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stringlengths 1
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college_math.PRECALCULUS
|
exercise.1.1.16
|
Write the set using interval notation: $\{x \mid x \leq-3$ or $x>0\}$
|
$(-\infty,-3] \cup(0, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-3] \cup(0, \infty)
|
college_math.PRECALCULUS
|
exercise.2.2.20
|
Solve the equation: $|2-5 x|=5|x+1|$
|
$x=-\frac{3}{10}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{3}{10}
|
college_math.PRECALCULUS
|
exercise.10.1.3
|
Convert the angle into the DMS system and round the answer to the nearest second: $-317.06^{\circ}$
|
$-317^{\circ} 3^{\prime} 36^{\prime \prime} \quad
|
Creative Commons License
|
college_math.precalculus
|
-317^{\circ} 3^{\prime} 36^{\prime \prime} \quad
|
college_math.PRECALCULUS
|
exercise.10.1.33
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $-315^{\circ}$
|
$-\frac{7 \pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
-\frac{7 \pi}{4}
|
college_math.PRECALCULUS
|
exercise.6.4.2
|
Solve the equation analytically: $\log _{2}\left(x^{3}\right)=\log _{2}(x)$
|
$x=1$
|
Creative Commons License
|
college_math.precalculus
|
x=1
|
college_math.PRECALCULUS
|
exercise.8.2.9
|
Solve the following system of linear equations: $\left\{\begin{aligned} 4 x-y+z & =5 \\ 2 y+6 z & =30 \\ x+z & =5\end{aligned}\right.$
|
$(-3 t+4,-6 t-6,2, t)$ for all real numbers $t$
|
Creative Commons License
|
college_math.precalculus
|
(-3 t+4,-6 t-6,2, t)$ for all real numbers $t
|
college_math.PRECALCULUS
|
exercise.6.1.25
|
Evaluate the expression: $\log \left(\frac{1}{1000000}\right)$
|
$\log \frac{1}{1000000}=-6$
|
Creative Commons License
|
college_math.precalculus
|
\log \frac{1}{1000000}=-6
|
college_math.PRECALCULUS
|
exercise.10.7.18
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\sin ^{2}(x)=\frac{3}{4}$
|
$x=\frac{\pi}{3}+\pi k$ or $x=\frac{2 \pi}{3}+\pi k ; x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{3}+\pi k$ or $x=\frac{2 \pi}{3}+\pi k ; x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.10.7.92
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\cos (2 x) \leq \sin (x)$
|
$\left[-\frac{11 \pi}{6},-\frac{7 \pi}{6}\right] \cup\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right] \cup,\left\{-\frac{\pi}{2}, \frac{3 \pi}{2}\right\}$
|
Creative Commons License
|
college_math.precalculus
|
\left[-\frac{11 \pi}{6},-\frac{7 \pi}{6}\right] \cup\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right] \cup,\left\{-\frac{\pi}{2}, \frac{3 \pi}{2}\right\}
|
college_math.PRECALCULUS
|
exercise.6.1.37
|
Evaluate the expression: $\log \left(\sqrt[3]{10^{5}}\right)$
|
$\log \left(\sqrt[3]{10^{5}}\right)=\frac{5}{3}$
|
Creative Commons License
|
college_math.precalculus
|
\log \left(\sqrt[3]{10^{5}}\right)=\frac{5}{3}
|
college_math.PRECALCULUS
|
exercise.6.3.31
|
Solve the equation analytically: $e^{x}-3 e^{-x}=2$
|
$x=\ln (3)$
|
Creative Commons License
|
college_math.precalculus
|
x=\ln (3)
|
college_math.PRECALCULUS
|
exercise.9.2.30
|
Express the repeating decimal as a fraction of integers: $0 . \overline{13}$
|
$\frac{13}{99}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{13}{99}
|
college_math.PRECALCULUS
|
exercise.9.1.6
|
Write out the first four terms of the given sequence: $\left\{\frac{\ln (n)}{n}\right\}_{n=1}^{\infty}$
|
$0, \frac{\ln (2)}{2}, \frac{\ln (3)}{3}, \frac{\ln (4)}{4}$
|
Creative Commons License
|
college_math.precalculus
|
0, \frac{\ln (2)}{2}, \frac{\ln (3)}{3}, \frac{\ln (4)}{4}
|
college_math.PRECALCULUS
|
exercise.6.3.26
|
Solve the equation analytically: $3^{(x-1)}=\left(\frac{1}{2}\right)^{(x+5)}$
|
$x=\frac{\ln (3)+5 \ln \left(\frac{1}{2}\right)}{\ln (3)-\ln \left(\frac{1}{2}\right)}=\frac{\ln (3)-5 \ln (2)}{\ln (3)+\ln (2)}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\ln (3)+5 \ln \left(\frac{1}{2}\right)}{\ln (3)-\ln \left(\frac{1}{2}\right)}=\frac{\ln (3)-5 \ln (2)}{\ln (3)+\ln (2)}
|
college_math.PRECALCULUS
|
exercise.3.3.47
|
Solve the polynomial inequality $(x-1)^{2} \geq 4$ and state your answer using interval notation.
