Fall 24 Final

Part I: Multiple Choice

  1. Find the derivative of \(\displaystyle y = \ln(5 + \arcsin(3x))\).

    1. \(\displaystyle y' = \frac{\sqrt{1-9x^2}}{2}\)
    2. \(\displaystyle y' = \frac{3}{(5 + \arcsin(3x))\sqrt{1-9x^2}}\)
    3. \(\displaystyle y' = \frac{3}{5 + \sqrt{1-9x^2}}\)
    4. \(\displaystyle y' = \frac{3}{5 + \sqrt{9x^2 - 1}}\)
    5. \(\displaystyle y' = 3\left(5+\sqrt{9x^2-1}\right)\)

  2. In which of the following intervals is the function \(\displaystyle f(x) = \frac{1}{2}x^4 - x^3 + 5x + 1\) concave down?

    1. \(\displaystyle (-\infty, \infty)\)
    2. \(\displaystyle (-\infty, 0)\)
    3. \(\displaystyle (0, \frac{1}{2})\)
    4. \(\displaystyle (0, 1)\)
    5. \(\displaystyle (1, \infty)\)

  3. The graph of \(\displaystyle y = \frac{x+2}{x^2 - 3x - 10}\) has

    1. vertical asymptotes at \(\displaystyle x = -2\) and \(\displaystyle x = 5\).
    2. a vertical asymptote at \(\displaystyle x = 2\) and a removable discontinuity at \(\displaystyle x = -5\).
    3. a vertical asymptote at \(\displaystyle x = -2\) and a removable discontinuity at \(\displaystyle x = 5\).
    4. a vertical asymptote at \(\displaystyle x = 5\) and a removable discontinuity at \(\displaystyle x = -2\).
    5. removable discontinuities at \(\displaystyle x = -2\) and \(\displaystyle x = 5\).

  4. The largest interval over which the function \(\displaystyle f(x) = \sqrt{12x + 12} - x\) is increasing is \(\displaystyle (-1, a)\). What is \(\displaystyle a\)?

    1. \(\displaystyle -\frac{1}{2}\)
    2. \(\displaystyle 0\)
    3. \(\displaystyle 1\)
    4. \(\displaystyle \frac{3}{2}\)
    5. \(\displaystyle 2\)
    6. \(\displaystyle +\infty\)

  5. Which of these lines is parallel to the line tangent to the graph of \(\displaystyle f(x) = \ln(\pi) + xe^{2x}\) at the point \(\displaystyle x = -1\)?

    1. \(\displaystyle y = -x + \pi\)
    2. \(\displaystyle y = \ln(\pi)\)
    3. \(\displaystyle y = -\frac{x}{e^2} + 2\ln(\pi)\)
    4. \(\displaystyle y = 3e^{2}x + \ln(\pi)\)
    5. \(\displaystyle\) None of those.

  6. Consider the two functions defined below:

    \[\displaystyle f(x) = \frac{1}{1+e^{x}} g(x) = \frac{5x^{7} + 4x^{3} + 9}{4x^{7} - 1}.\]

    Which of the following describes the number of horizontal asymptotes of the graph of each function?

    1. The graph of \(\displaystyle f(x)\) contains \textbf{two horizontal asymptotes}, while the graph of \(\displaystyle g(x)\) contains \textbf{one horizontal asymptote}.
    2. The graph of \(\displaystyle f(x)\) contains \textbf{one horizontal asymptote}, while the graph of \(\displaystyle g(x)\) contains \textbf{two horizontal asymptotes}.
    3. The graphs of both \(\displaystyle f(x)\) and \(\displaystyle g(x)\) contain exactly \textbf{one horizontal asymptote}.
    4. The graphs of both \(\displaystyle f(x)\) and \(\displaystyle g(x)\) contain exactly \textbf{zero horizontal asymptote}.
    5. None of the above are correct.

  7. Given the function \(\displaystyle f(x)\) defined as

    \[\displaystyle f(x) = \begin{cases} \ln(x + k), & \text{if } x \geq 0, \\ \frac{\sin(2x)}{x}, & \text{if } x < 0 \end{cases}\]

    find the value of \(\displaystyle k\) such that \(\displaystyle f(x)\) is continuous for all \(\displaystyle x\).

