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Question: <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... | https://math.stackexchange.com/questions/2089685/the-limit-comparison-test-v1 |
Question: <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 sect... | https://math.stackexchange.com/questions/4165567/can-i-find-the-surface-area-for-cone-by-the-surface-area-formula |
Question: <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">$f$... | https://math.stackexchange.com/questions/4093830/problem-about-conservative-vector |
Question: <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 $y$ ... | https://math.stackexchange.com/questions/850467/proving-the-monotonicity-of-a-function |
Question: <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... | https://math.stackexchange.com/questions/3463830/lim-h-to-0-int-xxh-lnt-dt |
Question: <p>I am trying to reproduce the result below</p>
<p><span class="math-container">$$\int\mathcal{D}V~e^{-\int dx~[a V^{2}(x)+iV(x)\int_{-\infty}^{\infty}dt~ \bar{\psi}^{a}(x,t)\gamma_{ab}\psi^{b}(x,t)]}= e^{-\frac{1}{4a}\int dx\int\int_{-\infty}^{\infty}dt dt'(\bar{\psi}^{a}(x,t)\psi^{a}(x,t))(\bar{\psi}^{b}(x... | https://math.stackexchange.com/questions/4169820/functional-gaussian-integral |
Question: <p>For example, one can apply $\cos x$ to number $a$ one time to get $\cos a$, two times $\cos \cos a$, three times $\cos \cos \cos a$, and so on. Is there a way to define fractional application for $\cos$? Or for any other function? Maybe exists general theory for that?</p>
Answer: <p>Obviously $f(x,n)$ def... | https://math.stackexchange.com/questions/1250978/apply-function-fractional-times |
Question: <p>I need help evaluating the integrals in Fourier Series.</p>
<p>For example, for the function <span class="math-container">$\cos^{2}x$</span>, I can evaluate <span class="math-container">$a_0$</span>, <span class="math-container">$a_n$</span>, and <span class="math-container">$b_n$</span>, where <span class... | https://math.stackexchange.com/questions/627718/fourier-series-of-cosnx |
Question: <p><strong>Background Information:</strong>
One of the most important ideas that Green discussed in his Essay is the connection between
what happens within a body and the properties of that body’s surface. He realized that, because the
boundary of an object is one dimension lower than the interior, the connec... | https://math.stackexchange.com/questions/4115061/the-fundamental-theorem-of-calculus-questions |
Question: <p>If $f(x)$ is a continuous function on the interval $[x, x+h]$, find $$\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} $$</p>
<p>W... | https://math.stackexchange.com/questions/2121801/what-is-the-lim-h-to0-of-the-average-value-of-fx-on-the-interval-x-x |
Question: <p>I know the meaning of <span class="math-container">$\int_{a}^{b}f(x)dx$</span>, which is <span class="math-container">$F(b)-F(a)$</span>. Geometrically, it gives us the area under the graph from <span class="math-container">$x=a$</span> to <span class="math-container">$x=b$</span></p>
<p>But what does <spa... | https://math.stackexchange.com/questions/4164659/meaning-of-int-abdx |
Question: <p>I had a practice midterm that had the following question: </p>
<p><span class="math-container">$A = \lim_{x \to\infty} R_n = \lim_{x \to\infty} (\sum_{i=1}^{n} f(x_i)\triangle x)$</span></p>
<p>Use this definition to find an expression for the area under the graph of <span class="math-container">$f(x) = ... | https://math.stackexchange.com/questions/2985141/riemann-sum-problem |
Question: <p>I want to find the derivative of the following:</p>
<p><span class="math-container">$$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)\f... | https://math.stackexchange.com/questions/4174763/derivative-using-fundamental-theorem-of-calculus-when-integrand-has-product-of-t |
Question: <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</co... | https://math.stackexchange.com/questions/4174327/solving-for-i-given-s-sum-n-1m-fraca-n1it-n |
Question: <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:... | https://math.stackexchange.com/questions/4171429/why-do-we-use-only-the-positive-root-when-differentiating-an-inverse-trig-functi |
Question: <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>
Answer: <p>Usually, a partial fraction decomposition is only possible for rational functions. The square root inside ... | https://math.stackexchange.com/questions/686382/decomposing-a-fraction-into-partial-fractions |
Question: <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-... | https://math.stackexchange.com/questions/4140640/multiplying-top-and-bottom-by-cos-x-to-solve-integral |
Question: <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 =... | https://math.stackexchange.com/questions/4140799/trig-substitution-reversion-issue-pm |
Question: <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... | https://math.stackexchange.com/questions/1834088/hyperbolic-trig-proofs-definitions |
