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 \(\displaystyle -\infty\).)
- \(\displaystyle \lim_{x \to 1} \frac{x^2 - 9}{x^2 + x - 6}\)
- \(\displaystyle \lim_{x \to 2^+} \frac{x^2 - 9}{x^2 + x - 6}\)
- \(\displaystyle \lim_{p \to 0} \frac{\ln(1 + 5p) - p}{\sin(3p)}\)
- \(\displaystyle \lim_{x \to \infty} xe^{-2x}\)
(a) State the limit definition of the derivative of \(\displaystyle f(x)\).
- Use the \textbf{limit definition of the derivative} to determine the derivative of \(\displaystyle f(x) = \frac{1}{3 - 2x}\). 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.
- \(\displaystyle f(x) = x^2 - \sin(x)\cos(x)\)
- \(\displaystyle g(t) = \arctan(\ln(t))\)
- \(\displaystyle h(x) = \frac{e^{3x}}{x+1}\)
Determine the \textbf{second derivative} of the function
Determine the following indefinite integrals.
- \(\displaystyle \int \left( x^3 - 5x + 7 \right) dx\)
- \(\displaystyle \int \frac{\left( \ln(x) + 4 \right)^{10}}{x} dx\)
- \(\displaystyle \int \frac{\sin(t)}{1 - 2\cos(t)} dt\)
Evaluate the following definite integrals.
- \(\displaystyle \int_{0}^{1} \left( e^{x} - \frac{3}{1 + x^{2}} \right) dx\)
- \(\displaystyle \int_{0}^{3} f(x) dx\), where \(\displaystyle f(x)\) is the function given by \(f(x) = \begin{cases}
\sin(x) & 0 \leq x < \frac{\pi}{2} \\
1 & \frac{\pi}{2} \leq x \leq 3
\end{cases}\)
Use calculus techniques to determine the \(\displaystyle x\)-coordinates of the local (relative) extrema for the function below. Be sure to label each extremum as a local (relative) maximum or as a local (relative) minimum.
The graph of \(\displaystyle y = f(x)\) is given below and consists of line segments and circle arcs. The domain of \(\displaystyle f\) is \(\displaystyle (0,6)\).
[Graph of y = f(x) as described in the problem, with labeled axes, intervals, and the function drawn.]
- Determine all values of \(\displaystyle x\) in \(\displaystyle (0,6)\) for which \(\displaystyle f'(x) > 0\). Write your answer using interval notation.
- Determine all values of \(\displaystyle x\) in \(\displaystyle (0,6)\) for which \(\displaystyle f'(x)\) is undefined.
- Estimate \(\displaystyle f'(1.5)\).
- Determine whether \(\displaystyle f''(3)\) is positive, negative, or zero. (Write “positive,” “negative,” or “zero.” No explanation is needed.)
- Determine \(\displaystyle \int_0^6 f(x)dx\).
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Determine all values of \(\displaystyle c\) for which the function below is continuous on \(\displaystyle (-\infty, \infty)\). Use the limit definition of continuity to explain your answer.
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Karina and Juan both leave an intersection at the same time, both driving on long, straight roads. Karina drives north at a speed of 50 miles per hour, and Juan drives east at 60 miles per hour. When 1.5 hours have elapsed since they left the intersection, how fast is the distance between Karina and Juan changing? Use calculus to justify your answer.
You plan to make a rectangular box having an open top. Each face of the box is a rectangle, and the top face of the box is missing so that the box is open. For the base of the box, you want the length to be twice the width; you also want the total surface area of the resulting box to be 200 square inches.
- Let \(\displaystyle w\) represent the width of the box in inches. Determine a formula \(\displaystyle V(w)\) for the total volume of the box as a function of \(\displaystyle w\) alone. Your final answer should include the variable \(\displaystyle w\) and can not include any other variables.
- Suppose you want to determine the dimensions of the box that will result in the maximum possible volume. Determine an appropriate domain for \(\displaystyle V(w)\). Briefly explain your answer. (Note: We will not actually maximize \(\displaystyle V(w)\) in this problem.)
Suppose you run a factory that processes raw materials in two possible ways.
Using method A, you can process \(\displaystyle x\) tons of material at a cost of \(\displaystyle x^2\) dollars.
Using method B, you can process \(\displaystyle y\) tons of material at a cost of \(\displaystyle 10y\) dollars.
You want to process a total of 200 tons of material, so your process is constrained by the equation \(\displaystyle x + y = 200\). The overall cost (in dollars) of processing the two materials is
You can process a partial ton of material using each method, so \(\displaystyle x\) and \(\displaystyle y\) do not have to be whole numbers. You plan to minimize the overall cost.
- Determine the overall cost \(\displaystyle C\) \(\displaystyle \textbf{as a function of } x\). (The variable \(\displaystyle y\) should not be used in your answer.)
- Determine an appropriate domain for your cost function \(\displaystyle C\) from part (a), and explain briefly.
[This problem continues on the next page.]
[Problem 15, continued]
- Use calculus techniques to determine the value of \(\displaystyle x\) which results in the minimum overall cost. Write a sentence, using appropriate units, to summarize your answer.