p>The rate of increase of the area must be close to</p>
<p><span class="math-container">$$\frac{20.008... | 256 |
calculus | Differential Notation Misunderstanding | https://math.stackexchange.com/questions/3104366/differential-notation-misunderstanding | <p>Consider I have a function <span class="math-container">$v=e^u$</span> where u is from the set of all Real numbers. Now, if I take the derivative here, I can get <span class="math-container">$dv/du = e^u$</span>. If I multiply both sides by the <span class="math-container">$du$</span>, I will get <span class="math-c... | <p><span class="math-container">$$\Delta v=e^{u+\Delta u}-e^u=e^u(e^{\Delta u}-1),$$</span> not <span class="math-container">$$e^{\Delta u}.$$</span></p>
<p>By the way, </p>
<p><span class="math-container">$$\lim_{\Delta u\to0}\frac{e^{\Delta u}-1}{\Delta u}=1,$$</span> and this justifies</p>
<p><span class="math-co... | 257 |
calculus | Finding the interval of when this function decreases | https://math.stackexchange.com/questions/3175234/finding-the-interval-of-when-this-function-decreases | <p>From an old math exam I found the question to find the interval for when a function is decreasing(so it can be used for the Integration test). But I can't seem to figure it out.</p>
<p>The function in question is:</p>
<p><span class="math-container">$f(x) =\dfrac{\sqrt{x}}{(x^\frac{3}{2} +2)^2}$</span></p>
<p>The... | <p>You are looking for the interval where the derivative is negative.</p>
<p>I will use a little trick, for comfort: as <span class="math-container">$x\ge0$</span>, I will replace <span class="math-container">$x$</span> by <span class="math-container">$z^2$</span> to get rid of the half-exponents. As the relation <spa... | 258 |
calculus | convergent series for $\sum_{n=1}^{\infty}\frac{n!}{n^n}$ | https://math.stackexchange.com/questions/3179505/convergent-series-for-sum-n-1-infty-fracnnn | <p>Help me please , I am not able to solve this problem.I have tried in many ways to figure out such as Ration test , Integral test , Comparison test , Limit Comparison Test , Root Test but i can't find the way out . This is my first question and i'm not good at English. If there is something wrong or you are not comfo... | <p>The usual approach to factorial-based problems is to use <a href="https://en.wikipedia.org/wiki/Stirling%27s_approximation" rel="nofollow noreferrer">Stirling's approximation</a> <span class="math-container">$n!\approx\sqrt{2\pi n}n^ne^{-n}$</span>, which shows this series converges provided <span class="math-contai... | 259 |
calculus | What do the following parametric curves represent? | https://math.stackexchange.com/questions/3205317/what-do-the-following-parametric-curves-represent | <p>(a) <span class="math-container">$x(v)= 3, y(v)= 4, z(v)= v$</span> for <span class="math-container">$−\infty < v < \infty$</span>,</p>
<p>(b) <span class="math-container">$x(t)= 3\cos(t), y(t)= 2\sin(t), z(t)= 3t−1$</span> for <span class="math-container">$0 \leq t < 2\pi$</span>.</p>
<p>I have no idea w... | <p>The graph of <span class="math-container">$$x(v)= 3, y(v)= 4, z(v)= v$$</span> is a vertical line passing through the point <span class="math-container">$(3,4,0)$</span> since <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are fixed and <span class="math-container">$z$</span> run... | 260 |
calculus | Can I use the partial implicit differentiation with $x = e^xy$? | https://math.stackexchange.com/questions/3575733/can-i-use-the-partial-implicit-differentiation-with-x-exy | <p>I want to know if I can use the partial implicit differentiation with this problem.</p>
<p>What is the derivative of <span class="math-container">$x = e^{xy}$</span>?</p>
| <p>Considering <span class="math-container">$y=f(x)$</span>, you get:
<span class="math-container">$$(x)'_x=(e^{xy})'_x \Rightarrow \\
1=e^{xy}\cdot (y+xy')\Rightarrow \\
y'=\frac{1-ye^{xy}}{xe^{xy}}$$</span>
Wolfram <a href="https://www.wolframalpha.com/input/?i=derivative%20x%3De%5Exy" rel="nofollow noreferrer">answe... | 261 |
calculus | How to solve $\frac{y}{y'}=ln(y)$ for $y$? | https://math.stackexchange.com/questions/3613859/how-to-solve-fracyy-lny-for-y | <p><span class="math-container">$\frac{y}{y'}=\ln(y)$</span></p>
<p><span class="math-container">$ydx=\ln(y)dy$</span></p>
<p><span class="math-container">$dx=\frac{\ln y}{y} dy$</span></p>
<p>]<span class="math-container">$\ln(y) =z$</span> => <span class="math-container">$dz=dy/y$</span></p>
<p>then <span class="... | <p>Almost: <span class="math-container">$x+C=\int zdz=\frac12z^2$</span>.</p>
