Knowledge Check: Chapters 7–12

Simplify$\sec x\cos x+\cos x-\frac{1}{\sec x}$.

Simplify$\tan^3 x-\tan x\sec^2 x$.

Find the exact value of$\tan\left(\frac{7\pi}{12}\right)$.

Find the exact value of$\sin(70^\circ)\cos(25^\circ)-\cos(70^\circ)\sin(25^\circ)$.

Find the exact value of$\tan\left(\sin^{-1}(0)+\sin^{-1}\left(\frac12\right)\right)$.

Given$\sec\theta=-\frac{5}{3}$with$\theta$in the interval$\left[\frac{\pi}{2},\pi\right]$, find$\sin(2\theta)$,$\cos(2\theta)$, and$\tan(2\theta)$, in that order, separated by commas.

Find the exact value of$\sec\left(\frac{3\pi}{8}\right)$.

In a right triangle, one acute angle is$\alpha$, the leg opposite$\alpha$has length$24$, the other leg has length$7$, and the hypotenuse has length$25$. Find$\tan\left(\frac{\alpha}{2}\right)$using a half-angle formula.

Evaluate$2\sin\left(\frac{2\pi}{3}\right)\sin\left(\frac{5\pi}{6}\right)$using a product-to-sum formula.

Use a product-to-sum formula to write$\sin(9x)\cos(3x)$as a sum.

Solve$2\sin\theta=-1$for all exact solutions on the interval$[0,2\pi)$. Enter both solutions, separated by a comma.

Solve$\csc^2x-3\csc x-4=0$on the interval$[0,2\pi)$. Round each solution to four decimal places, separated by commas.

A man’s eye level is$6$feet above the ground. He stands$3$feet from the base of a$15$-foot vertical ladder and looks up at the top of the ladder. Round to the nearest tenth of a degree: at what angle above horizontal is he looking?

Find a formula of the form$y=A\sin(Bx)+D$that fits the data in the table above.

For$y=3\cos(x\pi)$, find the amplitude, then the period, then the frequency (in Hz), in that order, separated by commas.

A population of lemmings varies periodically over the year, with a yearly low of$500$in March and an average yearly population of$950$. Modeling$t$as the month number (so$t=3$is March), write a function$P(t)$that models the population.

In the triangle above, find$c$, to the nearest tenth.

A pilot flying over a straight highway determines the angle of depression to milepost$A$to be$49^\circ$and to milepost$B$,$2.1$km farther along the highway from$A$, to be$25^\circ$. Find the distance of the plane from point$A$, to the nearest tenth, in kilometers.

In the triangle above, find$a$, to the nearest tenth.

To find the distance between two cities, a satellite calculates that one city is$210$km away, the other is$250$km away, and the angle at the satellite between the two lines of sight is$1.8^\circ$. Find the distance between the cities, to the nearest tenth, in kilometers.

Assume$\alpha$is opposite side$a$,$\beta$is opposite side$b$, and$\gamma$is opposite side$c$. Solve the triangle, given$\beta=68^\circ$,$b=21$,$c=16$. Find$a$, to the nearest tenth.

Convert the polar coordinates$\left(-2,\frac{3\pi}{2}\right)$to rectangular coordinates, as an ordered pair.

Convert the Cartesian equation$x^2+y^2=64$to a polar equation.

Convert$(-9,-4)$to polar coordinates, as an ordered pair$(r,\theta)$with$r>0$and$\theta$in degrees,$0^\circ\le\theta<360^\circ$. Round$r$to four decimal places and$\theta$to two decimal places, writing$\theta$with its degree symbol.

Find the absolute value of the complex number$4-3i$.

Convert$z=3\left(\cos40^\circ+i\sin40^\circ\right)$to rectangular form, rounding each part to the nearest tenth.

Find the square roots of$z=25\left(\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}\right)$. Enter both roots, each in the form$r(\cos\theta+i\sin\theta)$with$0\le\theta<2\pi$, separated by a comma.

Eliminate the parameter$t$to write a Cartesian equation for$x(t)=-\cos t$,$y(t)=2\sin^2 t$.

Parameterize the line from$(-2,3)$to$(4,7)$so that the line is at$(-2,3)$when$t=0$and at$(4,7)$when$t=1$. Enter the expression for$x(t)$, then the expression for$y(t)$, separated by a comma.

Eliminate the parameter$t$to write a Cartesian equation for$x(t)=e^t$,$y(t)=-2e^{5t}$.

The parametric equations$x(t)=-2\sin t$,$y(t)=5\cos t$trace an ellipse. Eliminate the parameter to write the Cartesian equation, in standard form.

