College of the Holy Cross, Fall 2007 Math 131, Midterm 2 (All Sections) Wednesday, October 24, 6 PM

Your Name:

Your Section:

Little (8:00am) Ballantine (9:00am) DeStefano (10:00am) DeStefano (noon)

Instructions: Please write your answers in the boxes provided, and show work on the test pages. Use the back of the preceding page if you need more space for scratch work. The numbers next to each part of the questions are their point values.

Please do not write in the space below

Problem Points/Poss
I  / 15
II  / 20
III  / 15
IV  / 20
V  / 10
VI  / 20
Total  /100

I. A. Consider the parametric curve x = 4 cos(t), y = 4 sin(t).

(4) 1. Give a Cartesian equation for this curve (eliminate the parameter t to obtain an equation in x and y) and describe the shape of the curve.

(4) 2. What portion of the curve is traced out for 0 ≤ t ≤ \frac{π}{4}, and in which direction is the curve being traced? (Either explain in words or draw a picture.)

Portion of curve: Direction:

B. Now consider the parametric curve x = cos(4t), y = sin(4t).

(3) 1. Give a Cartesian equation for this curve and describe the shape of the curve.

(4) 2. What portion of the curve is traced out of 0 ≤ t ≤ \frac{π}{4}, and in which direction is the curve being traced? (Either explain in words or draw a picture.)

Portion of curve: Direction:

II. Apply the rules for differentiation to find the derivatives of the functions given below. Do NOT simplify your answers after completing the derivative (except to clean up your answers).

(4) A. f(x) = x^{3} + 5x^{\frac{2}{3}} + \sqrt{2}

(4) B. g(x) = x^{\frac{5}{2}} − \frac{1}{x} + 7

(4) C. R(z) = (5z^{2} − 2z^{4})(z^{3} − 1)

(4) D. s(t) = \frac{\sqrt{t} − 2e^{t}}{e^{t} + t}

(4) E. P (x) = e^{x} − x^{e} + e^{π} III. The following graph shows y = f′(x) for some function f(x).

(3) A. Over what intervals is f(x) increasing? Over what intervals is f(x) decreasing?

(3) B. Over what intervals is f(x) concave up? Over what intervals is f(x) concave down?

(3) C. What happens at x = 1 on the graph y = f(x)?

(6) D. On the next page, sketch the graph y = f(x) if it is given that f(0) = 0.

This page for graph from Problem III D 5 (4) A. State the limit definition of the derivative f′(x).

(6) B. Use the definition to compute the derivative of f(x) = x^{2} + 2x + 1. (Do not use the derivative rules.)

(2) C. Check your answer to part (b) by finding f′(x) using the rules for differentiation. State each rule that you use.

(4) D. Find the equation of the tangent line to the curve y = x^{2} + 2x + 1 at x = −3.

(4) E. The position (in meters) of a moving object at time t (in seconds) is given by y = f(t) = t^{2} + 2t + 1. What is the physical meaning of the statement f′(2) = 6 and what are the units of f′(2)?

V. The following graph shows y = f(x) for some function f(x).

(5) A. At which x is f(x) discontinuous? Justify your answer using the definition of continuity.

(5) B. At which x is f(x) continuous but not differentiable? Explain.

VI. Compute the following limits. Show all work.

(5) A. lim_{x→π} sin (\frac{x}{2} − π)
(5) B. lim_{x→10} \frac{\sqrt{x − 6} − 2}{x − 10}
(5) C. lim_{x→1^{−}} \frac{|x − 1|}{x^{2} − 3x + 2}

(5) D. Find all vertical and horizontal asymptotes to the graph y = \frac{x^{2} + x − 2}{x^{2} + 5x + 6}.