Spring 19 Final

Problems

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 $\displaystyle -\infty$.)

Please circle your final answer.**
1. $\displaystyle \lim_{x \to 0} \frac{e^{2x} - 2}{e^x + 1}$
2. $\displaystyle \lim_{x \to 3^-} \frac{x^2 + 7}{x - 3}$
3. $\displaystyle \lim_{x \to \pi / 2} \frac{\cos(x)}{x - \frac{\pi}{2}}$
4. $\displaystyle \lim_{x \to \infty} \frac{3x^2(x-1)}{x^2 - 2}$
(a) State the limit definition of the derivative of $\displaystyle f(x)$.
1. Use the limit definition of the derivative to determine the derivative of $\displaystyle f(x) = \sqrt{x-3}$. No points will be awarded for the application of differentiation rules (and L’Hopital’s rule is not allowed).
Determine the first derivative of each of the following functions. Remember to use correct notation to write your final answer. Please circle your final answer.
1. $\displaystyle f(x) = \frac{x^3}{4} - \frac{2}{x^3} + \pi^2$
2. $\displaystyle f(x) = \left( \arctan(x) \right)^3$
3. $\displaystyle f(x) = e^{\sqrt{x} \cos(x)}$
4. $\displaystyle f(x) = x^{1/x}$
Consider the curve defined by the equation

\[\displaystyle (x^2 + y^2)^2 = 9(x^2 - y^2).\]

The graph of the curve is provided to the right.

The graph of the curve is provided to the right.
1. Determine $\displaystyle \frac{dy}{dx}$.
2. Determine an equation of the line tangent to the curve at the point $\displaystyle (x, y) = (\sqrt{5}, 1)$.
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Determine the following indefinite integrals. **Please circle your final answer**.
1. $\displaystyle \int (\cos(5x) + 1) dx$
2. $\displaystyle \int \left( \frac{1}{\sqrt{1-x^2}} + \frac{1}{e^x} \right) dx$
3. $\displaystyle \int \frac{3}{x (\ln(x))^2} dx$
Evaluate the following definite integrals. \textbf{Please circle your final answer}.

\[\displaystyle \text{(a) } \int_{0}^{1} (2x^2 + 3x + 5) dx\]

\[\displaystyle \text{(b) } \int_{1}^{2} \left( \frac{x}{\sqrt[3]{x^2 + 1}} \right) dx\]

\[\displaystyle \text{(c) } \int_{0}^{1/2} \left( \frac{e^{2x}}{1 + e^{2x}} \right) dx\]
For this problem, consider the function $\displaystyle f(x) = |x-3|$.

(a) Use a Riemann Sum with 5 sub-intervals of equal width and left endpoints to estimate the definite integral \[\displaystyle \int_0^4 |x-3| dx.\]

(b) Use areas to evaluate the definite integral \[\displaystyle \int_0^4 |x-3| dx.\]
Use calculus to determine the absolute maximum and absolute minimum values of the function below on the interval $\displaystyle [-10, 1]$:

\[\displaystyle f(x) = \frac{x}{x^2 + 9}\]
The graph of $\displaystyle y = g'(x)$, the derivative of $\displaystyle g$, is given below. Consecutive grid lines are one unit apart.

\[\displaystyle g'\]

This is the graph of $\displaystyle g'$.
1. State the intervals on which the function $\displaystyle g$ is increasing. (Briefly explain your reasoning.)
2. Assuming that $\displaystyle g(-3) = 1$, use the graph of $\displaystyle g'$ to make a rough sketch of the graph of $\displaystyle g$ on the axes below. Be sure to accurately show the increasing/decreasing behavior of $\displaystyle g$ and any horizontal tangents of $\displaystyle g$ on your graph.
\[\displaystyle g\]
1. Page 10 of 17
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On the axes below, sketch the graph of a function $\displaystyle f$ having the following properties.

(Consecutive grid lines are one unit apart.)

The domain of $\displaystyle f$ is $\displaystyle (-\infty, \infty)$.