A ball is launched with an initial velocity of$80$feet per second at an angle of$40^\circ$to the horizontal, and is released at a height of$4$feet above the ground. How long is the ball in the air, to the nearest tenth of a second?

For$\mathbf{u}=\mathbf{i}+4\mathbf{j}$and$\mathbf{v}=4\mathbf{i}+3\mathbf{j}$, calculate$\mathbf{u}\cdot\mathbf{v}$.

Find a unit vector in the same direction as$\mathbf{b}=-3\mathbf{i}-\mathbf{j}$. Give your exact answer, using a radical rather than a rounded decimal.

Solve the system by substitution, as an ordered pair$(x,y)$.$\begin{cases}10x+5y=-5\\3x-2y=-12\end{cases}$

A factory has a cost of production$C(x)=150x+15{,}000$and a revenue function$R(x)=200x$. What is the break-even point, as an ordered pair$(x,C(x))$?

Solve the system$\begin{cases}3x+2y-z=-10\\x-y+2z=7\\-x+3y+z=-2\end{cases}$, as an ordered triple$(x,y,z)$.

Solve the system$\begin{cases}2x-3y+z=0\\2x+4y-3z=0\\6x-2y-z=0\end{cases}$. Write the solution as an ordered triple$(x,y,z)$in terms of$x$.

Solve the system$\begin{cases}y=x^2-7\\y=5x-13\end{cases}$. Enter both solutions, as ordered pairs separated by a comma.

Solve the system$\begin{cases}y^2+x^2=25\\y^2-2x^2=1\end{cases}$. Enter all four solutions, as ordered pairs separated by commas.

Find the partial fraction decomposition of$\tfrac{7x+20}{x^2+10x+25}$.

Find the partial fraction decomposition of$\tfrac{-x^2+36x+70}{x^3-125}$.

Find the partial fraction decomposition of$\tfrac{x^3-4x^2+3x+11}{(x^2-2)^2}$.

Given$A=\begin{bmatrix}4&-2\\1&3\end{bmatrix}$, find$-4A$. Enter the second row as a comma-separated list of two numbers, left to right.

Given$C=\begin{bmatrix}6&7\\11&-2\\14&0\end{bmatrix}$and$B=\begin{bmatrix}6&7&-3\\11&-2&4\end{bmatrix}$, find$CB$. Enter the second row of$CB$as a comma-separated list of three numbers, left to right.

Gaussian elimination reduces a system’s augmented matrix to$\left[\begin{array}{ccc|c}1&0&-3&7\\0&1&2&-5\\0&0&0&0\end{array}\right]$, so the system has infinitely many solutions. Write$x$in terms of$z$.

Find$M^{-1}$for$M=\begin{bmatrix}-0.2&1.4\\1.2&-0.4\end{bmatrix}$. Enter the first row of$M^{-1}$as a comma-separated list of two numbers, left to right.

Use the inverse of a matrix to solve$\begin{cases}4x+3y-3z=-4.3\\5x-4y-z=-6.1\\x+z=-0.7\end{cases}$, as an ordered triple$(x,y,z)$.

Evaluate the determinant$\begin{vmatrix}100&0\\0&0\end{vmatrix}$.

Evaluate the determinant$\begin{vmatrix}-1&4&3\\0&2&3\\0&0&-3\end{vmatrix}$.

Use Cramer’s Rule to solve the system$\begin{cases}4x-2y=23\\-5x-10y=-35\end{cases}$, as an ordered pair$(x,y)$.

For the ellipse$\tfrac{x^2}{25}+\tfrac{y^2}{64}=1$, find the coordinates of the two foci. Enter them separated by a comma.

Write the standard form equation of the ellipse with center$(0,0)$, focus$(3,0)$, and vertex$(-5,0)$.

A whispering gallery is to be constructed so that the foci are located$35$feet from the center. If the length of the gallery is to be$100$feet, what should the height of the ceiling be, to the nearest hundredth of a foot?

For the hyperbola$\tfrac{(y+1)^2}{16}-\tfrac{(x-4)^2}{36}=1$, find the coordinates of the two vertices. Enter them separated by a comma.

For the hyperbola$\tfrac{x^2}{49}-\tfrac{y^2}{81}=1$, write the two asymptotes as equations, separated by a comma.

Write the standard form equation of the hyperbola with foci$(3,7)$and$(7,7)$, and vertex$(6,7)$.

For the parabola$(x+2)^2=\tfrac12(y-1)$, find the coordinates of the focus.