Question: <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>
Answer: <p>One of the properties of logarithms is the following:</p>... | https://math.stackexchange.com/questions/4159565/how-does-e-ln-x-e-ln1-x |
Question: <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>
Answer: <p>I'm guessing you're currently in high school so without beating around the ... | https://math.stackexchange.com/questions/4180376/what-is-meant-by-fx-is-function-of-x-or-fx-as-a-function-of-y |
Question: <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 + \delt... | https://math.stackexchange.com/questions/3118997/why-is-lim-delta-x-to0-frac-delta-x-delta-x-1 |
Question: <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}$$</span>
I nee... | https://math.stackexchange.com/questions/4180683/bound-to-sum-in-sqrta-i |
Question: <blockquote>
<p>Use the trapezoidal rule with $N=6$ to approximate the arc length of the curve $f(x) = \sin(x)$ from $x=0$ to $x=\pi$.</p>
</blockquote>
<p>So I found that $\Delta x = \frac{\pi}{6}$ which means that my interval points are $0,\frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}, \frac{5\pi}{6}$ and... | https://math.stackexchange.com/questions/393619/trap-rule-for-sinx |
Question: <blockquote>
<p>Force is generally a function of <span class="math-container">$\mathbf{r}(t)$</span>, <span class="math-container">$\mathbf{v}(t)$</span> and <span class="math-container">$t$</span>. <span class="math-container">$$1-)\begin{cases}
\mathbf F: \mathbb R^3\times\mathbb R^3 \times \mathbb R \ ... | https://math.stackexchange.com/questions/3001165/force-is-generally-a-function-of-mathbfrt-mathbfvt-and-t |
Question: <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>
Answer: <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 limit... | https://math.stackexchange.com/questions/2600840/can-lhopitals-rule-be-applied-only-for-a-part-of-a-function |
Question: <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 fun... | https://math.stackexchange.com/questions/4182392/cant-see-why-one-of-these-functions-is-conservative-and-the-other-isnt |
Question: <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:<... | https://math.stackexchange.com/questions/1617665/moment-of-inertia-around-z-axis |
Question: <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 ... | https://math.stackexchange.com/questions/4099535/first-fundamental-theorem-of-calculus-where-the-bounds-are-not-0-to-x |
Question: <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 ... | https://math.stackexchange.com/questions/2992196/the-intuitive-meaning-of-integrals |
Question: <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 choi... | https://math.stackexchange.com/questions/2484422/area-under-curve-limits |
Question: <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-con... | https://math.stackexchange.com/questions/4087649/how-to-find-a-function-fn-continuous-on-r-such-that-1fn-is-positiv |
Question: <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 th... | https://math.stackexchange.com/questions/164982/find-max-vertical-distance |
Question: <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>$$\lim _{ \theta \rightarrow 0 }{ \frac { -2sin^{ 2 }\theta }{ \theta } } +\lim _{ \theta \righta... | https://math.stackexchange.com/questions/1111672/limit-lim-theta-to-0-frac-cos2-theta-cos-theta-theta |
Question: <p>$\displaystyle \int \cosh ^2t\,\sinh ^5t \; \textrm{d}t \,$</p>
<p>Can't for the life of me figure this one out. I have tried various substitutions. The pythagorean hyperbolic identity, the double variable identity. Nothing. Could someone give me a push please. </p>
Answer: <p>With some manipulation usin... | https://math.stackexchange.com/questions/1156239/integral-issues |
Question: <p>How to prove this in a a smart way?</p>
<blockquote>
<p>If $y= \sin (m \sin^{-1} (x))$, then $(1-x^2)y^{(n+2)}-(2n+1)x{y^{(n+1)}}+(m^2-n^2)y^{(n)}=0$ derivative.</p>
</blockquote>
<p>I have been able to prove it by differentiating it twice and using Leibniz theorem, but thats very long, is there a nic... | https://math.stackexchange.com/questions/1161800/how-to-prove-this-in-smart-way |
Question: <p>Could someone please help me with the following question:</p>
<blockquote>
<p>Consider the function $g(x,y,z)=\ln(x^2-y+z^2)$. Find an equation of the level surface of the function through the point $(-1,2,1)$ which does not have $\ln$. Hint: first find $g(-1,2,1).$</p>
</blockquote>
<p>When I sub in t... | https://math.stackexchange.com/questions/1188100/equation-to-a-level-surface |
Question: <p>Find a parametrization for the circle centered around the origin, of radius 3 and contained in the xz-plane.</p>
<p>So from what I gathered you use the formula of sphere $x^2+y^2+z^2= r^2$ to solve this problem. So you know what the radius is 3 yet how does one find xyz just from having the radius?</p>
