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stringlengths 1
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college_math.PRECALCULUS
|
exercise.6.1.55
|
Find the domain of the function: $f(x)=\frac{1}{3-\log _{5}(x)}$
|
$(0,125) \cup(125, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(0,125) \cup(125, \infty)
|
college_math.PRECALCULUS
|
exercise.2.1.31
|
A plumber charges $\$ 50$ for a service call plus $\$ 80$ per hour. If she spends no longer than 8 hours a day at any one site, find a linear function that represents her total daily charges $C$ (in dollars) as a function of time $t$ (in hours) spent at any one given location.
|
$C(t)=80 t+50,0 \leq t \leq 8$.
|
Creative Commons License
|
college_math.precalculus
|
C(t)=80 t+50,0 \leq t \leq 8$.
|
college_math.PRECALCULUS
|
exercise.6.5.8
|
How much money needs to be invested now to obtain $\$ 2000$ in 3 years if the interest rate in a savings account is $0.25 \%$, compounded continuously? Round your answer to the nearest cent.
|
$P=\frac{2000}{e^{0.0025 \cdot 3}} \approx \$ 1985.06$
|
Creative Commons License
|
college_math.precalculus
|
P=\frac{2000}{e^{0.0025 \cdot 3}} \approx \$ 1985.06
|
college_math.PRECALCULUS
|
exercise.10.2.64
|
If $\theta=37.5^{\circ}$ and the side opposite $\theta$ has length 306 , how long is the side adjacent to $\theta$ ?
|
The hypotenuse has length $c=\frac{306}{\sin \left(37.5^{\circ}\right)} \approx 502.660$, so the side adjacent to $\theta$ has length $\sqrt{c^{2}-306^{2}} \approx 398.797$.
|
Creative Commons License
|
college_math.precalculus
|
The hypotenuse has length $c=\frac{306}{\sin \left(37.5^{\circ}\right)} \approx 502.660$, so the side adjacent to $\theta$ has length $\sqrt{c^{2}-306^{2}} \approx 398.797$.
|
college_math.PRECALCULUS
|
exercise.4.3.6
|
Solve the rational equation: $\frac{-x^{3}+4 x}{x^{2}-9}=4 x$
|
$x=0, x= \pm 2 \sqrt{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, x= \pm 2 \sqrt{2}
|
college_math.PRECALCULUS
|
exercise.8.2.21
|
Solve the following system of linear equations: $\left\{\begin{aligned} 2 x-y+z & =1 \\ 2 x+2 y-z & =1 \\ 3 x+6 y+4 z & =9\end{aligned}\right.$
|
$\left(\frac{1}{3}, \frac{2}{3}, 1\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(\frac{1}{3}, \frac{2}{3}, 1\right)
|
college_math.PRECALCULUS
|
exercise.11.7.78
|
Use a calculator to approximate the five fifth roots of 1.
|
$w_{0}=1$
|
Creative Commons License
|
college_math.precalculus
|
w_{0}=1
|
college_math.PRECALCULUS
|
exercise.8.2.7
|
Solve the following system of linear equations: $\left\{\begin{aligned}-5 x+y & =17 \\ x+y & =5\end{aligned}\right.$
|
$(-2,7)$
|
Creative Commons License
|
college_math.precalculus
|
(-2,7)
|
college_math.PRECALCULUS
|
exercise.4.3.15
|
Solve the rational inequality and express your answer using interval notation: $\frac{3 x-1}{x^{2}+1} \leq 1$
|
$(-\infty, 1] \cup[2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, 1] \cup[2, \infty)
|
college_math.PRECALCULUS
|
exercise.10.4.91
|
If $\tan (\theta)=\frac{x}{7}$ for $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, find an expression for $\sin (2 \theta)$ in terms of $x$.