| 262 |
calculus | Intuition behind integrating and differentiating determinants? | https://math.stackexchange.com/questions/3689270/intuition-behind-integrating-and-differentiating-determinants | <p><a href="https://byjus.com/jee/differentiation-integration-of-determinants/" rel="nofollow noreferrer">https://byjus.com/jee/differentiation-integration-of-determinants/</a></p>
<p>I saw this and I can't understand how this formula was derived, like why can we integrate row wise and add up determinants? Is there an... | <p><strong>Hint</strong>: expanding
<span class="math-container">$$
\det \left(
\begin{matrix}
f_1(x) & g_1(x) \\
f_2(x) & g_2(x)
\end{matrix}
\right) = f_1(x)g_2(x)-f_2(x)g_1(x)
$$</span>
and differentiating,
<span class="math-container">$$f_1'(x)g_2(x)+f_1(x)g_2'(x)-f_2'(x)g_1(x)-f_2(x)g_1'(x),$$</span>
you ... | 263 |
calculus | Questions about parametric equations | https://math.stackexchange.com/questions/3720326/questions-about-parametric-equations | <p>Consider the parametric equations: <span class="math-container">$$x=t^3-3t, \; \; y=t^2+t+1.$$</span></p>
<ol>
<li>What is the lowest point on this parametric curve?</li>
<li>For what values of <span class="math-container">$t$</span> does the curve move left, move right, move up and move down?</li>
<li>When is the c... | <p><span class="math-container">$$\dot x=3t^2-3$$</span> is negative for <span class="math-container">$-1<t<1$</span>, meaning that the curve is traversed from right to left in this range, and conversely.</p>
<p><span class="math-container">$$\dot y=2t+1$$</span> is negative when <span class="math-container">$t\l... | 264 |
calculus | Odd Intuitive proof for L'Hospital's rule | https://math.stackexchange.com/questions/3751075/odd-intuitive-proof-for-lhospitals-rule | <p>A professor of mine intuitively showed why L'Hospital's rule works for the <span class="math-container">$0/0$</span> case (by some simplifying assumptions). I understood that. He then contended that this is enough to prove that the rule works for the <span class="math-container">$\infty / \infty$</span> case. This ,... | <p>Yes, even the intuitive approach in this case will require some more work. We'll need the fact that <span class="math-container">$\left(\frac{1}{f}\right)'=-\frac{f'}{f^2}$</span>. Now apply the <span class="math-container">$0/0$</span> rule to <span class="math-container">$\frac{1/g}{1/f}$</span> to get that the li... | 265 |
calculus | Differentiating $V_c=V_s(1-e^{-t/T})$ | https://math.stackexchange.com/questions/3870146/differentiating-v-c-v-s1-e-t-t | <p>I have a formula for an electronic circuit as follows</p>
<p><span class="math-container">$$V_c=V_s(1-e^{-t/T})$$</span>
Apparently this differentiates to <span class="math-container">$$(V_s/T) e^{-t/T}$$</span></p>
<p>I say apparently because I looked up the answer which is a bit naughty but I can't figure it out. ... | <p><span class="math-container">\begin{align}
\frac{d}{dt} V_s(1-e^{-t/T})
&= V_s \frac{d}{dt} (1-e^{-t/T})
&\text{$V_s$ is a constant factor, can be pulled out of the derivative}
\\
&= V_s \frac{d}{dt}(- e^{-t/T}) & \text{$1$ is additive constant, has derivative zero}
\\
&= -V_s \frac{d}{dt} e^{-t... | 266 |
calculus | For what values of $c$ does the curve $ y = cx^{3} + e^{x} $ have inflection points? | https://math.stackexchange.com/questions/3924969/for-what-values-of-c-does-the-curve-y-cx3-ex-have-inflection-poi | <p>For what values of <span class="math-container">$c$</span> does the curve <span class="math-container">$ y = cx^{3} + e^{x} $</span> have inflection points?</p>
<p>at first I found first derivative <span class="math-container">$ f^{'}(x) = 3cx^2 + e^{x} $</span></p>
<p>then second derivative <span class="math-contai... | <p>Write the equation <span class="math-container">$y''=6cx+e^x=0$</span> as a system
<span class="math-container">$$\begin{cases}
y=e^x\\
y=-6cx\\
\end{cases}
$$</span></p>
<p>we see that for <span class="math-container">$c>0$</span> there is one and only one solution: one inflection point.</p>
<p>The tangent passi... | 267 |
calculus | Find the derivatives to $f(x)=4/x^2$ and $g(t)=(t-5)/(1+\sqrt{t}\,)$ | https://math.stackexchange.com/questions/979347/find-the-derivatives-to-fx-4-x2-and-gt-t-5-1-sqrtt | <p>I have these two assignments: </p>
<blockquote>
<p>Find the derivatives to (a) $f(x)=4/x^2$ and b) $g(t)=(t-5)/(1+\sqrt{t}\,)$ by using the definition $$\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}=f'(x)$$</p>
</blockquote>
<p>a) This is my attempt at (a); am I correct?