A... | https://math.stackexchange.com/questions/1423532/finding-the-parametrization-for-a-sphere |
Question: <p>Given that:</p>
<p>$$\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 q... | https://math.stackexchange.com/questions/1445477/computing-bigtriangledown21-r |
Question: <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>
Answer: <p>In every ... | https://math.stackexchange.com/questions/1536722/finding-discontinuities-points |
Question: <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... | https://math.stackexchange.com/questions/1691903/how-to-plot-fx-x2-3 |
Question: <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>
Answer: <p>Hint:</p>
<p>Consider $2$ cases, when the numerator is evaluated to $0$ and whe... | https://math.stackexchange.com/questions/1980858/calculus-problem-unknown-variable-in-a-quadratic |
Question: <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 ... | https://math.stackexchange.com/questions/2056607/dy-dx-problems-please-help |
Question: <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>
Answer: <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... | https://math.stackexchange.com/questions/2083127/how-to-show-lim-n-rightarrow-infty-fracan1an-a |
Question: <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... | https://math.stackexchange.com/questions/2246369/how-to-solve-this-complicated-integral |
Question: <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 ment... | https://math.stackexchange.com/questions/2271578/a-clue-to-solve-this-equation |
Question: <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 t... | https://math.stackexchange.com/questions/2273552/prove-that-the-following-expression-is-always-less-than-x-for-all-values-of-x-an |
Question: <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>
Answer: <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, onl... | https://math.stackexchange.com/questions/2275235/simple-algebra-radicals-prime-numbers |
Question: <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>$$\int m \frac{d^2x}{dt^2}dx = m\int\frac{d^2x}{dt^2}\frac{dx}{dt}dt = \frac{m}{2}\int\frac{d}{dt}\left(\frac{dx}{dt}\r... | https://math.stackexchange.com/questions/2439131/quick-questions-for-evaluating-an-integral |
Question: <blockquote>
<p>Let $p(x)$ be a $100$-degree polynomial with $100$ real and distinct roots, say $\alpha_1,\alpha_2,\cdots,\alpha_{100}$, and so $$p(x)=A(x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_{100}),$$
where $A\in\mathbb{R}\setminus\{0\}$ and $α_{i}\neq 0$ for all $i\in[1,100]$.
Find nature of roots of ... | https://math.stackexchange.com/questions/2667240/nature-of-roots-of-a-hectic-polynomial |
Question: <p>Given are two equations:</p>
<p>$$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$$</p>
<p>Given $t_1, v_1, t_2, v_2$, how can we solve for $\tau, v_0$... | https://math.stackexchange.com/questions/2713725/solving-exponential-equation-with-two-variables |
Question: <p>Evaluate $$\displaystyle\lim_{x\to 0} \frac{x\cos x - \ln (1+x)}{x^2}$$</p>
<p>Here's my method but that results into an error. </p>
<p>\begin{align}
\lim_{x\to 0} \frac{x\cos x - \ln (1+x)}{x^2}
&=\lim_{x\to 0}\frac{\cos x}{x} - \lim_{x\to 0}\left(\frac{1}{x}\right)\lim_{x\to 0}\left(\frac{\ln(1+x)}... | https://math.stackexchange.com/questions/2723256/error-in-evaluation-of-displaystyle-lim-x-to-0-fracx-cos-x-ln-1xx |
Question: <p>$$100(-10te^-0.1t + 10 \int e^{-0.1t}dt) = 100(-10te^-0.1t -100e^{-0.1t})+C$$</p>
<p>Why is the $+C$ outside of the brackets if the integration was done inside? I'm looking at my math book and I'm baffled.</p>
<p>Thanks for the help.</p>
Answer: <p>As far as I can see, you are multiplying the integral b... | https://math.stackexchange.com/questions/2741914/easy-question-why-is-c-outside-the-brackets |
Question: <p>No idea where to start on this question. Any help is appreciated:</p>
<blockquote>
<p>$$\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>
Answer: <p><strong>Hint:</strong></p>
<p>Define $f : \mathbb{R} \to \mathbb{... | https://math.stackexchange.com/questions/2840854/show-that-there-exists-a-x-in-mathbbr-such-that |
Question: <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/questi... | https://math.stackexchange.com/questions/2862222/curve-length-of-a-unit-sphere-which-intersect-with-a-plane |
Question: <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">$a... | https://math.stackexchange.com/questions/2958859/how-to-prove-the-following-statements-about-tangent-lines-to-y-ax2bxc |
Question: <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 th... | https://math.stackexchange.com/questions/3077488/how-fast-is-the-area-of-rectangle-increasing |
Question: <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 cla... | https://math.stackexchange.com/questions/3104366/differential-notation-misunderstanding |
Question: <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></... | https://math.stackexchange.com/questions/3175234/finding-the-interval-of-when-this-function-decreases |