|
$\frac{14 x}{x^{2}+49}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{14 x}{x^{2}+49}
|
college_math.PRECALCULUS
|
exercise.9.1.4
|
Write out the first four terms of the given sequence: $\left\{\frac{n^{2}+1}{n+1}\right\}_{n=0}^{\infty}$
|
$1,1, \frac{5}{3}, \frac{5}{2}$
|
Creative Commons License
|
college_math.precalculus
|
1,1, \frac{5}{3}, \frac{5}{2}
|
college_math.PRECALCULUS
|
exercise.10.1.37
|
Convert the angle from radian measure into degree measure: $\pi$
|
$180^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
180^{\circ}
|
college_math.PRECALCULUS
|
exercise.6.3.11
|
Solve the equation analytically: $5^{x}=-2$
|
No solution.
|
Creative Commons License
|
college_math.precalculus
|
No solution.
|
college_math.PRECALCULUS
|
exercise.11.1.1
|
The sounds we hear are made up of mechanical waves. The note ' $A$ ' above the note 'middle $\mathrm{C}^{\prime}$ is a sound wave with ordinary frequency $f=440 \mathrm{Hertz}=440 \frac{\mathrm{cycles}}{\text { second }}$. Find a sinusoid which models this note, assuming that the amplitude is 1 and the phase shift is 0 .
|
$S(t)=\sin (880 \pi t)$
|
Creative Commons License
|
college_math.precalculus
|
S(t)=\sin (880 \pi t)
|
college_math.PRECALCULUS
|
exercise.1.3.38
|
Determine whether or not the equation represents $y$ as a function of $x$: $x=-6$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.10.1.32
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $-270^{\circ}$
|
$-\frac{3 \pi}{2}$
|
Creative Commons License
|
college_math.precalculus
|
-\frac{3 \pi}{2}
|
college_math.PRECALCULUS
|
exercise.9.4.1
|
Simplify the expression: $(3 !)^{2}$
|
36
|
Creative Commons License
|
college_math.precalculus
|
36
|
college_math.PRECALCULUS
|
exercise.6.4.1
|
Solve the equation analytically: $\log (3 x-1)=\log (4-x)$
|
$x=\frac{5}{4}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{5}{4}
|
college_math.PRECALCULUS
|
exercise.11.4.27
|
Convert the point from polar coordinates into rectangular coordinates: $(6, \arctan (2))$
|
$\left(\frac{6 \sqrt{5}}{5}, \frac{12 \sqrt{5}}{5}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(\frac{6 \sqrt{5}}{5}, \frac{12 \sqrt{5}}{5}\right)
|
college_math.PRECALCULUS
|
exercise.6.1.53
|
Find the domain of the function: $f(x)=\log _{9}(|x+3|-4)$
|
$(-\infty,-7) \cup(1, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-7) \cup(1, \infty)
|
college_math.PRECALCULUS
|
exercise.10.7.83
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\sec (x) \leq 2$
|
$\left[-\frac{2 \pi}{3},-\frac{\pi}{3}\right) \cup\left(\frac{\pi}{3}, \frac{2 \pi}{3}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left[-\frac{2 \pi}{3},-\frac{\pi}{3}\right) \cup\left(\frac{\pi}{3}, \frac{2 \pi}{3}\right)
|
college_math.PRECALCULUS
|
exercise.9.1.21
|
Determine if the given sequence is arithmetic, geometric or neither. If it is arithmetic, find the common difference $d$; if it is geometric, find the common ratio $r$: $a_{n}=\frac{n !}{2}, n \geq 0$
|
neither
|
Creative Commons License
|
college_math.precalculus
|
neither
|
college_math.PRECALCULUS
|
exercise.10.1.31
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $135^{\circ}$
|
$\frac{3 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{3 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.10.1.30
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $240^{\circ}$
|
$\frac{4 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{4 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.10.4.90
|
If $\sin (\theta)=\frac{x}{2}$ for $-\frac{\pi}{2}<\theta<\frac{\pi}{2}$, find an expression for $\cos (2 \theta)$ in terms of $x$.