|
$(-\infty,-\sqrt[3]{3}) \cup(\sqrt[3]{2}, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-\sqrt[3]{3}) \cup(\sqrt[3]{2}, \infty)
|
college_math.PRECALCULUS
|
exercise.11.4.54
|
Convert the equation from rectangular coordinates into polar coordinates: $(x+2)^{2}+y^{2}=4$
|
$\left(15,2 \pi-\arctan \left(\frac{3}{4}\right)\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(15,2 \pi-\arctan \left(\frac{3}{4}\right)\right)
|
college_math.PRECALCULUS
|
exercise.6.2.13
|
Expand the given logarithm and simplify: $\log \left(\frac{100 x \sqrt{y}}{\sqrt[3]{10}}\right)$
|
$\frac{5}{3}+\log (x)+\frac{1}{2} \log (y)$
|
Creative Commons License
|
college_math.precalculus
|
\frac{5}{3}+\log (x)+\frac{1}{2} \log (y)
|
college_math.PRECALCULUS
|
exercise.3.1.30
|
Suppose the cost, in thousands of dollars, to produce $x$ hundred LCD TVs is given by $C(x)=200 x+25$ for $x \geq 0$. Find and simplify an expression for the profit function $P(x)$. (Remember: Profit $=$ Revenue - Cost.)
|
$p(t)=-t^{2}(3-5 t)\left(t^{2}+t+4\right)$
|
Creative Commons License
|
college_math.precalculus
|
p(t)=-t^{2}(3-5 t)\left(t^{2}+t+4\right)
|
college_math.PRECALCULUS
|
exercise.2.3.31
|
Solve the quadratic equation $x^{2}-10 y^{2}=0$ for $x$.
|
$x= \pm y \sqrt{10}$
|
Creative Commons License
|
college_math.precalculus
|
x= \pm y \sqrt{10}
|
college_math.PRECALCULUS
|
exercise.1.1.12
|
Write the set using interval notation: $\{x \mid x \neq 2,-2\}$
|
$(-\infty,-2) \cup(-2,2) \cup(2, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty,-2) \cup(-2,2) \cup(2, \infty)
|
college_math.PRECALCULUS
|
exercise.6.2.21
|
Use the properties of logarithms to write the expression as a single logarithm: $\log (x)-\frac{1}{3} \log (z)+\frac{1}{2} \log (y)$
|
$\log \left(\frac{x \sqrt{y}}{\sqrt[3]{z}}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\log \left(\frac{x \sqrt{y}}{\sqrt[3]{z}}\right)
|
college_math.PRECALCULUS
|
exercise.9.1.19
|
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 $r$: $2,22,222,2222, \ldots$
|
neither
|
Creative Commons License
|
college_math.precalculus
|
neither
|
college_math.PRECALCULUS
|
exercise.6.3.19
|
Solve the equation analytically: $\frac{100 e^{x}}{e^{x}+2}=50$
|
$x=\ln (2)$
|
Creative Commons License
|
college_math.precalculus
|
x=\ln (2)
|
college_math.PRECALCULUS
|
exercise.11.4.31
|
Convert the point from polar coordinates into rectangular coordinates: $\left(2, \pi-\arctan \left(\frac{1}{2}\right)\right)$
|
$\left(-\frac{4 \sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\frac{4 \sqrt{5}}{5}, \frac{2 \sqrt{5}}{5}\right)
|
college_math.PRECALCULUS
|
exercise.10.7.2
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos (3 x)=\frac{1}{2}$
|
$x=\frac{\pi}{9}+\frac{2 \pi k}{3}$ or $x=\frac{5 \pi}{9}+\frac{2 \pi k}{3} ; x=\frac{\pi}{9}, \frac{5 \pi}{9}, \frac{7 \pi}{9}, \frac{11 \pi}{9}, \frac{13 \pi}{9}, \frac{17 \pi}{9}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{9}+\frac{2 \pi k}{3}$ or $x=\frac{5 \pi}{9}+\frac{2 \pi k}{3} ; x=\frac{\pi}{9}, \frac{5 \pi}{9}, \frac{7 \pi}{9}, \frac{11 \pi}{9}, \frac{13 \pi}{9}, \frac{17 \pi}{9}
|
college_math.PRECALCULUS
|
exercise.2.2.4
|
Solve the equation: $4-|x|=3$
|
$x=-1$ or $x=1$
|
Creative Commons License
|
college_math.precalculus
|
x=-1$ or $x=1
|
college_math.PRECALCULUS
|
exercise.6.1.39
|
Evaluate the expression: $\log _{5}\left(3^{\log _{3}(5)}\right)$
|
$\log _{5}\left(3^{\log _{3} 5}\right)=1$
|
Creative Commons License
|
college_math.precalculus
|
\log _{5}\left(3^{\log _{3} 5}\right)=1
|
college_math.PRECALCULUS
|
exercise.10.7.42
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\tan (x)=\sec (x)$
|
No solution
|
Creative Commons License
|
college_math.precalculus
|
No solution
|
college_math.PRECALCULUS
|
exercise.10.7.34
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cot ^{4}(x)=4 \csc ^{2}(x)-7$
|
