Spring 25 Final

Part I: Multiple Choice

  1. If \(\displaystyle y = 2x^3 + 3x^2 - 12x - 10\), find \(\displaystyle y'(-1)\).

    1. \(\displaystyle -18\)
    2. \(\displaystyle -12\)
    3. \(\displaystyle -6\)
    4. \(\displaystyle 0\)
    5. \(\displaystyle 6\)

  2. If \(\displaystyle f(\theta) = \sin \theta\), find \(\displaystyle f^{(11)}(\pi)\), the 11th derivative of \(\displaystyle f(\theta)\) at \(\displaystyle \theta = \pi\).

    1. \(\displaystyle -\pi\)
    2. \(\displaystyle -1\)
    3. \(\displaystyle 0\)
    4. \(\displaystyle 1\)
    5. \(\displaystyle \pi\)

  3. How many critical points of \(\displaystyle y = x^5 - \frac{20}{3}x^3\) are NOT local extrema?

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

  4. If \(\displaystyle y(x) = \sqrt{1-x^2} + x \arcsin{x}\), compute \(\displaystyle y'(x)\).

    1. \(\displaystyle \frac{x}{\sqrt{1-x^2}}\)
    2. \(\displaystyle \frac{1}{\sqrt{1-x^2}}\)
    3. \(\displaystyle \arcsin{x} - \frac{2x}{\sqrt{1-x^2}}\)
    4. \(\displaystyle \arcsin{x} - \frac{x}{\sqrt{1-x^2}}\)
    5. \(\displaystyle \arcsin{x}\)

  5. A person walks away from a lamp post at a rate of 3 ft/sec. The lamp post is 10 ft tall, and the person is 5 ft tall. How fast is the length of the person's shadow increasing when the person is 15 ft from the lamp post?

    1. \(\displaystyle \frac{3}{10}\) ft/sec
    2. \(\displaystyle \frac{10}{3}\) ft/sec
    3. \(\displaystyle \frac{5}{2}\) ft/sec
    4. 2 ft/sec
    5. 3 ft/sec

  6. What is the slope of the tangent line to \(\displaystyle x y^2 + e^y = x^2\) at the point \(\displaystyle (1, 0)\)?

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

  7. Consider the function

    \[\displaystyle f(x) = \begin{cases} x^{2} + ax - 3x + 3 - a, & \text{if } x \leq 1 \\ \frac{2x}{x^{2} + 1}, & \text{if } x > 1 \end{cases}\]

    What should \(\displaystyle a\) be so that the function is differentiable everywhere?

    1. \(\displaystyle a = 0\)
    2. \(\displaystyle a = \frac{1}{2}\)
    3. \(\displaystyle a = \frac{1}{3}\)
    4. \(\displaystyle a = \frac{1}{4}\)
    5. \(\displaystyle a = 1\)

  8. If \(\displaystyle f(x) = x^{3} + 8x + \cos(3x)\) and \(\displaystyle g(x) = f^{-1}(x)\), find the slope of the tangent line to \(\displaystyle g(x)\) at the point \(\displaystyle (1, 0)\).

    1. \(\displaystyle \frac{1}{11}\)
    2. \(\displaystyle \frac{1}{8}\)
    3. \(\displaystyle -\frac{1}{8}\)
    4. \(\displaystyle \frac{1}{11 + 3 \sin 3}\)
    5. \(\displaystyle 8\)

  9. Let \(\displaystyle f\) and \(\displaystyle g\) be differentiable functions, and suppose the following values are known:

    \[\displaystyle \begin{aligned} &f(a) = -4, f'(a) = 8, g(a) = c, g'(a) = b, \\ &f(c) = 3, f'(c) = 2, g(c) = 5, g'(c) = 1. \end{aligned}\]

    Define \(\displaystyle h(x) = [ f(g(x)) ]^2\). What is the value of \(\displaystyle h'(a)\)?