|
$1-\frac{x^{2}}{2}$
|
Creative Commons License
|
college_math.precalculus
|
1-\frac{x^{2}}{2}
|
college_math.PRECALCULUS
|
exercise.11.4.20
|
Convert the point from polar coordinates into rectangular coordinates: $(-20,3 \pi)$
|
$(20,0)$
|
Creative Commons License
|
college_math.precalculus
|
(20,0)
|
college_math.PRECALCULUS
|
exercise.6.3.20
|
Solve the equation analytically: $\frac{5000}{1+2 e^{-3 t}}=2500$
|
$t=\frac{1}{3} \ln (2)$
|
Creative Commons License
|
college_math.precalculus
|
t=\frac{1}{3} \ln (2)
|
college_math.PRECALCULUS
|
exercise.10.7.26
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=5 \sin (x)-2$
|
$x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.2.2.1
|
Solve the equation: $|x|=6$
|
$x=-6$ or $x=6$
|
Creative Commons License
|
college_math.precalculus
|
x=-6$ or $x=6
|
college_math.PRECALCULUS
|
exercise.3.4.11
|
Simplify the quantity $\sqrt{-49}$
|
$7 i$
|
Creative Commons License
|
college_math.precalculus
|
7 i
|
college_math.PRECALCULUS
|
exercise.3.3.45
|
Solve the polynomial inequality $-2 x^{3}+19 x^{2}-49 x+20>0$ and state your answer using interval notation.
|
$(-\infty,-1) \cup(-1,0) \cup(2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-1) \cup(-1,0) \cup(2, \infty)
|
college_math.PRECALCULUS
|
exercise.10.2.5
|
Find the exact value of the cosine and sine of the given angle: $\theta=\frac{2 \pi}{3}$
|
$\cos \left(\frac{2 \pi}{3}\right)=-\frac{1}{2}, \sin \left(\frac{2 \pi}{3}\right)=\frac{\sqrt{3}}{2}$
|
Creative Commons License
|
college_math.precalculus
|
\cos \left(\frac{2 \pi}{3}\right)=-\frac{1}{2}, \sin \left(\frac{2 \pi}{3}\right)=\frac{\sqrt{3}}{2}
|
college_math.PRECALCULUS
|
exercise.6.3.12
|
Solve the equation analytically: $3^{(x-1)}=29$
|
$x=\frac{\ln (29)+\ln (3)}{\ln (3)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\ln (29)+\ln (3)}{\ln (3)}
|
college_math.PRECALCULUS
|
exercise.10.7.11
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin \left(2 x-\frac{\pi}{3}\right)=-\frac{1}{2}$
|
$x=\frac{3 \pi}{4}+\pi k$ or $x=\frac{13 \pi}{12}+\pi k ; x=\frac{\pi}{12}, \frac{3 \pi}{4}, \frac{13 \pi}{12}, \frac{7 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{3 \pi}{4}+\pi k$ or $x=\frac{13 \pi}{12}+\pi k ; x=\frac{\pi}{12}, \frac{3 \pi}{4}, \frac{13 \pi}{12}, \frac{7 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.6.2.5
|
Expand the given logarithm and simplify: $\ln \left(\frac{\sqrt{z}}{x y}\right)$
|
$\frac{1}{2} \ln (z)-\ln (x)-\ln (y)$
|
Creative Commons License
|
college_math.precalculus
|
\frac{1}{2} \ln (z)-\ln (x)-\ln (y)
|
college_math.PRECALCULUS
|
exercise.9.4.12
|
Expand the binomial: $\left(\frac{1}{3} x+y^{2}\right)^{3}$
|
$\left(\frac{1}{3} x+y^{2}\right)^{3}=\frac{1}{27} x^{3}+\frac{1}{3} x^{2} y^{2}+x y^{4}+y^{6}$
|
Creative Commons License
|
college_math.precalculus
|
\left(\frac{1}{3} x+y^{2}\right)^{3}=\frac{1}{27} x^{3}+\frac{1}{3} x^{2} y^{2}+x y^{4}+y^{6}
|
college_math.PRECALCULUS
|
exercise.10.7.23
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=\cos (x)$
|
$x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{2 \pi}{3}, \frac{4 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.2.3.21
|
What is the largest rectangular area one can enclose with 14 inches of string?
