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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 fun... | $C(p)=\left\{\begin{array}{rll}6 p+1.5 & \text { if } & 1 \leq p \leq 5 \\ 5.5 p & \text { if } & p \geq 6\end{array}\right.$ | 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}, \fra... | 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... |
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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