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 \((2,\infty)\). 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 \(\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\star (1,3).\)

  4. Take \(p=\left(\frac{1}{5},-\frac{\sqrt{24}}{5}\right).\) 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 \(\left(\frac{1}{5},-\frac{\sqrt{24}}{5}\right)\). 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 \(f(x)=\dfrac{(x-2)^2(x+5)}{x(x-9)^3}.\) Determine the zeros and poles of \(f\) and the orders of each zero and pole.

  10. Take \(f\) to be the function given by \(f(x)=\dfrac{(x-2)^2(x+5)}{x(x-9)^3}.\) 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. \(f(x)=\tan\left(\frac{3}{2}x-1\right)\)

  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 \[L(x)=\frac{3}{7}\left(x-1\right)+6.\] Determine \(P'(1).\)

  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 \(\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\star r=\left(\frac{1}{5},-\frac{\sqrt{24}}{5}\right).\)

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

  4. Take \(\theta\) to be an angle on the unit circle with coordinates \(\left(\frac{5}{7},y\right)\) 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 \(f(x)=\dfrac{(x-2)^2(x+5)}{x(x-9)^3}.\)

  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.

  1. 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. \(\cos(A-B)\)

  2. 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. \(f(x)=2\log_\frac{1}{2}(-x+5)\), \(g(x)=\log_\frac{1}{2}(x).\)

  3. 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. \(\log_b\left(bCD^2\right)\)
    2. \(\log_b\left(\frac{D^3}{b^4C}\right)\)

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

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

    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.

  6. Determine the equation of the line that is tangent to the circle \(C\) at the point \(\left(2,5\right)\), where \(C\) is the circle given by \[(x-1)^2+(y+3)^2=65.\]

  7. 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 \(\left(\tfrac{1}{3},4\right)\) 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 \(Q(x)\geq 0.\)
    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 \(Q(x)\leq 0.\)
    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 \(\left(2,5\right)\), 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 \[g(x)=\frac{1}{f(x)}.\]

  1. 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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