|
The largest rectangle has area 12.25 square inches.
|
Creative Commons License
|
college_math.precalculus
|
The largest rectangle has area 12.25 square inches.
|
college_math.PRECALCULUS
|
exercise.6.1.24
|
Evaluate the expression: $\log _{36}(36)$
|
$\log _{36}(36)=1$
|
Creative Commons License
|
college_math.precalculus
|
\log _{36}(36)=1
|
college_math.PRECALCULUS
|
exercise.10.7.71
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\sec ^{2}(x) \leq 4$
|
$\left(-\infty, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\infty, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right]
|
college_math.PRECALCULUS
|
exercise.7.2.15
|
Find the standard equation of the circle which satisfies the given criteria: endpoints of a diameter: $(3,6)$ and $(-1,4)$
|
$(x-1)^{2}+(y-5)^{2}=5$
|
Creative Commons License
|
college_math.precalculus
|
(x-1)^{2}+(y-5)^{2}=5
|
college_math.PRECALCULUS
|
exercise.1.3.41
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}+y^{2}=4$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.9.2.12
|
Rewrite the sum using summation notation: $1+2+4+\cdots+2^{29}$
|
$\sum_{k=1}^{30} 2^{k-1}$
|
Creative Commons License
|
college_math.precalculus
|
\sum_{k=1}^{30} 2^{k-1}
|
college_math.PRECALCULUS
|
exercise.9.4.6
|
Simplify the expression: $\frac{(k-1) !}{(k+2) !}, k \geq 1$.
|
$\frac{1}{k(k+1)(k+2)}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{1}{k(k+1)(k+2)}
|
college_math.PRECALCULUS
|
exercise.4.3.13
|
Solve the rational inequality and express your answer using interval notation: $\frac{x^{3}+2 x^{2}+x}{x^{2}-x-2} \geq 0$
|
$(-1,0] \cup(2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-1,0] \cup(2, \infty)
|
college_math.PRECALCULUS
|
exercise.1.3.43
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=4$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.1.1.11
|
Write the set using interval notation: $\{x \mid x \neq 0,2\}$
|
$(-\infty, 0) \cup(0,2) \cup(2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, 0) \cup(0,2) \cup(2, \infty)
|
college_math.PRECALCULUS
|
exercise.6.4.8
|
Solve the equation analytically: $\log \left(x^{2}-3 x\right)=1$
|
$x=-2,5$
|
Creative Commons License
|
college_math.precalculus
|
x=-2,5
|
college_math.PRECALCULUS
|
exercise.6.1.30
|
Evaluate the expression: $\log _{13}(\sqrt{13})$
|
$\log _{13}(\sqrt{13})=\frac{1}{2}$
|
Creative Commons License
|
college_math.precalculus
|
\log _{13}(\sqrt{13})=\frac{1}{2}
|
college_math.PRECALCULUS
|
exercise.7.3.15
|
Find an equation for the parabola which fits the given criteria: Vertex $(7,0)$, focus $(0,0)$
|
$y^{2}=-28(x-7)$
|
Creative Commons License
|
college_math.precalculus
|
y^{2}=-28(x-7)
|
college_math.PRECALCULUS
|
exercise.6.3.30
|
Solve the equation analytically: $4^{x}+2^{x}=12$
|
$x=\frac{\ln (3)}{\ln (2)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\ln (3)}{\ln (2)}
|
college_math.PRECALCULUS
|
exercise.3.4.17
|
Simplify the quantity $\sqrt{-(-9)}$