$x=\frac{\pi}{6}, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{4}, \frac{7 \pi}{4}, \frac{11 \pi}{6}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{6}, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{4}, \frac{7 \pi}{4}, \frac{11 \pi}{6}
|
college_math.PRECALCULUS
|
exercise.10.7.102
|
Express the domain of the function using the extended interval notation: $f(x)=\sqrt{2-\sec (x)}$
|
$\bigcup_{k=-\infty}^{\infty}\left\{\left[\frac{(6 k-1) \pi}{3}, \frac{(6 k+1) \pi}{3}\right] \cup\left(\frac{(4 k+1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)\right\}$
|
Creative Commons License
|
college_math.precalculus
|
\bigcup_{k=-\infty}^{\infty}\left\{\left[\frac{(6 k-1) \pi}{3}, \frac{(6 k+1) \pi}{3}\right] \cup\left(\frac{(4 k+1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)\right\}
|
college_math.PRECALCULUS
|
exercise.2.1.39
|
A local pizza store offers medium two-topping pizzas delivered for $\$ 6.00$ per pizza plus a $\$ 1.50$ delivery charge per order. On weekends, the store runs a 'game day' special: if six or more medium two-topping pizzas are ordered, they are $\$ 5.50$ each with no delivery charge. Write a piecewise-defined linear function which calculates the $\operatorname{cost} C$ (in dollars) of $p$ medium two-topping pizzas delivered during a weekend.
|
$C(p)=\left\{\begin{array}{rll}6 p+1.5 & \text { if } & 1 \leq p \leq 5 \\ 5.5 p & \text { if } & p \geq 6\end{array}\right.$
|
Creative Commons License
|
college_math.precalculus
|
C(p)=\left\{\begin{array}{rll}6 p+1.5 & \text { if } & 1 \leq p \leq 5 \\ 5.5 p & \text { if } & p \geq 6\end{array}\right.
|
college_math.PRECALCULUS
|
exercise.9.2.14
|
Rewrite the sum using summation notation: $-\ln (3)+\ln (4)-\ln (5)+\cdots+\ln (20)$
|
$\sum_{k=3}^{20}(-1)^{k} \ln (k)$
|
Creative Commons License
|
college_math.precalculus
|
\sum_{k=3}^{20}(-1)^{k} \ln (k)
|
college_math.PRECALCULUS
|
exercise.10.7.7
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cot (2 x)=-\frac{\sqrt{3}}{3}$
|
$x=\frac{\pi}{3}+\frac{\pi k}{2} ; x=\frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3}, \frac{11 \pi}{6}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{3}+\frac{\pi k}{2} ; x=\frac{\pi}{3}, \frac{5 \pi}{6}, \frac{4 \pi}{3}, \frac{11 \pi}{6}
|
college_math.PRECALCULUS
|
exercise.10.1.7
|
Convert the angle into decimal degrees and round the answer to three decimal places: $502^{\circ} 35^{\prime}$
|
$502.583^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
502.583^{\circ}
|
college_math.PRECALCULUS
|
exercise.9.4.13
|
Expand the binomial: $\left(x-x^{-1}\right)^{4}$
|
$\left(x-x^{-1}\right)^{4}=x^{4}-4 x^{2}+6-4 x^{-2}+x^{-4}$
|
Creative Commons License
|
college_math.precalculus
|
\left(x-x^{-1}\right)^{4}=x^{4}-4 x^{2}+6-4 x^{-2}+x^{-4}
|
college_math.PRECALCULUS
|
exercise.10.1.42
|
Convert the angle from radian measure into degree measure: $\frac{5 \pi}{3}$
|
$300^{\circ}$
|
Creative Commons License
|
college_math.precalculus
|
300^{\circ}
|
college_math.PRECALCULUS
|
exercise.10.7.89
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\cot (x) \geq 5$
|
$(-2 \pi, \operatorname{arccot}(5)-2 \pi] \cup(-\pi, \operatorname{arccot}(5)-\pi] \cup(0, \operatorname{arccot}(5)] \cup(\pi, \pi+\operatorname{arccot}(5)]$
|
Creative Commons License
|
college_math.precalculus
|
(-2 \pi, \operatorname{arccot}(5)-2 \pi] \cup(-\pi, \operatorname{arccot}(5)-\pi] \cup(0, \operatorname{arccot}(5)] \cup(\pi, \pi+\operatorname{arccot}(5)]
|
college_math.PRECALCULUS
|
exercise.9.1.13
|
Write out the first four terms of the given sequence: $F_{0}=1, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}, n \geq 2$ (This is the famous Fibonacci Sequence)
|
$1,1,2,3$
|
Creative Commons License
|
college_math.precalculus
|
1,1,2,3
|
college_math.PRECALCULUS
|
exercise.10.2.61
|
If $\theta=59^{\circ}$ and the side opposite $\theta$ has length 117.42 , how long is the hypotenuse?
