Mathematics 136, section 2 – Advanced Placement Calculus Exam 1 – October 1, 2009

Directions

Do all work in the blue exam booklet. There are 100 regular points, and 10 Extra Credit points possible. Please turn over this sheet for the graph for question I, and the Extra Credit question.

I. All parts of this question refer to the graph y = f(x) plotted in Figure 1 on the back of this sheet.

A. (5) Do lim_{x→2^{+}} f(x) and lim_{x→2^{−}} f(x) exist? If so, say what they are; if not say why not.

B. (5) Does f(x) appear to be differentiable at x = 3? Why or why not?

C. (10) On separate axes, plot the graphs y = 2f(x + 1) and y = f(2x) + 1. Label which graph is which.

II. At time t = 0, a steak at temperature 0^{∘} F is taken out of the freezer and left to thaw in a kitchen heated to 70^{∘} F. The steak’s temperature increases fastest at first, then at a slower and slower rate as it nears the room temperature; it never goes over 70^{∘} F.

A. (10) Sketch a graph of the temperature of the steak as a function of time. (For full credit, your graph should show the proper slopes and concavities.)

B. (5) A formula for the temperature as a function of t = time in hours is T = 70 − 70e^{−2t}. How long does it take for the temperature to reach 65^{∘}?

III. A. (5) What is the exact mathematical definition of the derivative of a function f(x) at x = a?

B. (10) Use the definition (not the “shortcut rules”) to compute the derivative of f(x) = \sqrt{x} at a general x > 0.

C. (5) What is the equation of the tangent line to y = \sqrt{x} at the point (4, 2)?

IV. Compute derivatives using the “shortcut rules”:

A. (5) f(x) = sin(x) + x^{5/3} − e^{x}.

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

C. (10) h(x) = sin^{−1}(ln(x)), and give the domain of h(x).

D. (10) Find \frac{dy}{dx} by implicit differentiation if cos(x + y^{2}) + tan^{−1}(x)y = 4.

V. (15) 100 cubic inches of cookie dough is being rolled out in a kitchen. The dough has the shape of a circular cylinder at all times, but the length of the cylinder increases and the radius decreases as times goes on. How fast is the radius of the dough cylinder changing if the length is increasing at 1 inch per second when the length is 10 inches?

Figure 1: Figure for Problem I

Extra Credit (10) The graph y = e^{x} is shifted 1 unit to the left. Can the same transformed graph be obtained by a vertical stretching or shrinking of y = e^{x}? If you say yes, give the stretching or shrinking factor. If you say no, explain why not.