|
3
|
Creative Commons License
|
college_math.precalculus
|
3
|
college_math.PRECALCULUS
|
exercise.1.1.18
|
Write the set using interval notation: $\{x \mid x>2$ or $x= \pm 1\}$
|
$\{-1\} \cup\{1\} \cup(2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
\{-1\} \cup\{1\} \cup(2, \infty)
|
college_math.PRECALCULUS
|
exercise.10.7.75
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cot ^{2}(x) \geq \frac{1}{3}$
|
$\left(0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \pi\right) \cup\left(\pi, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \pi\right) \cup\left(\pi, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right)
|
college_math.PRECALCULUS
|
exercise.6.3.37
|
Solve the inequality analytically: $25\left(\frac{4}{5}\right)^{x} \geq 10$
|
$\left(-\infty, \frac{\ln \left(\frac{2}{5}\right)}{\ln \left(\frac{4}{5}\right)}\right]=\left(-\infty, \frac{\ln (2)-\ln (5)}{\ln (4)-\ln (5)}\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\infty, \frac{\ln \left(\frac{2}{5}\right)}{\ln \left(\frac{4}{5}\right)}\right]=\left(-\infty, \frac{\ln (2)-\ln (5)}{\ln (4)-\ln (5)}\right]
|
college_math.PRECALCULUS
|
exercise.10.7.17
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos ^{2}(x)=\frac{1}{2}$
|
$x=\frac{\pi}{4}+\frac{\pi k}{2} ; x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{4}+\frac{\pi k}{2} ; x=\frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.6.4.20
|
Solve the equation analytically: $\log (x)-\log (2)=\log (x+8)-\log (x+2)$
|
$x=4$
|
Creative Commons License
|
college_math.precalculus
|
x=4
|
college_math.PRECALCULUS
|
exercise.10.1.43
|
Convert the angle from radian measure into degree measure: $-\frac{\pi}{6}$
|
$-30^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
-30^{\circ}
|
college_math.PRECALCULUS
|
exercise.11.8.58
|
A 600 pound Sasquatch statue is suspended by two cables from a gymnasium ceiling. If each cable makes a $60^{\circ}$ angle with the ceiling, find the tension on each cable. Round your answer to the nearest pound.
|
The resultant force is only about 296 pounds so the couch doesn't budge. Even if it did move, the stronger force on the third rope would have made the couch drift slightly to the south as it traveled down the street.
|
Creative Commons License
|
college_math.precalculus
|
The resultant force is only about 296 pounds so the couch doesn't budge. Even if it did move, the stronger force on the third rope would have made the couch drift slightly to the south as it traveled down the street.
|
college_math.PRECALCULUS
|
exercise.2.3.22
|
The height of an object dropped from the roof of an eight story building is modeled by $h(t)=-16 t^{2}+64,0 \leq t \leq 2$. Here, $h$ is the height of the object off the ground, in feet, $t$ seconds after the object is dropped. How long before the object hits the ground?
|
2 seconds.
|
Creative Commons License
|
college_math.precalculus
|
2 seconds.
