Math 2 Exam II Practice

These are a collection of problems you can use to practice for the exam. To see the answers go to this page.

P-level

1. Take $f$ to be an invertible function with a domain of $(-2,9]$ and a range of $(2,\mathrm{\infty })$. Determine the domain and range of $f^{-1}$.

2. The point $(-1,5)$ is in an invertible function $f$. Determine a point on $f^{-1}.$

3. Calculate $(\frac{1}{2},\frac{\sqrt{3}}{2})\star (1,3).$

4. Take $p=(\frac{1}{5},-\frac{\sqrt{24}}{5}).$ Determine $p^{-1}.$

5. Determine the fraction of a circle of the angle that is $\pi$ radians.

6. Take $\theta$ to be an angle on the unit circle with coordinates $(\frac{1}{5},-\frac{\sqrt{24}}{5})$. Calculate $\sin (\theta )$, $\cos (\theta )$ and $\tan (\theta ).$

7. Take $f$ to be the function given by $f(x)=-4(x-1{)}^2+4.$ Determine the vertex of $f$ and the maximum and minimum value of $f.$

8. Take $f$ to be the function given by $f(x)=2(x+3{)}^2(x-3{)}^5(x+1{)}^8.$ Determine the zeros of $f$ and the orders of each zero.

9. Take $f$ to be the function given by $f(x)={\displaystyle \frac{(x-2{)}^2(x+5)}{x(x-9{)}^3}}.$ Determine the zeros and poles of $f$ and the orders of each zero and pole.

10. Take $f$ to be the function given by $f(x)={\displaystyle \frac{(x-2{)}^2(x+5)}{x(x-9{)}^3}}.$ Determine the asymptotic behavior of $f$.

11. Take $f$ to be a function with fundamental period $11$ and $g$ to be the function $g(x)=5x+1$. Determine the fundamental period of the function $f\circ g$.

12. Take $f$ to be an odd function. Suppose that $(2,5)$ and $(-1,8)$ are points on $f$. Determine $f(-2)$ and $f(1)$.

13. Take $f$ to be an even function. Suppose that $f(5)=-1$ and $f(-3)=2$. Determine $f(-5)$ and $f(3)$.

14. Determine the fundamental period of each trigonometric function.

    1. $f(x)=\sin (-x)$
    2. $f(x)=\cos (3x)$
    3. $f(x)=\tan (\frac{3}{2}x-1)$

15. Determine the horizontal and vertical asymptotes of each function.

    1. $f(x)=3^x+4$
    2. $f(x)=-{\mathrm{e}}^{x-2}$
    3. $f(x)=-{\left(\frac{1}{2}\right)}^x-2$
    4. $f(x)=\ln (x)+2$
    5. $f(x)={\log }_3(x-2)$
    6. $f(x)=2{\log }_{\frac{1}{2}}(-x+5)$

16. Take $P$ to be a polynomial and $L$ to be the line that is tangent to $P$ at $(-2,5).$ The line $L$ is given by the equation $$L(x)=5(x-x_0)+y_0.$$ Determine $x_0$ and $y_0$.

17. Take $P$ to be a polynomial and $L$ to be the line that is tangent to $P$ at $(1,6).$ The line $L$ is given by the equation $$L(x)=\frac{3}{7}(x-1)+6.$$ Determine $P^{\prime }(1).$

18. Take $f$ to be an invertible function and $L$ to be the line that is tangent to $f$ at $(1,7)$. The line $L$ is given by $$L(x)=\frac{2}{11}(x-1)+7.$$ Determine the line that is tangent to $f^{-1}$ at $(7,1)$.

19. Take $f$ to be a function and $L$ to be the line that is tangent to $f$ at $(2,3)$. The line $L$ is given by $$L(x)=4(x-2)+3.$$ Determine the line that is tangent to $g$ at $(0,4)$ where $g$ is given by $$g(x)=f(x+2)+1.$$

C-level

1. Take $f$ to be the function given by $f(x)=(x+2{)}^3+1$. Find the inverse of $f$ and its domain and range.

2. Identify a point $r$ on the unit circle so that $(\frac{1}{2},\frac{\sqrt{3}}{2})\star r=(\frac{1}{5},-\frac{\sqrt{24}}{5}).$

3. Rotate the point $(1,3)$ about the point $(-3,1)$ by the angle $(\frac{1}{5},-\frac{\sqrt{24}}{5}).$

4. Take $\theta$ to be an angle on the unit circle with coordinates $(\frac{5}{7},y)$ and in quadrant IV. Determine $\sin (\theta )$, $\cos (\theta )$, $\tan (\theta )$, $\csc (\theta )$, $\sec (\theta )$ and $\cot (\theta )$.

5. Sketch the function $f$ given by $f(x)=2(x+3{)}^2(x-3{)}^5(x+1{)}^8.$

6. Sketch the function $f$ given by $f(x)={\displaystyle \frac{(x-2{)}^2(x+5)}{x(x-9{)}^3}}.$

7. Take $f$ to be an odd function whose graph is shown below on the restriction $[0,5]$. Sketch $f$ on the restriction $[-5,0].$ In other words, “complete” the graph.