    \(\displaystyle \bigcirc\) A 1

    \(\displaystyle \bigcirc\) B \(\displaystyle e\)

    \(\displaystyle \bigcirc\) C \(\displaystyle e^2\)

    \(\displaystyle \bigcirc\) D 2

    \(\displaystyle \bigcirc\) E \(\displaystyle -\infty\)

  8. Find the following limit provided that \(\displaystyle \lim_{x \to (-2)^-} f(x) = 0\) and \(\displaystyle \lim_{x \to (-2)^-} f'(x) = 2\).

    \[\displaystyle \lim_{x \to (-2)^-} \frac{\sin(\pi x)}{f(x)}\]

    \(\displaystyle \bigcirc\) A \(\displaystyle \frac{1}{2}\)

    \(\displaystyle \bigcirc\) B \(\displaystyle \frac{\pi}{2}\)

    \(\displaystyle \bigcirc\) C \(\displaystyle -\frac{\pi}{2}\)

    \(\displaystyle \bigcirc\) D None of the above, but the limit is well defined.

    \(\displaystyle \bigcirc\) E The limit does not exist.

  9. Let \(\displaystyle f(x) = x^{4/3} - 24x^{1/3}\). Which of the following best describes the critical points of \(\displaystyle f\)? Show your work in computing and labeling the critical points of \(\displaystyle f\).

    1. The graph of \(\displaystyle f(x)\) attains a local maximum at \(\displaystyle x = 0\) and a local minimum at \(\displaystyle x = 3\).
    2. The graph of \(\displaystyle f(x)\) attains a local minimum at \(\displaystyle x = 0\) and a local maximum at \(\displaystyle x = 3\).
    3. The graph of \(\displaystyle f(x)\) attains a local minimum at \(\displaystyle x = 0\) and a local minimum at \(\displaystyle x = 6\).
    4. The graph of \(\displaystyle f(x)\) attains a local maximum at \(\displaystyle x = 0\) and a local minimum at \(\displaystyle x = 6\).
    5. None of the above are correct.

  10. Determine the absolute maximum and minimum values of \(\displaystyle f(x) = \frac{\ln(x)}{x}\) on \(\displaystyle [1, e^4]\).

    1. This function does not attain a maximum value but has minimum value \(\displaystyle \frac{1}{e}\).
    2. This function has maximum value \(\displaystyle \frac{1}{e}\) and minimum value of \(\displaystyle \frac{4}{e^4}\).
    3. This function has maximum value \(\displaystyle \frac{1}{e}\) and minimum value 0.
    4. This function has maximum value 1 and minimum value \(\displaystyle e\).
    5. This function does not attain a maximum or minimum value.

  11. A car’s position along a straight road is modeled by the equation:

    \[\displaystyle e^{z} + \int_{3}^{z} v(t) dt = z^{3} + 5z\]

    where \(\displaystyle v(t)\) is the velocity of the car at time \(\displaystyle t\). Find the value of \(\displaystyle v(0)\).

    1. 0
    2. 1
    3. 3
    4. 4
    5. 5

  12. Lisa has plotted the graphs of \(\displaystyle f\), \(\displaystyle f'\) and \(\displaystyle f''\). Unfortunately, she accidentally lost track of what happens and labelled the functions \(\displaystyle a\), \(\displaystyle b\) and \(\displaystyle c\) in some random order. Which of these functions correspond to which? [You do not need to show any work.]

    The functions \(\displaystyle (a, b, c)\) correspond to ...

    1. \(\displaystyle (f, f', f'')\)
    2. \(\displaystyle (f, f'', f')\)
    3. \(\displaystyle (f', f, f'')\)
    4. \(\displaystyle (f', f'', f)\)
    5. \(\displaystyle (f'', f, f')\)
    6. \(\displaystyle (f'', f', f)\)

    IMAGE

    Referenced image used in the problem

  13. What is the area of the plane region bounded by the curves \(\displaystyle y = 2x^2\) and \(\displaystyle y = 12 - x^2\). Draw the graphs, shade the region, and show your work.

    1. 32
    2. 16
    3. 8
    4. 0
    5. None of the above

  14. Given the graph of \(\displaystyle f\) and \(\displaystyle g\) below, determine:

    \[\displaystyle (f \circ g)'(2) + (f g)'(-1)\]
    1. 0
    2. \(\displaystyle -1/2\)
    3. \(\displaystyle 1/2\)
    4. 1
    5. \(\displaystyle 3/2\)

    Graph of \(\displaystyle f(x)\) and \(\displaystyle g(x)\) is shown below the question.

    IMAGE

    Referenced image used in the problem

  15. Evaluate \(\displaystyle \int_{-4}^3 6x \sqrt{25 - x^2} dx\).

    1. \(\displaystyle -182\)
    2. \(\displaystyle -74\)
    3. \(\displaystyle 0\)
    4. \(\displaystyle 144\)
    5. \(\displaystyle 288\)

  16. Suppose \(\displaystyle h(x)\) is an even function for which \(\displaystyle \int_0^3 h(x) dx = 5\) and \(\displaystyle \int_2^3 h(x) dx = 1\). Find \(\displaystyle \int_{-2}^2 h(x) dx\).

