Fall 25 Final

Part I: Multiple Choice

  1. If \(\displaystyle y = \sin(x^2)\), find \(\displaystyle \frac{dy}{dx}\).

    1. \(\displaystyle 1\)
    2. \(\displaystyle x\)
    3. \(\displaystyle \cos x^2\)
    4. \(\displaystyle 2x \cos x^2\)
    5. Some other answer.

  2. Consider the function \(\displaystyle f(x) = x - \sin x\) on \(\displaystyle [0, 2\pi]\). This function has an absolute maximum at the point \(\displaystyle (a, b)\). What is \(\displaystyle b - a\)?

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

  3. Let \(\displaystyle f(x)\) and \(\displaystyle g(x)\) be differentiable functions with the following values and derivatives at specific points:

    \[\displaystyle \begin{array}{c|ccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & 4 & 1 & 3 & 2 & 0 \\ g(x) & 1 & 3 & 2 & 2 & 9 \\ f'(x) & 2 & 1 & 5 & 3 & 2 \\ g'(x) & 1 & 2 & 1 & 5 & 3 \\ \end{array}\]

    If \(\displaystyle h(x) = \frac{f(x) \cdot g(x)}{g(x)}\), find \(\displaystyle h'(1)\).

    a. 1

    b. 2

    c. 3

    d. 4

    e. 5

  4. Let \(\displaystyle f(x)\) be a function defined over the interval \(\displaystyle 1 \leq x \leq 2\). Find \(\displaystyle f(2)\) given that

    \[\displaystyle \int_1^2 f(x) dx = 2 \int_1^2 x dx\]

    a. 1

    b. 2

    c. ln(2)

    d. ln(2) - 1

    e. ln(2) + 1

  5. The surface area of an initially cubic salt block is increasing at a rate of 6 cm\(\displaystyle ^2\) per minute. How fast is its side length increasing when the volume is 27 cm\(\displaystyle ^3\)?

    1. \(\displaystyle \frac{1}{18}\) cm/min
    2. 1 cm/min
    3. 0
    4. \(\displaystyle \frac{2}{3}\) cm/min
    5. \(\displaystyle \frac{2}{9}\) cm/min

  6. Let \(\displaystyle f\) be a function that is defined and differentiable for all real numbers, and assume that \(\displaystyle f'(x)\) is increasing only over the interval (4, 8). Where is the function \(\displaystyle f\) necessarily increasing?

    1. Over the interval (4, 8).
    2. Over the interval (4, 8].
    3. Over the intervals \(\displaystyle (-\infty, 4)\) and (8, \(\displaystyle \infty\)).
    4. Everywhere \(\displaystyle f'\) is strictly increasing.
    5. We can't answer as we do not have enough information about \(\displaystyle f\).

  7. Find the equation of the line tangent to the graph of

    \[\displaystyle y = x^3 - x^2 - 9\]

    at the point (4, 39).

    [graph/curve image]

    1. \(\displaystyle y = 6x + 15\)
    2. \(\displaystyle y = 7x - 17\)
    3. \(\displaystyle y = 9x + 3\)
    4. \(\displaystyle y = 5x - 9\)
    5. \(\displaystyle y = 3x + 9\)

    IMAGE

    Referenced image used in the problem

  8. Determine the types of discontinuities of

    \[\displaystyle f(x) = \frac{(x^2 - 4x)}{(x-2)(x-5)}\]
    1. The function has infinite discontinuities at \(\displaystyle x = 2\) and \(\displaystyle x = 5\).
    2. The function has a removable discontinuity at \(\displaystyle x = 2\) and an infinite discontinuity at \(\displaystyle x = 5\).
    3. The function has a removable discontinuity at \(\displaystyle x = 5\) and an infinite discontinuity at \(\displaystyle x = 2\).
    4. The function has infinite discontinuities at \(\displaystyle x = 2\) and \(\displaystyle x = 5\).
    5. The function has removable discontinuities at \(\displaystyle x = 0\), \(\displaystyle x = 2\), and \(\displaystyle x = 5\).

  9. A vertical bottle, as depicted below, gets filled at a constant rate. What can you say about the graph of \(\displaystyle h(t)\), the function giving the height in terms of time \(\displaystyle t\)?

    Hint: You do not need to make any computations to solve this problem; think intuitively.