$$\lim_{h \to 0} \frac{\left(\displaystyle\frac{4}... | <ol>
<li>First function</li>
</ol>
<p>$$f'(x)=\lim_{h \to 0} \frac{\left(\frac{4}{(x+h)^2}-\frac{4}{x^2}\right)}{h}=\frac{1}{h}\left(\frac{4}{(x+h)^2}-\frac{4}{x^2}\right)=\lim_{h \to 0}\frac{1}{h}\frac{-4h^2-8xh}{(x+h)^2x^2}=\lim_{h \to 0}\frac{1}{h}\frac{-4h(h+2x)}{(x+h)^2x^2}= \lim_{h \to 0}\frac{-4(h+2x)}{(x+h)^2x... | 268 |
calculus | How to find the limit | https://math.stackexchange.com/questions/3813828/how-to-find-the-limit | <p>How can we find the limit
<span class="math-container">$$\lim_{x\to 0} \frac{(e^x-1-x)^2}{x(\sin x -x)}$$</span>?</p>
| <p><span class="math-container">$$\lim_{x\to 0} \frac{(e^x-1-x)^2}{x(\sin x -x)} = (\lim_{x\to 0} \frac{e^x-1-x}{x^2})^2\cdot\lim_{x\to 0} \frac{x^3}{\sin x -x} = (\frac{1}{2})^2\cdot(-6).$$</span></p>
<p>The last two limits are calculated with L'Hospital's rule.</p>
| 269 |
calculus | Find the only f=vt that has f(2t)=4f(t) | https://math.stackexchange.com/questions/4086587/find-the-only-f-vt-that-has-f2t-4ft | <p>I'm just starting calculus 1 and I don't know how to solve this. Can someone please help?</p>
<p>The problem below involves linear functions <span class="math-container">$f(t) = vt + C$</span>. Find the constants v and C.</p>
<p>Find the only <span class="math-container">$f=vt$</span> that has <span class="math-con... | <p>Assuming <span class="math-container">$v$</span> is constant: <span class="math-container">$f(2t) = 4f(t)$</span> implies <span class="math-container">$v(2t) = 4(vt)$</span> and therefore <span class="math-container">$2vt = 0$</span>.</p>
<p>This yields either <span class="math-container">$t$</span> or <span class="... | 270 |
calculus | $2^x+2^{-x} = 5$, solve $4^x+4^{-x}$ using the rules of exponents | https://math.stackexchange.com/questions/2017827/2x2-x-5-solve-4x4-x-using-the-rules-of-exponents | <blockquote>
<p>$2^x+2^{-x} = 5$</p>
<p>Solve:</p>
<p>$4^x+4^{-x}$</p>
</blockquote>
<p>I know I can solve this by solving the equation $2^x+2^{-x} = 5$ and then replacing $x$ on the second one with the result, but I found that to be too lengthy and overcomplicated.</p>
<p>Is there a faster and simpler wa... | <p>$4^x+4^{-x}=2^{2x}+2^{-2x}=(2^x+2^{-x})^2-2\cdot 2^x \cdot 2^{-x}=25-2=23$</p>
| 271 |
calculus | How to prove that sin(1/x) is continuous at x≠0 | https://math.stackexchange.com/questions/3672511/how-to-prove-that-sin1-x-is-continuous-at-x%e2%89%a00 | <p>Can someone help with the proof that sin(1/x) is continuous for all x≠0.(By the help of epsilon delta defination)</p>
<p>I am sharing what I have tried so far not much though.
