Mathematics 136, section 4 – Exam 1 – September 30, 2003

Directions: Answer questions I - V below in the blue exam booklet. There are 100 ordinary and 5 Extra Credit points, distributed as indicated.

I. A. (10) Find a possible formula for the function y(t) plotted below:

B. (10) Suppose you “zoomed in” on the graph of your function near t = 0. Would the graph look locally linear? If so, what would the equation of the line be? If not, say why not. (Use your formula from A, but don’t do any actual “zooming”.)

II. A population of wildebeest grows in a habitat that can support no more than P_{1} animals. At time t = 0, the number of wildebeest present is P_{0} < P_{1}. The population grows more and more rapidly at first, but then the rate of growth decreases and the population approaches P_{1}, without exceeding it. A. (10) Sketch a graph of the population of wildebeest as a function of time that fits this

description. B. (10) Give a qualitative graph of the derivative of the population function.

III. A. (10) 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) =

4x^{2} − 2x + 2 at a general x.

IV. Find the derivative functions of each of the following functions defined by formulas: A. (5) f(x) = 2x^{3} + 6\sqrt{x} − \frac{3}{\sqrt[3]{x}} + e^{π}

B. (7.5) f(x) = \frac{sin(x)}{1+cos(x)} C. (7.5) g(x) = arctan(e^{x} − 2^{x})

V. (20) Sketch the parametric curve whose coordinate functions x = f(t) and y = g(t) are given in the following plots:

(Over for extra credit problem.)

Extra Credit (10) The graph below shows y = f′(x), the derivative of a function f(x). Sketch the graph y = f(x) using the information given. Indicate on your sketch any local maximum, local minimum, or inflection points that f(x) has.