Spring 2006

Show all work in a neat and organized fashion. Clearly indicate your answers.

20 points possible.

1. (10 pts.) Given:

f(x) = unknown continuous function, with domain the set of all real numbers

f¢(x) = (5x+3)(x+3)3,             f¢¢(x) = 4(5x+6)(x+3)2

(a) Find the critical numbers of f.

(b) Find the intervals on which f is increasing or decreasing.

(c) Find the x-coordinates of all local maxima and local minima of f.

(d) Find the intervals of concavity.

(e) Find the x-coordinates of all points of inflection.

2. (5 pts.) Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval [a,b]. Then find all numbers c such that

f¢(c)= f(b)-f(a) b-a .

f(x)=x3-8x,       [-2,6]

3. (5 pts.) Find the following limits.

(a)

lim x®+¥  x+3 x2-9

(b)

lim x®-¥  x+3 x2-9

(c)

lim x® 3+  x+3 x2-9

(d)

lim x®3-  x+3 x2-9

(e)

lim x® -3  x+3 x2-9