|
The hypotenuse has length $\frac{117.42}{\sin \left(59^{\circ}\right)} \approx 136.99$.
|
Creative Commons License
|
college_math.precalculus
|
The hypotenuse has length $\frac{117.42}{\sin \left(59^{\circ}\right)} \approx 136.99$.
|
college_math.PRECALCULUS
|
exercise.10.7.95
|
Solve the given inequality: $6 \operatorname{arccot}(7 x) \geq \pi$
|
$\left(-\infty, \frac{\sqrt{3}}{7}\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\infty, \frac{\sqrt{3}}{7}\right]
|
college_math.PRECALCULUS
|
exercise.6.2.17
|
Use the properties of logarithms to write the expression as a single logarithm: $\log _{2}(x)+\log _{2}(y)-\log _{2}(z)$
|
$\log _{2}\left(\frac{x y}{z}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\log _{2}\left(\frac{x y}{z}\right)
|
college_math.PRECALCULUS
|
exercise.2.1.29
|
Carl can stuff 6 envelopes per minute. Find a linear function $E$ that represents the total number of envelopes Carl can stuff after $t$ hours, assuming he doesn't take any breaks.
|
$E(t)=360 t, t \geq 0$.
|
Creative Commons License
|
college_math.precalculus
|
E(t)=360 t, t \geq 0$.
|
college_math.PRECALCULUS
|
exercise.6.3.7
|
Solve the equation analytically: $3^{7 x}=81^{4-2 x}$
|
$x=\frac{16}{15}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{16}{15}
|
college_math.PRECALCULUS
|
exercise.2.4.23
|
The height $h$ in feet of a model rocket above the ground $t$ seconds after lift-off is given by $h(t)=-5 t^{2}+100 t$, for $0 \leq t \leq 20$. When does the rocket reach its maximum height above the ground? What is its maximum height?
|
$\left[-\frac{1}{3}, 4\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left[-\frac{1}{3}, 4\right]
|
college_math.PRECALCULUS
|
exercise.3.4.19
|
Simplify the given power of $i$: $i^5$
|
$i^{5}=i^{4} \cdot i=1 \cdot i=i$
|
Creative Commons License
|
college_math.precalculus
|
i^{5}=i^{4} \cdot i=1 \cdot i=i
|
college_math.PRECALCULUS
|
exercise.6.4.34
|
Solve the equation or inequality using your calculator: $\ln \left(-2 x^{3}-x^{2}+13 x-6\right)<0$
|
$\approx(-3.0281,-3) \cup(0.5,0.5991) \cup(1.9299,2)$
|
Creative Commons License
|
college_math.precalculus
|
\approx(-3.0281,-3) \cup(0.5,0.5991) \cup(1.9299,2)
|
college_math.PRECALCULUS
|
exercise.2.2.6
|
Solve the equation: $|7 x-1|+2=0$
|
no solution
|
Creative Commons License
|
college_math.precalculus
|
no solution
|
college_math.PRECALCULUS
|
exercise.6.1.33
|
Evaluate the expression: $36^{\log _{36}(216)}$
|
$36^{\log _{36}(216)}=216$
|
Creative Commons License
|
college_math.precalculus
|
36^{\log _{36}(216)}=216
|
college_math.PRECALCULUS
|
exercise.10.7.12
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $2 \cos \left(x+\frac{7 \pi}{4}\right)=\sqrt{3}$
|
$x=-\frac{19 \pi}{12}+2 \pi k$ or $x=\frac{\pi}{12}+2 \pi k ; x=\frac{\pi}{12}, \frac{5 \pi}{12}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{19 \pi}{12}+2 \pi k$ or $x=\frac{\pi}{12}+2 \pi k ; x=\frac{\pi}{12}, \frac{5 \pi}{12}
|
college_math.PRECALCULUS
|
exercise.1.3.44
|
Determine whether or not the equation represents $y$ as a function of $x$: $x^{3}+y^{3}=4$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
college_math.PRECALCULUS
|
exercise.6.4.32
|
Solve the equation or inequality using your calculator: $\ln (x)=\sqrt[4]{x}$
|
$x \approx 4.177, x \approx 5503.665$
|
Creative Commons License
|
college_math.precalculus
|
x \approx 4.177, x \approx 5503.665
|
college_math.PRECALCULUS
|
exercise.4.3.19
|
Solve the rational inequality and express your answer using interval notation: $\frac{x^{4}-4 x^{3}+x^{2}-2 x-15}{x^{3}-4 x^{2}} \geq x$
|
$[-3,0) \cup(0,4) \cup[5, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
[-3,0) \cup(0,4) \cup[5, \infty)
|
college_math.PRECALCULUS
|
exercise.3.3.42
|
Find the real solutions of the polynomial equation $x^{4}+2 x^{2}=15$.
