MATH 131, section 1 – Practice Questions for Exam 2 October 17, 2007

1. All parts of this question refer to the parametric curve x = 3 cos(2t), y = 5 sin(2t).

(a) Eliminate the parameter t and find a Cartesian equation for this curve.

(b) What portion of the curve is traced out for 0 ≤ t ≤ \frac{π}{2}, and in which direction is the curve being traced?

(c) What would change in your answer to part b if the curve above was replaced by x = 3 cos(−2t), y = 5 sin(−2t)

2. Use the sum, product, and/or quotient rules to compute the following derivatives. You may use any correct method, but must show work and simplify your answers for full credit.

(a) \frac{d}{dx}(5x\sqrt{x} − \frac{2}{x^{3}} + 11x − 4)

(b) \frac{d}{dt}(t^{2}e^{t})

(c) \frac{d}{dz} \frac{z^{2} − 2z + 4}{z^{2} + 1}
(d) \frac{d}{dx}(\frac{π^{2} + tan(e^{π}) − 2x^{e}}{4} )

3. The graph of a function f is shown below with several points marked. Find all the marked points at which the following are true, and give explanations for your answers.

A  B  C D E  F  G

(a) f is discontinuous.

(b) f is continuous, but the graph of f has a vertical tangent line.

(c) f is continuous, but the graph of f has no tangent line.

4. Compute the indicated limits. Show all work for full credit.

(a) lim_{x→1} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}
(b) lim_{x→2} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}
(c) lim_{x→∞} \frac{3x^{2} − 5x − 2}{x^{2} − 4x + 4}
(d) lim_{x→2^{+}} \frac{|x − 2|}{x^{2} − 5x + 6}

5. Let f(x) = x^{3} − x^{2}.

(a) Find all intervals on which f is decreasing.

(b) Find all intervals on which f is concave up.

(c) Find all intervals on which f is both increasing and concave down.

6. Do not use the differentiation rules from Chapter 3 in this question.

(a) State the limit definition of the derivative f′(x).

(b) Use the definition to compute the derivative function of f(x) = \frac{1}{3x}.

(c) Find the equation of the line tangent to the graph y = \frac{1}{3x} at x = 2.

7. The total cost (in $) of repaying a car loan at interest rate of r% per year is C = f(r). (a) What is the meaning of the statement f(7) = 20000?

(b) What is the meaning of the statement f′(7) = 3000? What are the units of f′(7)?