Question: <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... | https://math.stackexchange.com/questions/3179505/convergent-series-for-sum-n-1-infty-fracnnn |
Question: <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... | https://math.stackexchange.com/questions/3205317/what-do-the-following-parametric-curves-represent |
Question: <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>
Answer: <p>Considering <span class="math-container">$y=f(x)$</span>, you get:
<span class="math-container">$$(x)'_x=(e^{xy})'_x \Right... | https://math.stackexchange.com/questions/3575733/can-i-use-the-partial-implicit-differentiation-with-x-exy |
Question: <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 <sp... | https://math.stackexchange.com/questions/3613859/how-to-solve-fracyy-lny-for-y |
Question: <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? I... | https://math.stackexchange.com/questions/3689270/intuition-behind-integrating-and-differentiating-determinants |
Question: <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>Whe... | https://math.stackexchange.com/questions/3720326/questions-about-parametric-equations |
Question: <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> ca... | https://math.stackexchange.com/questions/3751075/odd-intuitive-proof-for-lhospitals-rule |
Question: <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 figur... | https://math.stackexchange.com/questions/3870146/differentiating-v-c-v-s1-e-t-t |
Question: <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="m... | https://math.stackexchange.com/questions/3924969/for-what-values-of-c-does-the-curve-y-cx3-ex-have-inflection-poi |
Question: <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?
$$\lim_{h \to 0} \frac{\left(\displaysty... | https://math.stackexchange.com/questions/979347/find-the-derivatives-to-fx-4-x2-and-gt-t-5-1-sqrtt |
Question: <p>How can we find the limit
<span class="math-container">$$\lim_{x\to 0} \frac{(e^x-1-x)^2}{x(\sin x -x)}$$</span>?</p>
Answer: <p><span class="math-container">$$\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\c... | https://math.stackexchange.com/questions/3813828/how-to-find-the-limit |
Question: <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... | https://math.stackexchange.com/questions/4086587/find-the-only-f-vt-that-has-f2t-4ft |
Question: <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 ... | https://math.stackexchange.com/questions/2017827/2x2-x-5-solve-4x4-x-using-the-rules-of-exponents |
Question: <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.F... | https://math.stackexchange.com/questions/3672511/how-to-prove-that-sin1-x-is-continuous-at-x%e2%89%a00 |
Question: <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">$f$<... | https://math.stackexchange.com/questions/1732642/function-increase-or-decrease |
Question: <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... | https://math.stackexchange.com/questions/4190525/is-there-anywhere-where-there-are-in-depth-walkthroughs-of-problems-on-stewarts |
Question: <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... | https://math.stackexchange.com/questions/3240322/taking-the-derivative-of-both-sides-of-an-equation |
Question: <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 va... | https://math.stackexchange.com/questions/3028636/determining-the-uniform-convergence |
Question: <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-fa... | https://math.stackexchange.com/questions/4192699/confused-about-variables |
Question: <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">... | https://math.stackexchange.com/questions/4193374/integral-over-circle-area-limited-by-two-straight-lines |
Question: <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>
Answer: <p>$$\sqrt{b}=20-\sqrt{a}$$
$$b=(20-\sqrt{a})^2=400-40\sqrt{a}+a$$
$$a-5b=a-5(400-40\sqrt{a}+a)=-4a+... | https://math.stackexchange.com/questions/1749111/sqrta-sqrtb-20-what-is-the-maximum-value-of-a-5b |
Question: <p>It is known that</p>
<p>$\displaystyle I_0(2)=\sum_{k=0}^{\infty}\frac{1}{(k!)^2} = \frac{1}{\pi}\int_{0}^{\pi}e^{2\cos\theta}d\theta$
(<a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html" rel="nofollow noreferrer">http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKi... | https://math.stackexchange.com/questions/2903380/how-to-obtain-the-integral-representation-of-modified-bessel-function-i-02 |
Question: <p>My thought was to:<br>
1) hypothesis there are 2 real roots for this equation,<br>
2) apply Rolle's theorem and come to a reductio ad absurdum<br>
and then if there aren't 2 real roots, it has to be 1. If there is 1 real root, this means that it has to have 3 non-real roots. But non real roots come in pair... | https://math.stackexchange.com/questions/2909560/proof-that-fx-4x4-2x1-has-no-real-roots |
Question: <p>How does one evaluate the following summation of $n^2$ terms by $x^n$ terms.