|
$x=-\frac{3}{2}, \pm \sqrt{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=-\frac{3}{2}, \pm \sqrt{3}
|
college_math.PRECALCULUS
|
exercise.6.2.16
|
Use the properties of logarithms to write the expression as a single logarithm: $4 \ln (x)+2 \ln (y)$
|
$\ln \left(x^{4} y^{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\ln \left(x^{4} y^{2}\right)
|
college_math.PRECALCULUS
|
exercise.6.1.43
|
Find the domain of the function: $f(x)=\ln \left(x^{2}+1\right)$
|
$(-\infty, \infty)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, \infty)
|
college_math.PRECALCULUS
|
exercise.6.4.14
|
Solve the equation analytically: $3 \ln (x)-2=1-\ln (x)$
|
$x=e^{3 / 4}$
|
Creative Commons License
|
college_math.precalculus
|
x=e^{3 / 4}
|
college_math.PRECALCULUS
|
exercise.1.1.29
|
Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(0,0),(x, y)$
|
$d=\sqrt{x^{2}+y^{2}}, M=\left(\frac{x}{2}, \frac{y}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
d=\sqrt{x^{2}+y^{2}}, M=\left(\frac{x}{2}, \frac{y}{2}\right)
|
college_math.PRECALCULUS
|
exercise.7.3.16
|
Find an equation for the parabola which fits the given criteria: Focus $(10,1)$, directrix $x=5$
|
$(y-1)^{2}=10\left(x-\frac{15}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
(y-1)^{2}=10\left(x-\frac{15}{2}\right)
|
college_math.PRECALCULUS
|
exercise.10.7.100
|
Express the domain of the function using the extended interval notation: $f(x)=\frac{\cos (x)}{\sin (x)+1}$
|
$\bigcup_{k=-\infty}^{\infty}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\bigcup_{k=-\infty}^{\infty}\left(\frac{(4 k-1) \pi}{2}, \frac{(4 k+3) \pi}{2}\right)
|
college_math.PRECALCULUS
|
exercise.10.7.45
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (2 x) \cos (x)+\sin (2 x) \sin (x)=1$
|
$x=\frac{\pi}{48}, \frac{11 \pi}{48}, \frac{13 \pi}{48}, \frac{23 \pi}{48}, \frac{25 \pi}{48}, \frac{35 \pi}{48}, \frac{37 \pi}{48}, \frac{47 \pi}{48}, \frac{49 \pi}{48}, \frac{59 \pi}{48}, \frac{61 \pi}{48}, \frac{71 \pi}{48}, \frac{73 \pi}{48}, \frac{83 \pi}{48}, \frac{85 \pi}{48}, \frac{95 \pi}{48}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{48}, \frac{11 \pi}{48}, \frac{13 \pi}{48}, \frac{23 \pi}{48}, \frac{25 \pi}{48}, \frac{35 \pi}{48}, \frac{37 \pi}{48}, \frac{47 \pi}{48}, \frac{49 \pi}{48}, \frac{59 \pi}{48}, \frac{61 \pi}{48}, \frac{71 \pi}{48}, \frac{73 \pi}{48}, \frac{83 \pi}{48}, \frac{85 \pi}{48}, \frac{95 \pi}{48}
|
college_math.PRECALCULUS
|
exercise.3.3.39
|
Find the real solutions of the polynomial equation $x^{3}-7 x^{2}=7-x$.
|
$x=7$
|
Creative Commons License
|
college_math.precalculus
|
x=7
|
college_math.PRECALCULUS
|
exercise.10.7.90
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\tan ^{2}(x) \geq 1$
|
$\left[-\frac{7 \pi}{4},-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\frac{5 \pi}{4}\right] \cup\left[-\frac{3 \pi}{4},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2},-\frac{\pi}{4}\right] \cup\left[\frac{\pi}{4}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right] \cup\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, \frac{7 \pi}{4}\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left[-\frac{7 \pi}{4},-\frac{3 \pi}{2}\right) \cup\left(-\frac{3 \pi}{2},-\frac{5 \pi}{4}\right] \cup\left[-\frac{3 \pi}{4},-\frac{\pi}{2}\right) \cup\left(-\frac{\pi}{2},-\frac{\pi}{4}\right] \cup\left[\frac{\pi}{4}, \frac{\pi}{2}\right) \cup\left(\frac{\pi}{2}, \frac{3 \pi}{4}\right] \cup\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right) \cup\left(\frac{3 \pi}{2}, \frac{7 \pi}{4}\right]
|
college_math.PRECALCULUS
|
exercise.10.7.15
|
Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\tan ^{2}(x)=3$
|
$x=\frac{\pi}{3}+\pi k$ or $x=\frac{2 \pi}{3}+\pi k ; x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{3}+\pi k$ or $x=\frac{2 \pi}{3}+\pi k ; x=\frac{\pi}{3}, \frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{5 \pi}{3}
|
college_math.PRECALCULUS
|
exercise.2.3.17
|
The temperature $T$, in degrees Fahrenheit, $t$ hours after $6 \mathrm{AM}$ is given by:
$T(t)=-\frac{1}{2} t^{2}+8 t+32, \quad 0 \leq t \leq 12$
What is the warmest temperature of the day? When does this happen?
