Mathematics 134 – Intensive Calculus for Science 2 Midterm Exam 3 – April 28, 2006

Directions: Do all work in the blue exam booklet. You may use an approved calculator, and the table of integrals provided. There are 100 possible points.

I. A) (10) Which of the following series is geometric:

series A : 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + ⋯ series B : \frac{3}{2} + \frac{3}{4} + \frac{3}{8} + \frac{3}{16} + ⋯

Explain how you can tell. B) (10) The sum of the geometric series

(b/2) + (b/2)^{2} + (b/2)^{3} + (b/2)^{4} + ⋯

is equal to 3. What is the value of b?

II. A) (15) Using the definition, compute the Taylor polynomial of degree n = 4 for f(x) =

cos(2x) at a = 0. B) (15) Using any applicable method, find the first three non-zero terms in the Taylor

series for g(x) = (1 + x)^{1/3} at a = 0.

III. A) (10) Which of the following two slope field plots corresponds to the differential equation

\frac{dy}{dx} = (y − 1)(y − 5). Explain how you can tell.

B) (10) Give qualitative sketches of the solutions with y(0) = 3 for each of the differential equations whose slope fields are shown in part B. (Show the graph in the blue book – you do not need to recopy the slope field).

C) (10) Find the general solution of the differential equation \frac{dy}{dx} = \frac{y}{x(x+1)}.

IV. A culture of bacteria contains 200 cells originally and grows at a rate proportional to its size. After half an hour the population has increased to 360 cells. A) (10) Find the function P (t) giving the population after t hours. B) (10) When will the population reach 10000 cells?