I have tried to do it, but couldn't figure it out as it is not the same as summing up $nx^n$ terms.</p>
<p>$$\sum_{n=0}^\infty n^2 x^n$$</p>
Answer: <p>$\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$</p>
<p>Differentiati... | https://math.stackexchange.com/questions/2923002/summation-of-n2xn-terms |
Question: <blockquote>
<p>If <span class="math-container">$$f(x) =
\begin{cases}
|x-2|+a^2-9a-9, &\text{if }x<2\\
2x-3, &\text{if } x\geqslant2
\end{cases}$$</span> has local minima at <span class="math-container">$x=2$</span>, then range of <span class="math-container">$a$</span> is… ?</p>
</bloc... | https://math.stackexchange.com/questions/2928763/finding-range-of-a |
Question: <p><span class="math-container">$$\int\limits_{0}^{\pi \over 6} {x \over \sqrt{1-2\sin{x}}}dx$$</span></p>
<p>I attempted lots of permutations but I can't solve it..
moreover, I don't know its convergence or divergence... please help!</p>
Answer: <p><em>This is not a serious answer. Just done for the fun of... | https://math.stackexchange.com/questions/2931296/how-to-integral-int-limits-0-pi-over-6-x-over-sqrt1-2-sinxdx |
Question: <p>Suppose that <span class="math-container">$f$</span> is a continuous function on <span class="math-container">$[0,1]$</span> and
<span class="math-container">$$\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... | https://math.stackexchange.com/questions/2944797/if-int-0x-f2tdt-le-fx-for-all-x-in-0-1-then-min-0-1-fx |
Question: <p>Evaluate <span class="math-container">$\lim_{x\to0} \frac{\sin(x^{30})}{\sin^{30}(5x)} $</span></p>
<p>I have tried applying L'Hospital's rule, but it took me a lot of time to factor the derivative. Is there any way can resolve this problem. Thanks.</p>
<p>The answer is <span class="math-container">$\fra... | https://math.stackexchange.com/questions/2946383/evaluate-limit-using-lhospital |
Question: <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">$x... | https://math.stackexchange.com/questions/2949154/maximum-value-on-a-circle |
Question: <blockquote>
<p>Evaluation of <span class="math-container">$\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 ... | https://math.stackexchange.com/questions/2952267/evaluation-of-integration-using-limit-as-a-sum |
Question: <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 thinki... | https://math.stackexchange.com/questions/2959847/show-that-c-is-in-interval-e-3-for-c-cdot-lnc-c-%e2%88%92-6-0 |
Question: <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>
Answer: <p>The inequality of arithmetic and geometric means says:</p>
<p><span class="math-conta... | https://math.stackexchange.com/questions/2971893/absolute-conditional-minimum-of-function-in-n-dimensional-space |
Question: <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>
Answer: <blockquote>
<p>So I multiplied by the conjugate and got</p>
</blockq... | https://math.stackexchange.com/questions/2979793/calculate-lim-x-to-infty-x-sqrt31-x3 |
Question: <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 o... | https://math.stackexchange.com/questions/2981463/int-limits-ab-fx-dx-leq-int-limits-ab-fxdx-for-f-continuous |
Question: <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>. ... | https://math.stackexchange.com/questions/2982498/number-of-solutions |
Question: <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 cl... | https://math.stackexchange.com/questions/2982727/how-to-differentiate-this-double-integral-from-christopher-bishops-pattern-reco |
Question: <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>
Answer: <p><span class="math-contain... | https://math.stackexchange.com/questions/2987471/approximate-0-99300-without-calculator |
Question: <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-conta... | https://math.stackexchange.com/questions/2992228/prove-that-int-infty-0-fracx-sinrxa2x2dx-frac-pi2e-ar |
Question: <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... | https://math.stackexchange.com/questions/3011769/show-fx-0-for-xx-0-if-its-f-f-and-fx-0-0 |
Question: <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">\... | https://math.stackexchange.com/questions/3012499/getting-the-rate-of-drain-from-a-tank |
Question: <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>
Answer: <p>Yes, lower limit can be <span class="math-container">$-\infty$</span> but ... | https://math.stackexchange.com/questions/3030569/can-the-lower-limit-of-fracddx-intx-a-ftdt-fx-be-infty |
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