I have figured out that modulus value of</p>
<p>sin(1/x)-sin(1/a) is less than modulus value of</p>
<p>(1/x)-(1/a) for all a≠0.From here I... | <p><span class="math-container">$|\frac 1 x -\frac 1 a|=\frac {|x-a|} {|a||x|} \leq \frac {|x-a|} {|a|(|a|-|x-a|)} <\frac {|x-a|} {|a|(|a|/2)}$</span> if <span class="math-container">$|x-a| <|a|/2$</span>. Hence <span class="math-container">$|\frac 1 x -\frac 1 a|<\epsilon$</span> if <span class="math-contai... | 272 |
calculus | Function increase or decrease | https://math.stackexchange.com/questions/1732642/function-increase-or-decrease | <p>The question is</p>
<blockquote>
<p><span class="math-container">$$\text{Let } f(r) = r^{1/3} + \frac 1r \text{ for } r>0$$</span>
a) Determine where the function <span class="math-container">$f$</span> is increasing or decreasing.</p>
<p>b) Determine where the function <span class="math-container">$f$</span> is ... | <p>I'll help you with part a). We can see that $f'(r)$ is defined for all $r>0$, so we just need to find where $f'(r)=0$. Let's start by rewriting that as</p>
<p>$$\frac 13r^{-2/3} -r^{-2}=0$$</p>
<p>Move the second term to the other side to get</p>
<p>$$\frac 13r^{-2/3}=r^{-2}$$</p>
<p>Taking the reciprocal of ... | 273 |
calculus | Is there anywhere where there are in-depth walkthroughs of problems on Stewart's Calculus? | https://math.stackexchange.com/questions/4190525/is-there-anywhere-where-there-are-in-depth-walkthroughs-of-problems-on-stewarts | <p>I'm currently taking calc 2 and using Stewart's calculus. My major qualms with the book is the lack of examples. On a scale of 1-10, the practice problems in the chapters are like 1-3, then the example problems are all very difficult without walkthroughs at the end of the book. I'm struggling to figure out where I w... | 274 | |
calculus | Taking the Derivative of Both Sides of an Equation | https://math.stackexchange.com/questions/3240322/taking-the-derivative-of-both-sides-of-an-equation | <p>If we have an equation like </p>
<p>y = x^2</p>
<p>This implies that </p>
<p>y’ = 2x</p>
<p>If we have an equation like </p>
<p>x = 4x^2</p>
<p>and we take the derivative of both sides we get</p>
<p>1 = 8x</p>
<p>With the solution x = 1/8, which is not the solution to the original equation. This is instead t... | <p>Your problem comes from thinking that you take derivatives of equations. But you don't. You take derivatives of <em>functions</em>.</p>
<p>For example, you can think of the equation
<span class="math-container">$$
x^2 -1 = (x-1)(x+1)
$$</span>
as telling you two different ways to write the same function of <span cl... | 275 |
calculus | Determining the uniform convergence | https://math.stackexchange.com/questions/3028636/determining-the-uniform-convergence | <p>Show that the series ,whose partial sum of n terms is <span class="math-container">$S_n=\frac{x}{(1+nx^2)}$</span>, converges uniformly for all real x.</p>
<p>I found that the series is pointwise convergent to 0 for all x.
For showing uniform convergence, I found out that the function S attains maximum value at <sp... | <p>Note the sum of the first <span class="math-container">$n$</span> terms as <span class="math-container">$S_n(x)$</span>. We have </p>
<p><span class="math-container">$$ S_n(x) = \frac{x}{1+nx^2}$$</span>
and <span class="math-container">$ \lim_{n \to \infty} S_n(x) = 0 \forall x \in \mathbb{R}.$</span> Therefore th... | 276 |
calculus | Confused about variables | https://math.stackexchange.com/questions/4192699/confused-about-variables | <p>This is probably a very dumb question but after trying to review some calculus after years not using it, I am confused by variables in the equation for a tangent line.