|
$64^{\circ}$ at 2 PM (8 hours after 6 AM.)
|
Creative Commons License
|
college_math.precalculus
|
64^{\circ}$ at 2 PM (8 hours after 6 AM.)
|
college_math.PRECALCULUS
|
exercise.10.7.80
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cot (x) \leq 4$
|
$\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\frac{\pi}{6}, \frac{\pi}{6}\right)
|
college_math.PRECALCULUS
|
exercise.9.2.13
|
Rewrite the sum using summation notation: $2+\frac{3}{2}+\frac{4}{3}+\frac{5}{4}+\frac{6}{5}$
|
$\sum_{k=1}^{5} \frac{k+1}{k}$
|
Creative Commons License
|
college_math.precalculus
|
\sum_{k=1}^{5} \frac{k+1}{k}
|
college_math.PRECALCULUS
|
exercise.11.4.35
|
Convert the point from polar coordinates into rectangular coordinates: $(\pi, \arctan (\pi))$
|
$\left(\frac{\pi}{\sqrt{1+\pi^{2}}}, \frac{\pi^{2}}{\sqrt{1+\pi^{2}}}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(\frac{\pi}{\sqrt{1+\pi^{2}}}, \frac{\pi^{2}}{\sqrt{1+\pi^{2}}}\right)
|
college_math.PRECALCULUS
|
exercise.9.1.3
|
Write out the first four terms of the given sequence: $\{5 k-2\}_{k=1}^{\infty}$
|
$3,8,13,18$
|
Creative Commons License
|
college_math.precalculus
|
3,8,13,18
|
college_math.PRECALCULUS
|
exercise.6.2.22
|
Use the properties of logarithms to write the expression as a single logarithm: $-\frac{1}{3} \ln (x)-\frac{1}{3} \ln (y)+\frac{1}{3} \ln (z)$
|
$\ln \left(\sqrt[3]{\frac{z}{x y}}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\ln \left(\sqrt[3]{\frac{z}{x y}}\right)
|
college_math.PRECALCULUS
|
exercise.1.1.19
|
Write the set using interval notation: $\{x \mid-3<x<3$ or $x=4\}$
|
$(-3,3) \cup\{4\}$
|
Creative Commons License
|
college_math.precalculus
|
(-3,3) \cup\{4\}
|
college_math.PRECALCULUS
|
exercise.8.4.8
|
Find the inverse of the matrix or state that the matrix is not invertible: $H=\left[\begin{array}{rrrr}1 & 0 & -3 & 0 \\ 2 & -2 & 8 & 7 \\ -5 & 0 & 16 & 0 \\ 1 & 0 & 4 & 1\end{array}\right]$
|
$H^{-1}=\left[\begin{array}{rrrr}16 & 0 & 3 & 0 \\ -90 & -\frac{1}{2} & -\frac{35}{2} & \frac{7}{2} \\ 5 & 0 & 1 & 0 \\ -36 & 0 & -7 & 1\end{array}\right]$
|
Creative Commons License
|
college_math.precalculus
|
H^{-1}=\left[\begin{array}{rrrr}16 & 0 & 3 & 0 \\ -90 & -\frac{1}{2} & -\frac{35}{2} & \frac{7}{2} \\ 5 & 0 & 1 & 0 \\ -36 & 0 & -7 & 1\end{array}\right]
|
college_math.PRECALCULUS
|
exercise.8.7.27
|
Solve the system of nonlinear equations: $\left\{\begin{aligned} 2 y+2 z & =\lambda y z \\ 2 x+2 z & =\lambda x z \\ 2 y+2 x & =\lambda x y \\ x y z & =1000\end{aligned}\right.$
|
$x=10, y=10, z=10, \lambda=\frac{2}{5}$
|
Creative Commons License
|
college_math.precalculus
|
x=10, y=10, z=10, \lambda=\frac{2}{5}
|
college_math.PRECALCULUS
|
exercise.10.7.81
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\cos (x)>\frac{\sqrt{3}}{2}$
|
$\left[-\pi,-\frac{\pi}{2}\right) \cup\left[-\frac{\pi}{3}, \frac{\pi}{3}\right] \cup\left(\frac{\pi}{2}, \pi\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left[-\pi,-\frac{\pi}{2}\right) \cup\left[-\frac{\pi}{3}, \frac{\pi}{3}\right] \cup\left(\frac{\pi}{2}, \pi\right]
|
college_math.PRECALCULUS
|
exercise.1.3.2
|
Determine whether or not the relation represents $y$ as a function of $x$ and find the domain and range of those relations which are functions:
$\{(-3,0),(1,6),(2,-3),(4,2),(-5,6),(4,-9),(6,2)\}$
|
Not a function
|
Creative Commons License
|
college_math.precalculus
|
Not a function
|
college_math.PRECALCULUS
|
exercise.1.1.24
|
Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\left(\frac{1}{2}, 4\right),\left(\frac{3}{2},-1\right)$
|
$d=\sqrt{26}, M=\left(1, \frac{3}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
d=\sqrt{26}, M=\left(1, \frac{3}{2}\right)
|
college_math.PRECALCULUS
|
exercise.6.2.2
|
Expand the given logarithm and simplify: $\log _{2}\left(\frac{128}{x^{2}+4}\right)$
|
$7-\log _{2}\left(x^{2}+4\right)$
|
Creative Commons License
|
college_math.precalculus
|
7-\log _{2}\left(x^{2}+4\right)
|
college_math.PRECALCULUS
|
exercise.10.7.27
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $3 \cos (2 x)=\sin (x)+2$
|
$x=\arctan (2), \pi+\arctan (2)$
|
Creative Commons License
|
college_math.precalculus
|
