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 \(\displaystyle -\infty\).)
- \(\displaystyle \lim_{x \to 2} \sqrt[3]{\frac{x^2 + x - 6}{x^2 - x - 2}} =\)
- \(\displaystyle \lim_{x \to 7} \frac{\sqrt{x+2} - 3}{x-7} =\)
- \(\displaystyle \lim_{x \to 0} \frac{1 - \cos(x)}{x^2} =\)
- \(\displaystyle \lim_{x \to \infty} x e^{-3x} =\)
Consider the graph of \(\displaystyle y = f(x)\) defined on the closed interval \(\displaystyle [-5, 4]\) given below. The grid lines are one unit apart. Based on the graph, answer the following questions.
- Determine all values of \(\displaystyle a\) such that \(\displaystyle f(a)\) exists and \(\displaystyle \lim_{x \to a} f(x)\) does not exist.
- Determine all values of \(\displaystyle x\) in the interval \(\displaystyle (-5, 4)\) such that \(\displaystyle f(x)\) exists and \(\displaystyle f'(x)\) does not exist.
- Determine all values of \(\displaystyle x\) in the interval \(\displaystyle (-5, 4)\) such that \(\displaystyle f'(x) < 0\).
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Use the limit definition of the derivative to determine the derivative of
No points will be awarded for the application of differentiation rules (and L’Hopital’s Rule is not allowed.)
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Determine the first derivative of each of the following functions. Remember to use correct notation to write your final answer.
- \(\displaystyle f(t) = \frac{4t^5 - 3}{7t} + e^{9t}\)
- \(\displaystyle h(x) = \cos(x) \arcsin(x)\)
- \(\displaystyle k(y) = \tan(9y) - \ln(5y^3 - y^2 + 4)\)
Determine \(\displaystyle \frac{dy}{dx}\).
- \(\displaystyle y = x^{\sin(x)}\)
- \(\displaystyle y^3 - 4y = x^2 e^y\)
Consider \(\displaystyle f(x) = e^x\). It has been graphed below.
[Graph of \(\displaystyle y = e^x\) with labeled axes: \(\displaystyle x\) from -5 to 5, \(\displaystyle y\) from -2 to 5, and gridlines.]
- Write the equation of the tangent line of \(\displaystyle y = f(x)\) at the point (0, 1) and then \textbf{graph the line} on the grid provided above.
- Use the linearization of \(\displaystyle y = f(x)\) at \(\displaystyle x = 0\) to approximate \(\displaystyle e^{0.05}\). (You don't need to simplify.) Use the graph to explain whether or not your approximation is an overestimate or underestimate of \(\displaystyle e^{0.05}\).
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Evaluate the following. You do not need to simplify your answers.
- \(\displaystyle \int \left( \frac{2}{x^3} + \frac{8}{x} + \frac{1}{1+x^2} \right) dx\)
- \(\displaystyle \int \frac{\sqrt{\ln(x)} + 4}{x} dx\)
- \(\displaystyle \int_{-2}^{1} (x^2 + x + 1) dx\)
- \(\displaystyle \frac{d}{dx} \int_{2}^{x} \left( \csc^2(t) + 2^t \right) dt\)
Suppose that from a height of 6 feet above the ground, a ball is tossed vertically in such a way that its velocity function is given by \(\displaystyle v(t) = 32 - 32t\), where \(\displaystyle t\) is measured in seconds and \(\displaystyle v\) in feet per second. Assume that this function is valid for \(\displaystyle t \geq 0\).
- After how many seconds will the ball change direction and begin to fall to the ground?
- Determine the distance between the ball and the ground after 1.5 seconds.
- After how many seconds will the ball hit the ground?
Air is leaking from a spherical balloon at a rate of 5 cm\(\displaystyle ^3\)/min. Determine the rate at which the radius of the balloon is changing when the radius of the balloon is 10 cm. (You do not need to simplify your answer; decimal approximations will not be accepted.)
Hint: The volume of a sphere is \(\displaystyle V = \frac{4}{3} \pi r^3\).
A rectangular box with a square base and an open top is to be constructed.
- Suppose the volume of the box is 200 cubic inches. Write a formula for the surface area of the box as a function of \(\displaystyle x\), the lengths of its base, and determine the domain of the function.
(A diagram of a box labeled with base dimensions \(\displaystyle x\) and height \(\displaystyle y\).)
- The manufacturing company has decided that the box will need to have a volume of 216 cubic inches. The material chosen to construct the box will cost \$0.20/square inch for the base and \$0.30/square inch for the sides. Then the cost (in dollars) of the box is given by
where \(\displaystyle x\) is represents the lengths of the base of the box. What should the dimensions of the box be to minimize the cost of making the box?
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