Spring 22 Final

Problems

  1. Determine the following limits; briefly explain your thinking on each one. If you apply L'Hopital's rule, indicate where you have applied it and why you can apply it. If your final answer is "does not exist," \(\displaystyle \infty\), or \(\displaystyle -\infty\), briefly explain your answer. (You will not receive full credit for a "does not exist" answer if the answer is \(\displaystyle \infty\) or \(\displaystyle -\infty\).)

    1. \(\displaystyle \lim_{x \to 2} \left( x^3 - 7x + 17 \right)\)
    2. \(\displaystyle \lim_{x \to 0} \frac{2x}{e^{3x} - 7x - 1}\)
    3. \(\displaystyle \lim_{x \to \infty} \frac{\ln(x)}{7x^2}\)

  2. Consider the graph of \(\displaystyle y = f(x)\) given below. The grid lines are one unit apart. Based on the graph, answer the following questions.

    [Graph of \(\displaystyle y = f(x)\)]

    1. Determine \(\displaystyle \lim_{x \to -2} f(x)\).
    2. Determine \(\displaystyle f(1)\).
    3. Determine all values of \(\displaystyle x\) in the interval \(\displaystyle (-5, 4)\) at which \(\displaystyle f'(x) < 0\).
    4. Determine all values of \(\displaystyle x\) in the interval \(\displaystyle (-5, 4)\) at which \(\displaystyle f'(x)\) is undefined. Briefly explain your thinking.

    IMAGE

    Graph of the derivative of a function

  3. (a) State the limit definition of the derivative of \(\displaystyle f(x)\).

    1. Use the \textbf{limit definition of the derivative} to determine the derivative of \(\displaystyle f(x) = \frac{1}{1 - 3x}\). No points will be awarded for the application of differentiation rules (and L'Hopital's rule is not allowed).

  4. Determine the first derivative of each of the following functions. Remember to use correct notation to write your final answer.

    1. \(\displaystyle f(x) = \frac{x^5}{3} - \frac{1}{\sqrt{x}} + \pi^7\)
    2. \(\displaystyle f(x) = \frac{\sin(x)}{3x+4}\)
    3. \(\displaystyle f(x) = [x^2 + \arcsin(x)]^5\)
    4. \(\displaystyle y = x^{\cos(x)}\)

  5. Determine the following indefinite integrals.

    1. \(\displaystyle \int \left( x^5 + 2\sqrt{x} - e^x \right) dx\)
    2. \(\displaystyle \int \left( \frac{3}{x} + \frac{4}{1 + x^2} \right) dx\)
    3. \(\displaystyle \int \sin(x) e^{3 \cos(x)} dx\)

  6. Evaluate the following definite integrals.

    1. \(\displaystyle \int_{\pi/12}^{\pi/6} (\cos(3x)) dx\)
    2. \(\displaystyle \int_0^1 (1-2x)^6 dx\)

  7. An object moves along a straight line with acceleration function \(\displaystyle a(t) = -10t\) meters per second squared, where \(\displaystyle t \geq 0\). The object's initial velocity is 70 meters/second. Give appropriate units with each answer.

    1. Determine the velocity function \(\displaystyle v(t)\).
    2. When is the particle at rest?
    3. When is the object's velocity 50 meters per second?
    4. What is the displacement of the object from \(\displaystyle t = 0\) seconds to \(\displaystyle t = 5\) seconds?

  8. A rocket sitting on the ground is launched vertically upward and is tracked by an observer who is sitting on the ground 1000 feet from the launch site. Determine the velocity of the rocket when the angle of observation \(\displaystyle \theta\) from the observer to the rocket is \(\displaystyle \pi/6\) radians and is increasing at a rate of 0.2 radians per second.

    ![A diagram showing an observer, the rocket's starting point, a vertical ascent, and the angle \(\displaystyle \theta\) from the observer to the rocket.]

    IMAGE

    Diagram showing an observer measuring the angle to a rocket

  9. Consider the point (3, 1) and the parabola \(\displaystyle y = x^2\) graphed below. (Adjacent grid lines are one unit apart.)

    1. Let \(\displaystyle (x, y)\) represent a point on the parabola \(\displaystyle y = x^2\). Determine a function \(\displaystyle f(x)\) for the distance from the point \(\displaystyle (x, y)\) to the point (3, 1). Your function should include the variable \(\displaystyle x\) and can not include any other variables.
    2. Suppose you want to determine the closest point on the parabola to the point (3, 1). Determine an appropriate domain for the function \(\displaystyle f(x)\), and briefly explain your answer. (Do not determine the closest point.)

    IMAGE

    Diagram of a box with labeled dimensions

  10. You plan to make an open-top box with a square base; you want to construct the box to have volume 5 ft\(\displaystyle ^3\) and the minimum possible cost. The cost of the material for the base of the box is $3 per square foot, and the cost of the material for the sides of the box is $1 per square foot.

    Let \(\displaystyle w\) represent the (base) width of the box, respectively. Then the cost (in dollars) of the box is

    \[\displaystyle C(w) = 3w^2 + \frac{20}{w}.\]

    Using the function \(\displaystyle C(w)\) with domain \(\displaystyle (0, \infty)\), determine the width \(\displaystyle w\) that results in the minimum possible cost for your box. You must use calculus techniques to verify that your \(\displaystyle w\) results in the minimum possible cost.