A parabola has focus$\left(2,\tfrac98\right)$and directrix$y=\tfrac78$. Write its equation in standard form.

A searchlight is shaped like a paraboloid of revolution. The light source is located$1.5$feet from the base along the axis of symmetry, and the depth of the searchlight is$3$feet. What should the width of the opening be, to the nearest hundredth of a foot?

The equation$x^2-xy+y^2-6=0$contains an$xy$-term. Through what acute angle$\theta$should the axes be rotated to eliminate it?

Using that same rotation, write the resulting equation, relative to the$x'y'$-system, in standard form.

For the polar equation$r=\tfrac{1}{4+3\sin\theta}$, what is the eccentricity, as a fraction?

Find a polar equation of the conic with focus at the origin, eccentricity$e=1$, and directrix$x=3$.

Write the first four terms of the sequence defined by the recursive formula$a_1=2,\ a_n=a_{n-1}+n$. Enter the first four terms, in order, separated by commas.

Write the first four terms of the sequence defined by the explicit formula$a_n=10^n+3$. Enter the first four terms, in order, separated by commas.

Find the common difference of that same sequence,$\tfrac{4}{7},\tfrac{47}{21},\tfrac{82}{21},\tfrac{39}{7},\ldots$, as a fraction.

An arithmetic sequence has the first term$a_1=18$and common difference$d=-8$. What are the first five terms? Enter them in order, separated by commas.

Find the common ratio of that same sequence,$-2,-1,-\tfrac12,-\tfrac14,\ldots$, as a fraction.

What are the first five terms of the geometric sequence$a_1=3,\ a_n=4\cdot a_{n-1}$? Enter them in order, separated by commas.

Write an explicit formula for the geometric sequence$-\tfrac15,-\tfrac{1}{15},-\tfrac{1}{45},-\tfrac{1}{135},\ldots$

Use summation notation to write the sum of the terms$\tfrac12m+5$from$m=0$to$m=5$.

Use the formula for the sum of the first$n$terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series$2.5,4,5.5,\ldots$

Find the sum of the infinite geometric series$\sum_{k=1}^{\infty}45\cdot\left(-\tfrac13\right)^{k-1}$, as a fraction.

Alejandro deposits $80 of his monthly earnings into an annuity that earns$6.25\%$annual interest, compounded monthly. To the nearest cent, how much money will he have saved after$5$years? Enter the number only, without a dollar sign.

How many ways are there to choose a number from the set$\{-10,-6,4,10,12,18,24,32\}$that is divisible by either$4$or$6$?

Calculate$P(18,4)$.

Calculate$C(15,6)$.

How many distinct ways can the word DEADWOOD be arranged?

Evaluate the binomial coefficient$\binom{23}{8}$.

Use the Binomial Theorem to write the first three terms of$(2a+b)^{17}$.

Find the seventh term of$\left(x^2-\tfrac12\right)^{13}$without fully expanding the binomial.

Suppose two six-sided dice are rolled. What is the probability of rolling a pair (both dice showing the same number)? Enter your answer as a fraction.

Using the same experiment (two six-sided dice rolled), what is the probability that a roll includes neither a$5$nor a$6$on either die? Enter your answer as a fraction.

A bowl of candy holds$16$peppermint,$14$butterscotch, and$10$strawberry flavored candies. Suppose a person grabs a handful of$7$candies. To the nearest tenth of a percent, what is the percent chance that exactly$3$are butterscotch? Express your answer as a percentage.

For the function$f$graphed above, find$\lim_{x\to-1^+}f(x)$.

Given$\lim_{x\to c}f(x)=-3$and$\lim_{x\to c}g(x)=5$, find$\lim_{x\to c}\tfrac{f(x)}{g(x)}$, as a fraction.

Find$\lim_{x\to25}\left(\tfrac{x^2-625}{\sqrt{x}-5}\right)$.

Find$\lim_{x\to4}\left(\tfrac{7-\sqrt{12x+1}}{x-4}\right)$, as a fraction.

Find all$x$-values at which$f(x)=\tfrac{x^3-125}{2x^2-12x+10}$is discontinuous. Enter only the numbers, separated by a comma.

Use the definition of the derivative to find$f'(x)$for$f(x)=4x^2-7$.

Use the definition of the derivative to find$f'(x)$for$f(x)=\tfrac{1}{x+2}$.

Use the definition of the derivative to find$f'(x)$for$f(x)=\sqrt{x-1}$.

Find an equation of the tangent line to the graph of$f(x)=3x^2-2x-6$at$x=-2$. Write the equation in slope-intercept form.