So I watched the very first lecture on calculus by MIT ( <a href="https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006" r... | <p>Well, the underlying concept is simple. We take a certain point <span class="math-container">$(x_0, y_0)$</span> lying on the given curve. At this point, slope of tangent is obviously given by:
<span class="math-container">$$m=\left(\frac {dy}{dx}\right)_{x=x_0}$$</span>
Hence, for your given curve, at <span class="... | 277 |
calculus | Integral over circle area limited by two straight lines | https://math.stackexchange.com/questions/4193374/integral-over-circle-area-limited-by-two-straight-lines | <p>I need to integrate a function over the area limited by the circle and two straight lines, i.e.<span class="math-container">$x^2+y^2<R^2$</span> and <span class="math-container">$x<-b, y>a$</span>. For this I integrate over <span class="math-container">$y$</span> from <span class="math-container">$a$</span>... | <p>Consider the lines <span class="math-container">$x = -b$</span> and <span class="math-container">$y = a.$</span>
These two lines are perpendicular and divide the plane into three quadrants
around the point <span class="math-container">$(-b,a).$</span></p>
<p>The two conditions <span class="math-container">$x < -b... | 278 |
calculus | $\sqrt{a} +\sqrt{b} = 20$. What is the maximum value of $a-5b$? | https://math.stackexchange.com/questions/1749111/sqrta-sqrtb-20-what-is-the-maximum-value-of-a-5b | <p>It is given that <span class="math-container">$$\sqrt{a} +\sqrt{b} = 20$$</span>
Where a and b are real numbers.</p>
<p>What is the maximum value of <span class="math-container">$a-5b$</span>?</p>
| <p>$$\sqrt{b}=20-\sqrt{a}$$
$$b=(20-\sqrt{a})^2=400-40\sqrt{a}+a$$
$$a-5b=a-5(400-40\sqrt{a}+a)=-4a+200\sqrt{a}-2000$$</p>
<p>$$\frac{d}{da}(a-5b)=-4+\frac{100}{\sqrt{a}}$$</p>
<p>Since $\sqrt{a}<20$, </p>
<p>$$\frac{d}{da}(a-5b)>0$$</p>
<p>Hence, $a-5b$ is maximum when $a$ is maximum, viz., when $a=400$ (and... | 279 |
calculus | How to obtain the integral representation of Modified Bessel function $I_0(2)$? | https://math.stackexchange.com/questions/2903380/how-to-obtain-the-integral-representation-of-modified-bessel-function-i-02 | <p>It is known that</p>
<p>$\displaystyle I_0(2)=\sum_{k=0}^{\infty}\frac{1}{(k!)^2} = \frac{1}{\pi}\int_{0}^{\pi}e^{2\cos\theta}d\theta$
(<a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html</a... | <p>Exchange the order of integration:
$$\frac{1}{\pi}\int_{0}^{\pi}e^{2\cos\theta}d\theta$$
$$=\frac{1}{\pi}\int_{0}^{\pi}\sum_{n=0}^{\infty}\frac{\cos^n \theta}{n!}2^nd\theta$$
$$=\frac{1}{\pi}\sum_{n=0}^{\infty}\frac{2^n}{n!}\int_{0}^{\pi}\cos^n \theta d\theta.$$
For the integral $\int_{0}^{\pi}\cos^n \theta d\theta$... | 280 |
calculus | Proof that $f(x)=4x^4-2x+1$ has no real roots. | https://math.stackexchange.com/questions/2909560/proof-that-fx-4x4-2x1-has-no-real-roots | <p>My thought was to:<br>
1) hypothesis there are 2 real roots for this equation,<br>
2) apply Rolle's theorem and come to a reductio ad absurdum<br>
and then if there aren't 2 real roots, it has to be 1. If there is 1 real root, this means that it has to have 3 non-real roots. But non real roots come in pairs, so eith... | <ul>
<li>$f'(x)=2(8x^3-1)$, so there's a single critical point: $\; x=\frac12$.</li>
<li>$f''(x)=48x^2\ge 0$, so by the second derivative test, this critical point is a <em>minimum</em>, and this minimum is an absolute minimum.</li>
<li>$f(\frac12)=\frac14>0$.</li>
</ul>
| 281 |
calculus | Summation of $n^2x^n$ terms | https://math.stackexchange.com/questions/2923002/summation-of-n2xn-terms | <p>How does one evaluate the following summation of $n^2$ terms by $x^n$ terms.
I have tried to do it, but couldn't figure it out as it is not the same as summing up $nx^n$ terms.</p>
<p>$$\sum_{n=0}^\infty n^2 x^n$$</p>
| <p>$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$</p>
<p>Differentiating (and multiplying with $x$)we have,</p>
<p>$\displaystyle \frac{x}{(1-x)^2}=\sum_{n=0}^{\infty}nx^n$</p>
<p>Differentiating(and multiplying with $x$) we have, </p>
<p>$\displaystyle \frac{[(1-x)^2(1)-(x)2(1-x)(-1)]x}{(1-x)^4}= \frac{x^2... | 282 |
calculus | Finding range of $a$ | https://math.stackexchange.com/questions/2928763/finding-range-of-a | <blockquote>
<p>If <span class="math-container">$$f(x) =
\begin{cases}
|x-2|+a^2-9a-9, &\text{if }x<2\\
2x-3, &\text{if } x\geqslant2
\end{cases}$$</span> has local minima at <span class="math-container">$x=2$</span>, then range of <span class="math-container">$a$</span> is… ?</p>
</blockquote>
<... | <p>A non-continuous function can also have the minimum at <span class="math-container">$x=2$</span></p>
<p>Here you want a local minimum at <span class="math-container">$x=2$</span> thus
<span class="math-container">$$
|2-2|+a^2-9a-9\ge 2\cdot2-3
$$</span></p>
<p><span class="math-container">$$
a^2-9a-9\ge 1
$$</sp... | 283 |
calculus | How to integral $\int\limits_{0}^{\pi \over 6} {x \over \sqrt{1-2\sin{x}}}dx$ ..? | https://math.stackexchange.com/questions/2931296/how-to-integral-int-limits-0-pi-over-6-x-over-sqrt1-2-sinxdx | <p><span class="math-container">$$\int\limits_{0}^{\pi \over 6} {x \over \sqrt{1-2\sin{x}}}dx$$</span></p>
<p>I attempted lots of permutations but I can't solve it..