$\displaystyle f(-2) = 4$

\[\displaystyle \lim_{x \to -2} f(x) = -3\]

The line $\displaystyle y = 1$ is a horizontal asymptote.

\[\displaystyle \lim_{x \to 2} f(x) = \infty\]

\[\displaystyle \lim_{x \to \infty} f(x) = 0\]

(Cartesian axes and grid provided for sketching the graph)

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Suppose $\displaystyle f$ and $\displaystyle g$ are continuous on $\displaystyle [0, 5]$; some values of $\displaystyle f(x)$, $\displaystyle f'(x)$, $\displaystyle g(x)$, and $\displaystyle g'(x)$ are given in the table below.

\[\displaystyle \begin{array}{|c|c|c|c|c|} \hline x & f(x) & f'(x) & g(x) & g'(x) \\ \hline 1 & -3 & 2 & -1 & -2 \\ 2 & 5 & -1 & -3 & 2 \\ 3 & 1 & -3 & 4 & -1 \\ \hline \end{array}\]
1. Let $\displaystyle h(x) = f(x)g(x)$. Determine $\displaystyle h'(2)$.
2. Determine the linearization $\displaystyle L(x)$ of $\displaystyle h(x) = f(x)g(x)$ at $\displaystyle x = 2$.
3. For the function $\displaystyle h(x) = f(x)g(x)$, use the linearization to approximate the value of $\displaystyle h(2.1)$.
A particle moves along a straight line with position function $\displaystyle s(t) = \frac{15 t^2}{2} - t^3$, for $\displaystyle t > 0$, where $\displaystyle s$ is in feet and $\displaystyle t$ is in seconds.
1. Determine the velocity of the particle when the acceleration is zero.
2. On the interval $\displaystyle (0, \infty)$, when is the particle moving in the positive direction? In the negative direction?
3. Determine all local (relative) extrema of the position function on the interval $\displaystyle (0, \infty)$. (You may use any relevant work from previous parts.)
4. Determine $\displaystyle \frac{d}{dt} \left( \int_{1}^{t} s(u) du \right)$.
According to one model, the average global temperature $\displaystyle x$ of the earth (in degrees Celsius) is related to the atmospheric concentration $\displaystyle y$ of carbon dioxide (in parts per million, or ppm) by the equation

\[\displaystyle x^2 + 30x + 2125 = 140\sqrt{y}.\]

When the global temperature is 15 degrees Celsius and the concentration of carbon dioxide is increasing at a rate of 5 ppm/year, determine the rate of change of the global temperature with respect to time.
You plan to build a box with a square base and no lid. The material for the base of the box costs $0.20 per square foot and the material for the sides costs $0.05 per square foot. You can spend at most $5.40 total on the materials for the box. You want to use these materials to construct the box of maximum possible volume which is within your budget.
1. Let $\displaystyle x$ be the length, in feet, of each of the sides of the square base. Determine a formula for the function $\displaystyle V(x)$ representing the volume of the box having base dimension $\displaystyle x$ feet. Your final answer $\displaystyle V(x)$ should include the variable $\displaystyle x$ and may not include any other variables.
2. Determine an appropriate domain for $\displaystyle V(x)$. Briefly explain your answer. (Note: There may be more than one correct answer; choose one domain you believe is a reasonable choice, and explain your thinking.)
A rectangle with base $\displaystyle x$ units has a segment of length 4 cm joining one vertex to the midpoint $\displaystyle M$ of an opposite side, as shown in the diagram below. The area $\displaystyle A(x)$ of the rectangle is given by the function $\displaystyle A(x) = 2x\sqrt{16 - x^2}$.

[Diagram of a rectangle with a base labeled $\displaystyle x$, and a diagonal segment of length 4 from one vertex to the midpoint $\displaystyle M$ of the opposite side.]
1. Determine an appropriate domain for the function $\displaystyle A(x)$. Briefly explain your answer.
2. Determine the maximum possible area of the rectangle. Use calculus to justify your answer.
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