|
college_math.PRECALCULUS
|
exercise.11.4.73
|
Convert the equation from polar coordinates into rectangular coordinates: $r=-\csc (\theta) \cot (\theta)$
|
$r=7 \sin (\theta)$
|
Creative Commons License
|
college_math.precalculus
|
r=7 \sin (\theta)
|
college_math.PRECALCULUS
|
exercise.6.2.26
|
Use the properties of logarithms to write the expression as a single logarithm: $\ln (x)+\frac{1}{2}$
|
$\ln (x \sqrt{e})$
|
Creative Commons License
|
college_math.precalculus
|
\ln (x \sqrt{e})
|
college_math.PRECALCULUS
|
exercise.5.3.26
|
Solve the equation or inequality: $5-(4-2 x)^{\frac{2}{3}}=1$
|
$x=-2,6$
|
Creative Commons License
|
college_math.precalculus
|
x=-2,6
|
college_math.PRECALCULUS
|
exercise.6.1.22
|
Evaluate the expression: $\log _{\frac{1}{5}}(625)$
|
$\log _{\frac{1}{5}}(625)=-4$
|
Creative Commons License
|
college_math.precalculus
|
\log _{\frac{1}{5}}(625)=-4
|
college_math.PRECALCULUS
|
exercise.6.4.10
|
Solve the equation analytically: $\log \left(\frac{x}{10^{-3}}\right)=4.7$
|
$x=10^{1.7}$
|
Creative Commons License
|
college_math.precalculus
|
x=10^{1.7}
|
college_math.PRECALCULUS
|
exercise.9.2.9
|
Rewrite the sum using summation notation: $8+11+14+17+20$
|
$\sum_{k=1}^{5}(3 k+5)$
|
Creative Commons License
|
college_math.precalculus
|
\sum_{k=1}^{5}(3 k+5)
|
college_math.PRECALCULUS
|
exercise.6.1.44
|
Find the domain of the function: $f(x)=\log _{7}(4 x+8)$
|
$(-2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-2, \infty)
|
college_math.PRECALCULUS
|
exercise.10.7.4
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\tan (6 x)=1$
|
$x=\frac{\pi}{24}+\frac{\pi k}{6} ; x=\frac{\pi}{24}, \frac{5 \pi}{24}, \frac{3 \pi}{8}, \frac{13 \pi}{24}, \frac{17 \pi}{24}, \frac{7 \pi}{8}, \frac{25 \pi}{24}, \frac{29 \pi}{24}, \frac{11 \pi}{8}, \frac{37 \pi}{24}, \frac{41 \pi}{24}, \frac{15 \pi}{8}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{24}+\frac{\pi k}{6} ; x=\frac{\pi}{24}, \frac{5 \pi}{24}, \frac{3 \pi}{8}, \frac{13 \pi}{24}, \frac{17 \pi}{24}, \frac{7 \pi}{8}, \frac{25 \pi}{24}, \frac{29 \pi}{24}, \frac{11 \pi}{8}, \frac{37 \pi}{24}, \frac{41 \pi}{24}, \frac{15 \pi}{8}
|
college_math.PRECALCULUS
|
exercise.6.1.32
|
Evaluate the expression: $7^{\log _{7}(3)}$
|
$7^{\log _{7}(3)}=3$
|
Creative Commons License
|
college_math.precalculus
|
7^{\log _{7}(3)}=3
|
college_math.PRECALCULUS
|
exercise.6.2.23
|
Use the properties of logarithms to write the expression as a single logarithm: $\log _{5}(x)-3$
|
$\log _{5}\left(\frac{x}{125}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\log _{5}\left(\frac{x}{125}\right)
|
college_math.PRECALCULUS
|
exercise.6.4.31
|
Solve the equation or inequality using your calculator: $\ln (x)=e^{-x}$
|
$x \approx 1.3098$
|
Creative Commons License
|
college_math.precalculus
|
x \approx 1.3098
|
college_math.PRECALCULUS
|
exercise.6.3.39
|
Solve the inequality analytically: $70+90 e^{-0.1 t} \leq 75$
|
$\left[\frac{\ln \left(\frac{1}{18}\right)}{-0.1}, \infty\right)=[10 \ln (18), \infty)$
|
Creative Commons License
|
college_math.precalculus
|
\left[\frac{\ln \left(\frac{1}{18}\right)}{-0.1}, \infty\right)=[10 \ln (18), \infty)
|
college_math.PRECALCULUS
|
exercise.3.4.26
|
Simplify the given power of $i$: $i^{304}$