moreover, I don't know its convergence or divergence... please help!</p>
| <p><em>This is not a serious answer. Just done for the fun of it.</em></p>
<p>As Ekesh answered, there is no solution even using special functions.</p>
<p>For the fun of it, I tried to see what would give the magnificent approximation
<span class="math-container">$$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi ... | 284 |
calculus | If $\int_0^x f^2(t)dt \le f(x)$ for all $x \in [0,1]$, then $\min_{[0,1]} f(x) \le 1$? | https://math.stackexchange.com/questions/2944797/if-int-0x-f2tdt-le-fx-for-all-x-in-0-1-then-min-0-1-fx | <p>Suppose that <span class="math-container">$f$</span> is a continuous function on <span class="math-container">$[0,1]$</span> and
<span class="math-container">$$\int_0^x [f(t)]^2dt \le f(x) \quad \text{for all} \quad x \in[0,1].$$</span>
Prove or disprove
<span class="math-container">$$\min_{0\le x\le 1} f(x) \le 1.... | <p>I assume that <span class="math-container">$f^2(t)$</span> means <span class="math-container">$\big(f(t)\big)^2$</span>. I have a very weak bound
<span class="math-container">$$\min_{x\in[0,1]}\,f(x)<2\sqrt{2}\,,$$</span>
and do not know how to improve it. Maybe somebody can use my proof to get a better bound.... | 285 |
calculus | Evaluate limit using L'Hospital | https://math.stackexchange.com/questions/2946383/evaluate-limit-using-lhospital | <p>Evaluate <span class="math-container">$\lim_{x\to0} \frac{\sin(x^{30})}{\sin^{30}(5x)} $</span></p>
<p>I have tried applying L'Hospital's rule, but it took me a lot of time to factor the derivative. Is there any way can resolve this problem. Thanks.</p>
<p>The answer is <span class="math-container">$\frac{1}{5^{30... | <p>Use Taylor series so you don't have to differentiate <span class="math-container">$30$</span> times. We have <span class="math-container">$\sin(t)=t-\frac16t^3+\cdots$</span>. Inserting <span class="math-container">$t=x^{30}$</span> gives us the numerator, and inserting <span class="math-container">$t=5x$</span> and... | 286 |
calculus | Maximum value on a circle | https://math.stackexchange.com/questions/2949154/maximum-value-on-a-circle | <p>I need to find the maximum value of a function on a circle: Let <span class="math-container">$C$</span> denote the circle of radius <span class="math-container">$6$</span> centered at the origin in the <span class="math-container">$xy$</span>-plane. Find the maximum value of <span class="math-container">$x^2y$</span... | <p>Hint: For <span class="math-container">$(x,y)$</span> on the circle of radius <span class="math-container">$6$</span>, we have
<span class="math-container">$$
x^2=36-y^2
$$</span>
So you can find a single variable function to maximize.</p>
| 287 |
calculus | Evaluation of Integration using limit as a sum | https://math.stackexchange.com/questions/2952267/evaluation-of-integration-using-limit-as-a-sum | <blockquote>
<p>Evaluation of <span class="math-container">$\displaystyle \int^{2}_{1}\frac{1}{x}dx$</span> using limit as a sum</p>
</blockquote>
<p>Try: Using The formula <span class="math-container">$$\int^{b}_{a}f(x)dx = \lim_{h\rightarrow 0}h\times \sum^{n-1}_{r=1}f(a+rh)$$</span></p>
<p>where <span class="mat... | <p>Consider the points <span class="math-container">$x=c^k$</span> with <span class="math-container">$c^n=2$</span>.</p>
<p><span class="math-container">$$\int_1^2\frac{dx}{x}\approx\sum_{k=0}^{n+1} \frac{\Delta c^k}{c^k}=\sum_{k=1}^n \frac{ c^{k+1}-c^k}{c^k}=n(c-1)=n\left(\sqrt[n]2-1\right).$$</span></p>
<p>Then,</p... | 288 |
calculus | Show that $c$ is in interval $[e,3]$ for $c\cdot \ln{c} + c − 6 = 0$ | https://math.stackexchange.com/questions/2959847/show-that-c-is-in-interval-e-3-for-c-cdot-lnc-c-%e2%88%92-6-0 | <p>Show that there is a unique number <span class="math-container">$c \in \mathbb{R}$</span> that fulfills the equation and that this number is in the interval <span class="math-container">$[e, 3]$</span>.</p>