    1. \(\displaystyle 4b\)
    2. \(\displaystyle 6b\)
    3. \(\displaystyle 12b\)
    4. \(\displaystyle 15b\)
    5. \(\displaystyle 24b\)
    6. \(\displaystyle 36b\)

  10. Below is the graph of the derivative \(\displaystyle f'\) of a continuous function \(\displaystyle f\).

    Below is the graph of the derivative \(\displaystyle f'\) of a continuous function \(\displaystyle f\).

    At what value(s) of \(\displaystyle x\) does \(\displaystyle f\) have a local maximum?

    1. 0 only
    2. 2 only
    3. 4 only
    4. 0 and 6
    5. 2 and 4
    6. 4 and 6
    7. 2 and 6
    8. 4 and 8

    IMAGE

    Graph of the derivative of a function

  11. If \(\displaystyle H(x) = \int_{0}^{2x} e^{3t^2} dt\), find \(\displaystyle H'(1)\).

    1. \(\displaystyle e^3\)
    2. \(\displaystyle e^6\)
    3. \(\displaystyle e^{12}\)
    4. \(\displaystyle 2e^6\)
    5. \(\displaystyle 2e^{12}\)

  12. Let \(\displaystyle f(x)\) be a continuous function defined for all real numbers. We are given that

    \[\displaystyle \int_{-1}^{1} f(x) dx = 3, \text{and} \int_{-1}^{2} f(x) dx = 5.\]

    Calculate \(\displaystyle \int_{1}^{2} (3 - 2f(x)) dx\).

    1. \(\displaystyle -1\)
    2. \(\displaystyle 0\)
    3. \(\displaystyle 1\)
    4. \(\displaystyle 5\)
    5. \(\displaystyle 7\)

  13. Compute \(\displaystyle \int_{-2}^{0} 1 + \sqrt{4 - x^2} dx\) by recognizing it as the area of a region made up of simple geometric shapes.

    1. \(\displaystyle \pi + 2\)
    2. \(\displaystyle -\pi - 2\)
    3. \(\displaystyle 2\pi\)
    4. \(\displaystyle \pi + 4\)
    5. \(\displaystyle -\pi\)

  14. Compute the area bounded by \(\displaystyle y = e^x\), \(\displaystyle x = 0\), and \(\displaystyle y = 2\).

    1. \(\displaystyle 2 \ln(2) - 1\)
    2. \(\displaystyle 2 \ln(2) + 1\)
    3. \(\displaystyle \ln(2) - 1\)
    4. \(\displaystyle 2 - \ln(2)\)
    5. \(\displaystyle 2 \ln(2)\)

  15. A scientist is studying how the concentration of a chemical in a solution changes with temperature. The concentration \(\displaystyle C\), measured in moles per liter, is modeled by the function

    \[\displaystyle C(T) = \ln(T^2 + 5)\]

    where \(\displaystyle T\) is the temperature in degrees Celsius. To make quick predictions without using a calculator, the scientist wants to use a linear approximation of \(\displaystyle C(T)\) near \(\displaystyle T = 2^\circ C\). Estimate the concentration at \(\displaystyle T = 2.1^\circ C\).

    1. \(\displaystyle C(2.1) \approx \ln(9) + \frac{1}{45}\)
    2. \(\displaystyle C(2.1) \approx \ln(9) + \frac{1}{90}\)
    3. \(\displaystyle C(2.1) \approx \ln(9) + \frac{2}{45}\)
    4. \(\displaystyle C(2.1) \approx \ln(9) + \frac{4}{81}\)
    5. \(\displaystyle C(2.1) \approx \ln(9) + \frac{1}{9}\)

  16. \(\displaystyle \int_{0}^{2} e^{-x^2} dx\) is to be approximated using a Riemann sum with 8 subintervals of equal width, where the height of each rectangle is determined by the right endpoint of the subinterval. If \(\displaystyle i\) is an integer between 1 and 8, what is the area of the \(\displaystyle i\)th rectangle?