|
$i^{304}=\left(i^{4}\right)^{76}=1^{76}=1$
|
Creative Commons License
|
college_math.precalculus
|
i^{304}=\left(i^{4}\right)^{76}=1^{76}=1
|
college_math.PRECALCULUS
|
exercise.6.3.32
|
Solve the equation analytically: $e^{x}+15 e^{-x}=8$
|
$x=\ln (3), \ln (5)$
|
Creative Commons License
|
college_math.precalculus
|
x=\ln (3), \ln (5)
|
college_math.PRECALCULUS
|
exercise.6.4.22
|
Solve the equation analytically: $\ln (\ln (x))=3$
|
$x=e^{e^{3}}$
|
Creative Commons License
|
college_math.precalculus
|
x=e^{e^{3}}
|
college_math.PRECALCULUS
|
exercise.10.2.51
|
Approximate the given value to three decimal places: $\sin (392.994)$
|
$\sin (392.994) \approx-0.291$
|
Creative Commons License
|
college_math.precalculus
|
\sin (392.994) \approx-0.291
|
college_math.PRECALCULUS
|
exercise.9.4.15
|
Simplify the power of a complex number: $(-1+i \sqrt{3})^{3}$
|
8
|
Creative Commons License
|
college_math.precalculus
|
8
|
college_math.PRECALCULUS
|
exercise.2.2.2
|
Solve the equation: $|3 x-1|=10$
|
$x=-3$ or $x=\frac{11}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=-3$ or $x=\frac{11}{3}
|
college_math.PRECALCULUS
|
exercise.6.1.16
|
Evaluate the expression: $\log _{3}(27)$
|
$\log _{3}(27)=3$
|
Creative Commons License
|
college_math.precalculus
|
\log _{3}(27)=3
|
college_math.PRECALCULUS
|
exercise.10.2.1
|
Find the exact value of the cosine and sine of the given angle: $\theta=0$
|
$\cos (0)=1, \sin (0)=0$
|
Creative Commons License
|
college_math.precalculus
|
\cos (0)=1, \sin (0)=0
|
college_math.PRECALCULUS
|
exercise.10.1.54
|
A rock got stuck in the tread of my tire and when I was driving 70 miles per hour, the rock came loose and hit the inside of the wheel well of the car. How fast, in miles per hour, was the rock traveling when it came out of the tread? (The tire has a diameter of 23 inches.)
|
70 miles per hour
|
Creative Commons License
|
college_math.precalculus
|
70 miles per hour
|
college_math.PRECALCULUS
|
exercise.2.3.35
|
Solve the quadratic equation $y^{2}-4 y=x^{2}-4$ for $y$.
|
$y=2 \pm x$
|
Creative Commons License
|
college_math.precalculus
|
y=2 \pm x
|
college_math.PRECALCULUS
|
exercise.1.3.42
|
Determine whether or not the equation represents $y$ as a function of $x$: $y=\sqrt{4-x^{2}}$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
college_math.PRECALCULUS
|
exercise.2.2.7
|
Solve the equation: $\frac{5-|x|}{2}=1$
|
$x=-3$ or $x=3$
|
Creative Commons License
|
college_math.precalculus
|
x=-3$ or $x=3
|
college_math.PRECALCULUS
|
exercise.6.1.50
|
Find the domain of the function: $f(x)=\ln (4 x-20)+\ln \left(x^{2}+9 x+18\right)$
|
$(5, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(5, \infty)
|
college_math.PRECALCULUS
|
exercise.10.1.4
|
Convert the angle into the DMS system and round the answer to the nearest second: $179.999^{\circ}$
|
$179^{\circ} 59^{\prime} 56^{\prime \prime}$
|
Creative Commons License
|
college_math.precalculus
|
179^{\circ} 59^{\prime} 56^{\prime \prime}
|
college_math.PRECALCULUS
|
exercise.1.3.47
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}=y^{2}$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.6.4.36
|
Solve the equation analytically: $\ln (3-y)-\ln (y)=2 x+\ln (5)$ for $y$.