8. Take $A$ and $B$ to be two angles in quadrant I with $\sin (A)=\frac{1}{5}$ and $\cos (B)=\frac{3}{7}$. Calculate the following.

   1. $\sin (A+B)$
   2. $\cos (A-B)$

9. Sketch the function $f$ by using transformations on the given function $g$. Indicate what the transformations are and the order of the transformations. Include also the domain and range and any asymptotes.

   1. $f(x)=3^x+4$, $g(x)=3^x$
   2. $f(x)=-e^{x-2}$, $g(x)=e^x$
   3. $f(x)=-{\left(\frac{1}{2}\right)}^x-2$, $g(x)={\left(\frac{1}{2}\right)}^x$
   4. $f(x)=-\ln (x)+2$, $g(x)=\ln (x)$
   5. $f(x)={\log }_3(x-2)$, $g(x)={\log }_3(x)$
   6. $f(x)=2{\log }_{\frac{1}{2}}(-x+5)$, $g(x)={\log }_{\frac{1}{2}}(x).$

10. Take $C$, $D$ and $b$ to be positive real numbers so that ${\log }_b(C)=11$ and ${\log }_b(D)=-3.$ Calculate the following.

    1. ${\log }_b\left(bC{D}^2\right)$
    2. ${\log }_b\left(\frac{D^3}{b^{4}C}\right)$

11. A quantity $A$ changes according to an exponential model for change and $$\{\begin{array}{l}A(2)=4\\ A(5)=14.\end{array}$$ Identify a formula for $A(t).$

12. Take $P$ and $L$ to be given by $$P(x)=x^2+4x+1\text{and}L(x)=mx+b.$$

    1. Identify a quadratic equation that determines $m$ so that $L$ is tangent to $P$ at $(2,13)$.
    2. Use part a to determine the equation of the line tangent to $P$ at $(2,13).$ You must use part a in order to receive credit.

13. Determine the equation of the line that is tangent to the circle $C$ at the point $(2,5)$, where $C$ is the circle given by $$(x-1{)}^2+(y+3{)}^2=65.$$

14. Take $f$ to be a function and $L$ to be the line that is tangent to $f$ at $(2,3)$. The line $L$ is given by $$L(x)=4(x-2)+3.$$ Determine the line that is tangent to $g$ at $(\frac{1}{3},4)$ where $g$ is given by $$g(x)=f(3x+1)+1.$$

B-Level

1. Take $Q$ to be the rational function that is given by $$Q(x)=\frac{x^2(x+3)(x-4{)}^3(x+4{)}^4}{(x-3{)}^2(x-5{)}^3}.$$
   1. Sketch $Q$.
   2. Use the sketch of $Q$ to determine all solutions to the inequality $Q(x)\ge 0.$
   3. Use the sketch of $Q$ to determine all solutions to the inequality $Q(x)>0.$
   4. Use the sketch of $Q$ to determine all solutions to the inequality $Q(x)\le 0.$
   5. Use the sketch of $Q$ to determine all solutions to the inequality $Q(x)<0.$
2. A mass of bacteria experiences exponential growth. At time $2$ an experimenter has $20$ grams of bacteria. At time $10$, the mass has grown to a mass of $45$.
   1. Determine the doubling time of the bacteria.
   2. Determine the growth rate of the bacteria.
3. Take $A$ to be a particle that moves on the line segment with endpoints $(1,2)$ and $(5,9)$. It is at $(1,2)$ at time $0$ and it moves along the line segment at unit speed. Take $B$ to be a particle that rotates around $A$ in a counter-clockwise direction as $A$ travels. It is a distance of $3$ to the right of $A$ at time $0$ and has a speed of $6$.
   1. Write down equations that model the motion for $A$ and $B$.
   2. Determine how long it takes for $A$ to reach $(5,9)$.
   3. Determine how many full orbits $B$ has made by the time $A$ reached $(5,9)$.

A-level

1. Take $f$ to be a function given by $f(x)=x^2$ defined on $\mathcal{D}(f)=(-4,-3]\cup (-2,-1)\cup [0,1].$

   1. Sketch $f$ and $f^{-1}$.
   2. Write a formula for $f^{-1}$.

2. Determine the equation of the line that is tangent to the ellipse $E$ at the point $(2,5)$, where $E$ is the ellipse given by $$(x-1{)}^2+\frac{(y+3{)}^2}{8}=9.$$

3. Take $f$ to be the function whose graph is given below. Use $y$-axis inversion to graph the function $g$ that is given by $$g(x)=\frac{1}{f(x)}.$$

4. Take $C$ to be a circle and $p$ to be a point on $C$. A graph of $C$ and $p$ is given below. Use the graph to determine the equation of the line that is tangent to $C$ at $p$.

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