    1. \(\displaystyle \frac{1}{8} e^{-\frac{i}{4}}\)
    2. \(\displaystyle \frac{1}{8} e^{-\frac{i^2}{4}}\)
    3. \(\displaystyle \frac{1}{4} e^{-\frac{i^2}{4}}\)
    4. \(\displaystyle \frac{1}{4} e^{-(\frac{i}{4})^2}\)
    5. \(\displaystyle e^{-\frac{i}{4}}\)

  17. Evaluate \(\displaystyle \int_{0}^{1} \frac{x}{1 + x^2} dx\).

    1. \(\displaystyle \ln(2)\)
    2. \(\displaystyle \frac{1}{2}\)
    3. \(\displaystyle \frac{1}{4} \ln(2)\)
    4. \(\displaystyle \arctan(1) = \frac{\pi}{4}\)
    5. \(\displaystyle \frac{1}{2} \ln(2)\)

  18. If \(\displaystyle f(x)\) is continuous and it is known that \(\displaystyle \int_{0}^{2} f(x) dx = 6\), evaluate \(\displaystyle \int_{0}^{\frac{\pi}{2}} f(2 \sin t) \cos t dt\).

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

Part II: Free Response

  1. Compute the following limits:

    1. \(\displaystyle \lim_{t \to 1^-} \sqrt{2t + 5}\)
    2. \(\displaystyle \lim_{x \to 2} \frac{x^2 - 7x + 10}{x^2 - 4}\)
    3. \(\displaystyle \lim_{x \to 1} \frac{3^x - 3}{x - 1}\)
    4. \(\displaystyle \lim_{y \to +\infty} \sqrt{y^2 + 6y - 1} - y\)
    1. Write the limit definition of the derivative of a function \(\displaystyle f(x)\).
    2. Use that definition to compute the derivative of \(\displaystyle f(x) = \frac{1}{x+5}\). (No credit will be awarded for any other method.)

  3. Consider the function

    \[\displaystyle f(x) = \frac{x}{x^2+1}\]

    with first derivative

    \[\displaystyle f'(x) = \frac{1-x^2}{(x^2+1)^2}\]

    and second derivative:

    \[\displaystyle f''(x) = \frac{2x(x^2-3)}{(x^2+1)^3}.\]
    1. Find the domain of \(\displaystyle f(x)\).
    2. Find the \(\displaystyle x\)- and \(\displaystyle y\)- intercepts of \(\displaystyle f(x)\).
    3. Does \(\displaystyle f(x)\) possess any symmetry? (Even/Odd/Neither)
    4. Does \(\displaystyle f(x)\) have any vertical or horizontal asymptotes? Justify.
    5. Using the first and/or the second derivative of \(\displaystyle f(x)\), find all critical points.
    6. Using the first and/or the second derivative of \(\displaystyle f(x)\), find all inflection points.
    7. Organize the above findings in a table that encodes the signs of the first and second derivative of \(\displaystyle f(x)\). Clearly state the interval(s) where the function is increasing, decreasing, concave up or concave down. If not applicable, state “none”.
    8. Sketch the function, putting in evidence intercepts, local extrema, inflection points and asymptotes, if any. Do not forget to label your graph.

    (Graph with labeled axes from -4 to 4 on x and -2 to 2 on y)

    IMAGE

    Diagram of a box with labeled dimensions

  4. A closed box with a lid is to be formed by trimming identical rectangles (shaded below) from two adjacent corners of a flat, 6-foot by 6-foot cardboard sheet with negligible thickness. The sides are then folded up to form the box. A diagram is provided here:

    1. Write a formula for the volume of the box in terms of \(\displaystyle x\) along with a reasonable domain. Simplify the volume formula as much as you can.
    2. Use calculus to find the value of \(\displaystyle x\) that gives the largest volume and justify your finding.

    IMAGE

    Diagram of a box with labeled dimensions