    (image of the vertical bottle)

    a. It is increasing concave up.

    b. It is increasing concave down.

    c. It is decreasing concave up.

    d. It is decreasing concave down.

    e. It is increasing linearly.

    IMAGE

    Outline of a vertical bottle

  10. Consider the family of piecewise functions

    \[\displaystyle f(x) = \begin{cases} 2x, & x < 1 \\ \lambda x + 1, & x \geq 1 \end{cases}\]

    depending on a parameter \(\displaystyle \lambda\). For what values of \(\displaystyle \lambda\) is \(\displaystyle f\) differentiable at \(\displaystyle x = 1\)?

    a. \(\displaystyle \lambda = 2\)

    b. \(\displaystyle \lambda = 1\) and \(\displaystyle \lambda = 1\)

    c. \(\displaystyle \lambda = 2\) and \(\displaystyle \lambda = 1\)

    d. \(\displaystyle \lambda = 1\)

    e. \(\displaystyle \lambda = 0\) and \(\displaystyle \lambda = 1\)

  11. Estimate the area under the curve, plot of \(\displaystyle y = 16 - x^2\), over the interval \(\displaystyle [1, 5]\). Use a Right Riemann sum with four subintervals of equal width.

    1. 25
    2. 18
    3. 30
    4. 36
    5. 68

    [graph of the parabola is shown]

    IMAGE

    Graph of a parabola on coordinate axes

  12. Compute \(\displaystyle \int_{0}^{\pi} \sin\left(\frac{x}{10}\right) dx\).

    1. 10
    2. 0
    3. \(\displaystyle \frac{10}{\pi}\)
    4. \(\displaystyle 10\pi\)
    5. Some other value

  13. The graph of \(\displaystyle f(x) = \frac{x^2}{2}\) is given below. What is the area of the region bounded by the graph of \(\displaystyle f(x)\), the x-axis, and the lines \(\displaystyle x = 1\) and \(\displaystyle x = 2\)?

    (graph of \(\displaystyle f(x)\) with x from 0 to 3 and y from 0 to 5, curve labelled \(\displaystyle f(x)\), and vertical lines at x = 1 and x = 2)

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

    IMAGE

    Referenced image used in the problem

  14. If \(\displaystyle f(x) = 4x\), and the derivative of \(\displaystyle f\) at \(\displaystyle x = 5\) is \(\displaystyle f'(5)\), what is \(\displaystyle f'(5)\)?

    1. 2
    2. 5
    3. 25
    4. 4
    5. 20
    6. The answer is not in the list.

  15. If

    \[\displaystyle \int f(x) dx = 5 \text{ and } \int f(x) dx = 1,\]

    find

    \[\displaystyle \int 4f(x) dx\]

    a. 2

    b. 5

    c. 1

    d. 6

    e. 20

    f. 4

  16. Find

    \[\displaystyle \frac{d}{dx} \left( \int_1^x \frac{1}{1 + t^2} dt \right)\]

    a. \(\displaystyle \sin x\)

    b. \(\displaystyle \sin x + C\)

    c. \(\displaystyle \frac{\sin x}{\cos x}\)

    d. \(\displaystyle \frac{1}{\cos x}\)

    e. \(\displaystyle \frac{1}{1 + x^2}\)

    f. \(\displaystyle \sin x \cos x\)

Part II: Free Response

  1. For this problem, the graph of \(\displaystyle f(x)\) is given below. Note that all curves shown are either straight lines or arcs of a circle, and the ends of the graph extends linearly towards positive or negative infinity. No need to show your work for this question.

    (Graph of \(\displaystyle f(x)\) shown, labeled with x and y axes)

    \[\displaystyle \begin{array}{cc} \text{()} & (a)\ \lim_{x \to 2} f(x) \\ \end{array}\]

    ANSWER

    \[\displaystyle \begin{array}{cc} \text{()} & (b)\ \lim_{x \to 3^-} f(x) \\ \end{array}\]

    ANSWER

    \[\displaystyle \begin{array}{cc} \text{()} & (c)\ \lim_{x \to 2} (5f(x) - 4x + e) \\ \end{array}\]

    ANSWER

    \[\displaystyle \begin{array}{cc} \text{()} & (d)\ \lim_{x \to -4} \frac{f(x) - f(-4)}{x-(-4)} \\ \end{array}\]

    ANSWER

    The graph of \(\displaystyle f(x)\) from the previous page is shown again:

    [Graph of \(\displaystyle f(x)\) appears here.]