<p><span class="math-container">$$ c\cdot \ln{c} + c − 6 = 0$$</span></p>
<p>At first I was thinking about u... | <p>Note <span class="math-container">$c>0$</span>.</p>
<p>Set <span class="math-container">$e^y:=c$</span>, then</p>
<p>1)<span class="math-container">$ye^y +e^y -6=0$</span>,</p>
<p><span class="math-container">$e^y(1+y)-6=0$</span>.</p>
<p><span class="math-container">$f(y):=(1+y)e^y-6.$</span></p>
<p><span c... | 289 |
calculus | Absolute conditional minimum of function in n-dimensional space | https://math.stackexchange.com/questions/2971893/absolute-conditional-minimum-of-function-in-n-dimensional-space | <p>Function</p>
<p><span class="math-container">$$F(x_1,x_2,...,x_n) = \sum_{i=1}^n x_i$$</span></p>
<p>on the constraint</p>
<p><span class="math-container">$$G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$$</span></p>
| <p>The inequality of arithmetic and geometric means says:</p>
<p><span class="math-container">$(x_1x_2....x_n)^{1/n} \le \frac{F(x_1,...,x_n)}{n}$</span>.</p>
<p>If <span class="math-container">$x_1x_2....x_n=1$</span>, then we get</p>
<p><span class="math-container">$n=F(1,...,1) \le F(x_1,...,x_n)$</span>.</p>
| 290 |
calculus | calculate $\lim_{x\to\infty} x + \sqrt[3]{1-x^3}$ | https://math.stackexchange.com/questions/2979793/calculate-lim-x-to-infty-x-sqrt31-x3 | <p>So I multiplied by the conjugate and got <span class="math-container">$$\lim_{x\to\infty} \frac{x^2-(1-x^3)^\frac{2}{3} + x(1-x^3)^\frac{1}{3}-(1-x^3)}{x-(1-x^3)^\frac{2}{3}}$$</span></p>
<p>and this is where I got stuck.</p>
| <blockquote>
<p>So I multiplied by the conjugate and got</p>
</blockquote>
<p>What conjugate expression was that exactly...?</p>
<p>You want to get rid of the cube root by using:
<span class="math-container">$$a+b=\frac{\left(a+b\right)\left(a^2-ab+b^2\right)}{a^2-ab+b^2}=\frac{a^3+b^3}{a^2-ab+b^2}$$</span>
with, i... | 291 |
calculus | $|\int \limits_a^b f(x) dx|\leq\int \limits_a^b |f(x)|dx$ for f continuous | https://math.stackexchange.com/questions/2981463/int-limits-ab-fx-dx-leq-int-limits-ab-fxdx-for-f-continuous | <p>How to prove <span class="math-container">$|\int \limits_a^b f(x) dx|\leq\int \limits_a^b |f(x)|dx$</span> for f continuous? This is a step in the solution of a problem from Mendelson's introduction to topology. This book assumes the reader has only a background in first-year calculus, not measure theory or advanced... | <p>You can write: <span class="math-container">$$f(x)=f_+(x)-f_-(x)$$</span> where <span class="math-container">$f_+(x)=\max(f(x),0)$</span> and <span class="math-container">$f_-(x)=-\min(f(x),0)$</span></p>
<p>Note that: <span class="math-container">$$|f(x)|=f_+(x)+f_-(x)$$</span></p>
<p>The functions <span class="m... | 292 |
calculus | Number of solutions. | https://math.stackexchange.com/questions/2982498/number-of-solutions | <blockquote>
<p>For each positive real number <span class="math-container">$\lambda$</span>, let <span class="math-container">$A_\lambda$</span> be the set of all natural numbers <span class="math-container">$n$</span> such that <span class="math-container">$|\sin\sqrt{n+1}-\sin\sqrt n|<\lambda$</span>. Let <span ... | <p><span class="math-container">$$\sin \sqrt{n+1} - \sin \sqrt{n}=2 \sin \frac12 (\sqrt{n+1}-\sqrt{n}) \cos \frac12 (\sqrt{n+1}+\sqrt{n})= \\ = 2 \sin \frac{1}{2(\sqrt{n+1}+\sqrt{n})} \cos \frac12 (\sqrt{n+1}+\sqrt{n})$$</span></p>
<p>From this we can see that the limit for <span class="math-container">$n \to \infty$<... | 293 |
calculus | How to differentiate this double integral from Christopher Bishop's Pattern Recognition book. | https://math.stackexchange.com/questions/2982727/how-to-differentiate-this-double-integral-from-christopher-bishops-pattern-reco | <p>In Bishop's book Pattern Recognition and Machine Learning, the following can be found on page 46:</p>