    1. \(\displaystyle 0\)
    2. \(\displaystyle 3\)
    3. \(\displaystyle 4\)
    4. \(\displaystyle 6\)
    5. \(\displaystyle 8\)

Part II: Free Response

  1. Compute the following limits:

    1. \(\displaystyle \lim_{x \to 1} \sin\left(\frac{5\pi x}{4}\right)\)
    2. \(\displaystyle \lim_{x \to 0} \frac{\sin(x)}{\arcsin(2x)}\)
    3. \(\displaystyle \lim_{x \to 1^+} x^{1/(x-1)}\)

  2. Consider the function \(\displaystyle f(x) = \cos(x)\).

    1. Find its linearization at \(\displaystyle x = \pi\).
    2. Use the above linearization to estimate \(\displaystyle \cos(3)\).

  3. The limit below represents the \textbf{derivative} of which function at which point? Justify your answer.

    \[\displaystyle \lim_{h \to 0} \frac{\ln(e+h) - 1}{h}\]

  4. Express the limit below as a \textbf{definite integral} over the interval \(\displaystyle [0, \frac{\pi}{4}]\). Justify your answer. (Note that you do not need to compute the integral nor the limit.)

    \[\displaystyle \lim_{n \to \infty} \sum_{i=1}^n \frac{\pi}{4n} \tan\left( \frac{i\pi}{4n} \right)\]

  5. On the axes provided, sketch the graph of a single function \(\displaystyle h(x)\) which has all of the following properties :

    1. The domain of \(\displaystyle h(x)\) is \(\displaystyle [-3, 5)\).
    2. \(\displaystyle h(-1) = -4, h(0) = 2\).
    3. \(\displaystyle h(x)\) has a removable discontinuity at \(\displaystyle x = -1\), and \(\displaystyle h(x)\) is continuous through the rest of the domain.
    4. \(\displaystyle \lim_{x \to -1^-} h(x) = 2\).
    5. On the interval \(\displaystyle (-3, -1)\), \(\displaystyle h'(x) > 0\) and \(\displaystyle h''(x) > 0\).
    6. On the interval \(\displaystyle (-1, 3)\), \(\displaystyle h'(x) = 0\).
    7. \(\displaystyle h'(3)\) does not exist.
    8. \(\displaystyle \lim_{x \to 5^-} h(x) = \infty\).

    [Graph with labeled axes and grid shown]

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    Coordinate axes with grid for sketching or reading a graph

  6. Evaluate the following integrals:

    1. \(\displaystyle \int \left( 4x^3 + \sqrt[4]{x^4} + \frac{1}{x^3} \right) dx\)
    2. \(\displaystyle \int \frac{\ln(x)}{x} dx\)
    3. \(\displaystyle \int_{-3}^{5} 1 - |x-2| dx\). (Hint: The integrand is graphed below:)

    [Graph of \(\displaystyle y = 1 - |x-2|\) shown with labeled axes and domain from approximately \(\displaystyle -3\) to \(\displaystyle 5\).]

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    Diagram of a cylindrical tank with labeled dimensions

  7. A space shuttle is launched vertically from the JFK Center, and its position is being tracked by a radar device located 2 miles horizontally from the launch site. The shuttle is ascending vertically at a rate of 4 miles per second. Let \(\displaystyle \theta\) represent the angle between the horizontal ground and the line of sight from the radar to the shuttle. At what rate is \(\displaystyle \theta\) changing when the shuttle’s altitude is 3 miles?

  8. A company is designing a new underground water tank in the shape of a right circular cylinder. The tank will be constructed using two circular steel plates for the top and bottom and a curved steel sheet for the sides. The steel sheets are joined at the seams, which must be minimized to reduce construction costs and improve durability. The tank must hold exactly \(\displaystyle 4 \mathrm{m}^3\) of water. Find the dimensions (radius and height) of the tank that minimize the total length of the seams. Note that the seams are indicated in bold.

    1. Label the diagram of the tank, clearly marking all variables used to describe the function that you are optimizing.

    [Diagram of a right circular cylinder/tank]

    1. Write all the constraints and equations relevant to the situation.
    2. Write a formula of the function you want to optimize in terms of a single variable, and give a reasonable domain for the function (this is the domain where the problem makes sense).
    3. Use calculus to minimize the function and find the tank dimensions that result in the lowest construction cost.

    IMAGE

    Diagram of a cylindrical tank with labeled dimensions