|
$y=\frac{3}{5 e^{2 x}+1}$
|
Creative Commons License
|
college_math.precalculus
|
y=\frac{3}{5 e^{2 x}+1}
|
college_math.PRECALCULUS
|
exercise.10.7.25
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \cos (2 x)+\cos (x)+2=0$
|
$x=\frac{7 \pi}{6}, \frac{11 \pi}{6}, \arcsin \left(\frac{1}{3}\right), \pi-\arcsin \left(\frac{1}{3}\right)$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{7 \pi}{6}, \frac{11 \pi}{6}, \arcsin \left(\frac{1}{3}\right), \pi-\arcsin \left(\frac{1}{3}\right)
|
college_math.PRECALCULUS
|
exercise.1.3.36
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{2}-y^{2}=1$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.10.7.24
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x)=2-5 \cos (x)$
|
$x=\frac{2 \pi}{3}, \frac{4 \pi}{3}, \arccos \left(\frac{1}{3}\right), 2 \pi-\arccos \left(\frac{1}{3}\right)$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{2 \pi}{3}, \frac{4 \pi}{3}, \arccos \left(\frac{1}{3}\right), 2 \pi-\arccos \left(\frac{1}{3}\right)
|
college_math.PRECALCULUS
|
exercise.10.7.59
|
Solve the equation: $\arccos (2 x)=\pi$
|
$x=-\frac{1}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{1}{2}
|
college_math.PRECALCULUS
|
exercise.6.4.9
|
Solve the equation analytically: $\log _{125}\left(\frac{3 x-2}{2 x+3}\right)=\frac{1}{3}$
|
$x=-\frac{17}{7}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{17}{7}
|
college_math.PRECALCULUS
|
exercise.6.3.18
|
Solve the equation analytically: $30-6 e^{-0.1 x}=20$
|
$x=-10 \ln \left(\frac{5}{3}\right)=10 \ln \left(\frac{3}{5}\right)$
|
Creative Commons License
|
college_math.precalculus
|
x=-10 \ln \left(\frac{5}{3}\right)=10 \ln \left(\frac{3}{5}\right)
|
college_math.PRECALCULUS
|
exercise.1.1.17
|
Write the set using interval notation: $\{x \mid x \leq 5$ or $x=6\}$
|
$(-\infty, 5] \cup\{6\}$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, 5] \cup\{6\}
|
college_math.PRECALCULUS
|
exercise.8.2.27
|
At a local buffet, 22 diners (5 of whom were children) feasted for $\$162.25$, before taxes. If the kids buffet is $\$4.50$, the basic buffet is $\$7.50$, and the deluxe buffet (with crab legs) is $\$9.25$, find out how many diners chose the deluxe buffet.
|
This time, 7 diners chose the deluxe buffet.
|
Creative Commons License
|
college_math.precalculus
|
This time, 7 diners chose the deluxe buffet.
|
college_math.PRECALCULUS
|
exercise.10.7.67
|
Solve the equation: $8 \operatorname{arccot}^{2}(x)+3 \pi^{2}=10 \pi \operatorname{arccot}(x)$
|
$x=-1,0$
|
Creative Commons License
|
college_math.precalculus
|
x=-1,0
|
college_math.PRECALCULUS
|
exercise.3.4.22
|
Simplify the given power of $i$: $i^8$
|
$i^{8}=i^{4} \cdot i^{4}=\left(i^{4}\right)^{2}=(1)^{2}=1$
|
Creative Commons License
|
college_math.precalculus
|
i^{8}=i^{4} \cdot i^{4}=\left(i^{4}\right)^{2}=(1)^{2}=1
|
college_math.PRECALCULUS
|
exercise.1.1.8
|
Write the set using interval notation: $\{x \mid x \neq 5\}$
|
$(-\infty, 5) \cup(5, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, 5) \cup(5, \infty)
|
college_math.PRECALCULUS
|
exercise.2.3.34
|
Solve the quadratic equation $y^{2}-3 y=4 x$ for $y$.
|
$y=\frac{3 \pm \sqrt{16 x+9}}{2}$
|
Creative Commons License
|
college_math.precalculus
|
y=\frac{3 \pm \sqrt{16 x+9}}{2}
|
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