x=\arctan (2), \pi+\arctan (2)
|
college_math.PRECALCULUS
|
exercise.8.4.3
|
Find the inverse of the matrix or state that the matrix is not invertible: $C=\left[\begin{array}{rr}6 & 15 \\ 14 & 35\end{array}\right]$
|
$C$ is not invertible
|
Creative Commons License
|
college_math.precalculus
|
C$ is not invertible
|
college_math.PRECALCULUS
|
exercise.6.1.40
|
Evaluate the expression: $\log \left(e^{\ln (100)}\right)$
|
$\log \left(e^{\ln (100)}\right)=2$
|
Creative Commons License
|
college_math.precalculus
|
\log \left(e^{\ln (100)}\right)=2
|
college_math.PRECALCULUS
|
exercise.10.7.64
|
Solve the equation: $12 \operatorname{arccsc}\left(\frac{x}{3}\right)=2 \pi$
|
$x=\frac{1}{2}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{1}{2}
|
college_math.PRECALCULUS
|
exercise.8.2.29
|
Find the quadratic function passing through the points $(-2,1),(1,4),(3,-2)$
|
$f(x)=-\frac{4}{5} x^{2}+\frac{1}{5} x+\frac{23}{5}$
|
Creative Commons License
|
college_math.precalculus
|
f(x)=-\frac{4}{5} x^{2}+\frac{1}{5} x+\frac{23}{5}
|
college_math.PRECALCULUS
|
exercise.2.1.32
|
A salesperson is paid $\$ 200$ per week plus $5 \%$ commission on her weekly sales of $x$ dollars. Find a linear function that represents her total weekly pay, $W$ (in dollars) in terms of $x$. What must her weekly sales be in order for her to earn $\$ 475.00$ for the week?
|
$W(x)=200+.05 x, x \geq 0$ She must make $\$ 5500$ in weekly sales.
|
Creative Commons License
|
college_math.precalculus
|
W(x)=200+.05 x, x \geq 0$ She must make $\$ 5500$ in weekly sales.
|
college_math.PRECALCULUS
|
exercise.10.2.54
|
Approximate the given value to three decimal places: $\cos (e)$
|
$\cos (e) \approx-0.912$
|
Creative Commons License
|
college_math.precalculus
|
\cos (e) \approx-0.912
|
college_math.PRECALCULUS
|
exercise.10.1.35
|
Convert the angle from degree measure into radian measure, giving the exact value in terms of $\pi$: $45^{\circ}$
|
$\frac{\pi}{4}$
|
Creative Commons License
|
college_math.precalculus
|
\frac{\pi}{4}
|
college_math.PRECALCULUS
|
exercise.10.7.56
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (5 x)=-\cos (2 x)$
|
$x=\frac{\pi}{7}, \frac{\pi}{3}, \frac{3 \pi}{7}, \frac{5 \pi}{7}, \pi, \frac{9 \pi}{7}, \frac{11 \pi}{7}, \frac{5 \pi}{3}, \frac{13 \pi}{7}$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{\pi}{7}, \frac{\pi}{3}, \frac{3 \pi}{7}, \frac{5 \pi}{7}, \pi, \frac{9 \pi}{7}, \frac{11 \pi}{7}, \frac{5 \pi}{3}, \frac{13 \pi}{7}
|
college_math.PRECALCULUS
|
exercise.11.4.19
|
Convert the point from polar coordinates into rectangular coordinates: $\left(11,-\frac{7 \pi}{6}\right)$
|
$\left(-\frac{11 \sqrt{3}}{2}, \frac{11}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(-\frac{11 \sqrt{3}}{2}, \frac{11}{2}\right)
|
college_math.PRECALCULUS
|
exercise.10.2.9
|
Find the exact value of the cosine and sine of the given angle: $\theta=\frac{5 \pi}{4}$
|
$\cos \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}, \sin \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}$
|
Creative Commons License
|
college_math.precalculus
|
\cos \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}, \sin \left(\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}
|
college_math.PRECALCULUS
|
exercise.1.3.46
|
Determine whether or not the equation represents $y$ as a function of $x$: $2 x y=4$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
college_math.PRECALCULUS
|
exercise.8.7.20
|
Solve the following system: $\left\{\begin{aligned} x^{2}+\sqrt{y}+\log _{2}(z) & =6 \\ 3 x^{2}-2 \sqrt{y}+2 \log _{2}(z) & =5 \\ -5 x^{2}+3 \sqrt{y}+4 \log _{2}(z) & =13\end{aligned}\right.$
|
$(1,4,8),(-1,4,8)$
|
Creative Commons License
|
college_math.precalculus
|
(1,4,8),(-1,4,8)
|
college_math.PRECALCULUS
|
exercise.4.3.5
|
Solve the rational equation: $\frac{x^{2}-2 x+1}{x^{3}+x^{2}-2 x}=1$
|
No solution
|
Creative Commons License
|
college_math.precalculus
|
No solution
|
college_math.PRECALCULUS
|
exercise.2.3.18
|
Suppose $C(x)=x^{2}-10 x+27$ represents the costs, in hundreds, to produce $x$ thousand pens. How many pens should be produced to minimize the cost? What is this minimum cost?