<p>(1):
<span class="math-container">$$
J[f] = \iint\{f(\mathbf{x}) - t\}^2p(\mathbf{x},t)\mathrm{d}t \mathrm{d}\mathbf{x}
$$</span></p>
<p>He then differentiates this expression with respect to <span class="math-... | 294 | |
calculus | Approximate $(0.99)^{300}$ without calculator | https://math.stackexchange.com/questions/2987471/approximate-0-99300-without-calculator | <blockquote>
<p>Approximate <span class="math-container">$(0.99)^{300}$</span> without calculator.</p>
</blockquote>
<p>This question is in my textbook but i don't know how to approximate without calculator. How can i evaluate without calculator? Thanks in advance.</p>
| <p><span class="math-container">$$300 \ln (1-1/100) \approx 300 (-1/100-1/20000) \approx -3$$</span></p>
<p><span class="math-container">$$e^{-3} = (3-(3-e))^{-3} \approx \frac{1}{27} \left(1+(3-e)\right)=\frac{4-e}{27}=0.0475...$$</span></p>
<p><span class="math-container">$$0.99^{300}=0.0490...$$</span></p>
<p>As ... | 295 |
calculus | Prove that, $\int^{\infty}_0\frac{x\sin(rx)}{a^2+x^2}dx=\frac{\pi}{2}e^{-ar}$ | https://math.stackexchange.com/questions/2992228/prove-that-int-infty-0-fracx-sinrxa2x2dx-frac-pi2e-ar | <p>The question is: prove that</p>
<p><span class="math-container">$$\int^{\infty}_0\frac{x\sin(rx)}{a^2+x^2}dx=\frac{\pi}{2}e^{-ar}$$</span></p>
<p>This is what I've got so far:</p>
<p>Let <span class="math-container">$I(r)=\int^{\infty}_0\frac{x\sin(rx)}{a^2+x^2}dx$</span></p>
<p><span class="math-container">$I'(... | 296 | |
calculus | Show $f(x) >0$ for $x>x_0$ if its $f' >f$ and $f(x_0)=0$ | https://math.stackexchange.com/questions/3011769/show-fx-0-for-xx-0-if-its-f-f-and-fx-0-0 | <blockquote>
<p>Let <span class="math-container">$f: \mathbb{R} \rightarrow \mathbb{R}$</span> be a differentiable function. Suppose that <span class="math-container">$f'(x)>f(x)$</span> for all <span class="math-container">$x \in \mathbb{R}$</span>, and <span class="math-container">$f(x_0)=0$</span> for some <spa... | <p>Proceed by contradiction. First since <span class="math-container">$f'(x_0) > f(x_0) = 0$</span>, there is some <span class="math-container">$t > 0$</span> s.t. <span class="math-container">$f (x)>0$</span> on <span class="math-container">$(x_0, x_0+t)$</span>. Assume <span class="math-container">$f(x) \leq... | 297 |
calculus | Getting the rate of drain from a tank | https://math.stackexchange.com/questions/3012499/getting-the-rate-of-drain-from-a-tank | <p>A tank with a top radius of <strong>1m</strong>, a bottom radius of <strong>0.5m</strong> and a height of <strong>2m</strong> is initially filled with water. Water drains through a square hole of side <strong>3cm</strong> in the bottom.</p>
<p>How do I get the rate of drain,
<span class="math-container">\begin{equa... | <p>Conservation of mass and constant density tells us change in volume in the tank equals change in volume out of tank.</p>
<p>Thinking area times velocity gives you rate of volume change out of tank, by dimensional analysis.</p>
<p>Bernouli s principle says</p>
<p><span class="math-container">$\frac{1}{2}v^2=gh$</s... | 298 |
calculus | Can the lower limit of $\frac{d}{dx} \int^x_a f(t)dt = f(x)$ be $-\infty$? | https://math.stackexchange.com/questions/3030569/can-the-lower-limit-of-fracddx-intx-a-ftdt-fx-be-infty | <p>I'm self studying math, based on the fundamental theorem of Calculus, <span class="math-container">$$\frac{d}{dx} \int^x_a f(t)dt = f(x)$$</span> can the lower limit be <span class="math-container">$-\infty$</span>?</p>
| <p>Yes, lower limit can be <span class="math-container">$-\infty$</span> but only <em>provided</em> the resulting improper integral converges. This only means if you first integrate from <span class="math-container">$a$</span> to <span class="math-container">$x,$</span> get that answer, and then let <span class="math-c... | 299 |
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