    () (e) List the x-coordinates of all critical points of \(\displaystyle f(x)\):

    ANSWER

    () (f) Find the average rate of change of \(\displaystyle f(x)\) on the interval \(\displaystyle [-2, 2]\).

    ANSWER

    () (g) Find the instantaneous rate of change of \(\displaystyle f(x)\) at \(\displaystyle x = 2\).

    ANSWER

    The graph of \(\displaystyle f(x)\) from the previous page is given again.

    () (h) Can the Mean Value Theorem be applied on the interval \(\displaystyle [-1, 1]\)? \textbf{Justify}.

    () (i) Can the Intermediate Value Theorem be applied on the interval \(\displaystyle [0, 2]\)? \textbf{Justify}.

    () (j) \[\displaystyle \int_{-1}^{2} f(x) dx\]

    ANSWER

    IMAGE

    Coordinate axes with grid for sketching or reading a graph

    IMAGE

    Coordinate axes with grid for sketching or reading a graph

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    Coordinate axes with grid for sketching or reading a graph

  2. Let \(\displaystyle f(x) = \sqrt{x}\).

    () (a) Using the limit definition of the derivative, and \textbf{no other method}, compute the derivative of \(\displaystyle f(x)\).

    ANSWER

    () (b) Check your answer in part (a) using the power rule.

    () (c) Use linearization to estimate \(\displaystyle \sqrt{25.04}\).

    ANSWER

  3. Consider a right triangle with side lengths 30, 40 and 50. A rectangle is inscribed in the triangle so that two of the sides of the rectangle lie along the legs (the sides forming the right angle) of the triangle. Determine the maximum possible area of the rectangle, and justify why this value represents an absolute maximum.

    () (a) Make a sketch of the situation.

    () (b) Write a formula for the area of the rectangle in terms of one variable along with a reasonable domain.

    () (c) Solve the problem using calculus and justify your finding.

    ANSWER

  4. Consider the function

    \[\displaystyle f(x) = \frac{-3e^x + 1}{e^x - 1},\]

    with first derivative

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

    and second derivative:

    \[\displaystyle f''(x) = -\frac{2e^x (e^x + 1)}{(e^x - 1)^3}.\]

    The function has a single x-intercept at \(\displaystyle (-\ln 3, 0)\).

    () (a) Find the domain of \(\displaystyle f(x)\).

    ANSWER

    () (b) Find all vertical asymptotes of \(\displaystyle f(x)\) or show that there are none. Justify your findings.

    ANSWER

    () (c) The graph of \(\displaystyle f(x)\) has a horizontal asymptote of \(\displaystyle y = -3\) as \(\displaystyle x \to +\infty\); find any additional horizontal asymptotes, or show that there are no others. Justify your answer.

    ANSWER

    Recall that

    \[\displaystyle f(x) = \frac{-3e^x + 1}{e^x - 1},\]

    with first derivative

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

    and second derivative:

    \[\displaystyle f''(x) = -\frac{2e^x(e^x + 1)}{(e^x - 1)^3}.\]

    () (d) Using the first and/or the second derivative of \(\displaystyle f(x)\), find all critical points or show that no critical points exist.

    ANSWER

    () (e) Using the first and/or the second derivative of \(\displaystyle f(x)\), find all inflection points or show that no inflection points exist.

    ANSWER

    1. Organize your work in the table below. For each interval, indicate:

    i. the sign of \(\displaystyle f'(x)\) by writing + or –,

    ii. whether the function is increasing or decreasing by drawing \(\displaystyle /\) or \(\displaystyle \backslash\),

    iii. the sign of \(\displaystyle f''(x)\) by writing + or –,

    iv. whether the graph is concave up or down by drawing \(\displaystyle \cup\) or \(\displaystyle \cap\).

    Use as many interval columns as needed.

    Interval :
    Sign of \(\displaystyle f'\)
    Increasing/Decreasing
    Sign of \(\displaystyle f''\)
    Concave up/down
    1. Sketch the function. Include intercepts, local extrema, inflection points and asymptotes, if any.

    [Grid for sketching the function, labeled axes from -4 to 4 (x-axis) and -6 to 2 (y-axis)]

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    Blank coordinate grid for sketching a graph