|
5000 pens should be produced for a cost of $\$ 200$.
|
Creative Commons License
|
college_math.precalculus
|
5000 pens should be produced for a cost of $\$ 200$.
|
college_math.PRECALCULUS
|
exercise.11.4.39
|
Convert the equation from rectangular coordinates into polar coordinates: $y=7$
|
$\left(7 \sqrt{2}, \frac{7 \pi}{4}\right)$
|
Creative Commons License
|
college_math.precalculus
|
\left(7 \sqrt{2}, \frac{7 \pi}{4}\right)
|
college_math.PRECALCULUS
|
exercise.6.1.38
|
Evaluate the expression: $\ln \left(\frac{1}{\sqrt{e}}\right)$
|
$\ln \left(\frac{1}{\sqrt{e}}\right)=-\frac{1}{2}$
|
Creative Commons License
|
college_math.precalculus
|
\ln \left(\frac{1}{\sqrt{e}}\right)=-\frac{1}{2}
|
college_math.PRECALCULUS
|
exercise.1.3.33
|
Determine whether or not the equation represents $y$ as a function of $x$: $y=x^{3}-x$
|
Function
|
Creative Commons License
|
college_math.precalculus
|
Function
|
college_math.PRECALCULUS
|
exercise.6.1.18
|
Evaluate the expression: $\log _{2}(32)$
|
$\log _{2}(32)=5$
|
Creative Commons License
|
college_math.precalculus
|
\log _{2}(32)=5
|
college_math.PRECALCULUS
|
exercise.6.4.5
|
Solve the equation analytically: $\log _{3}(7-2 x)=2$
|
$x=-1$
|
Creative Commons License
|
college_math.precalculus
|
x=-1
|
college_math.PRECALCULUS
|
exercise.10.7.43
|
Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (6 x) \cos (x)=-\cos (6 x) \sin (x)$
|
$x=0, \frac{\pi}{7}, \frac{2 \pi}{7}, \frac{3 \pi}{7}, \frac{4 \pi}{7}, \frac{5 \pi}{7}, \frac{6 \pi}{7}, \pi, \frac{8 \pi}{7}, \frac{9 \pi}{7}, \frac{10 \pi}{7}, \frac{11 \pi}{7}, \frac{12 \pi}{7}, \frac{13 \pi}{7}$
|
Creative Commons License
|
college_math.precalculus
|
x=0, \frac{\pi}{7}, \frac{2 \pi}{7}, \frac{3 \pi}{7}, \frac{4 \pi}{7}, \frac{5 \pi}{7}, \frac{6 \pi}{7}, \pi, \frac{8 \pi}{7}, \frac{9 \pi}{7}, \frac{10 \pi}{7}, \frac{11 \pi}{7}, \frac{12 \pi}{7}, \frac{13 \pi}{7}
|
college_math.PRECALCULUS
|
exercise.10.7.72
|
Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cos ^{2}(x)>\frac{1}{2}$
|
$\left[0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right) \cup\left(\frac{7 \pi}{4}, 2 \pi\right]$
|
Creative Commons License
|
college_math.precalculus
|
\left[0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right) \cup\left(\frac{7 \pi}{4}, 2 \pi\right]
|
college_math.PRECALCULUS
|
exercise.6.3.16
|
Solve the equation analytically: $500\left(1-e^{2 x}\right)=250$
|
$x=\frac{1}{2} \ln \left(\frac{1}{2}\right)=-\frac{1}{2} \ln (2)$
|
Creative Commons License
|
college_math.precalculus
|
x=\frac{1}{2} \ln \left(\frac{1}{2}\right)=-\frac{1}{2} \ln (2)
|
college_math.PRECALCULUS
|
exercise.2.4.33
|
Solve the quadratic equation $x^{2}-m x=1$ for $x$.
|
$(-\infty, 1) \cup\left(2, \frac{3+\sqrt{17}}{2}\right)$
|
Creative Commons License
|
college_math.precalculus
|
(-\infty, 1) \cup\left(2, \frac{3+\sqrt{17}}{2}\right)
|
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