Spring 2006

100 points possible.

1. (10 pts.) Let f(x)=x³-9x²-48x+5. Find the absolute maximum and absolute minimum values of f on the interval [-4,3].

2. (10 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+7x,       [-6,2]

3. (15 pts.) Given:

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

f¢(x) = (7x+4)(x+4)5,             f¢¢(x) = 6(7x+8)(x+4)4

(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.

4. (10 pts.) Find the following limits exactly. Provide all major algebraic/symbolic steps to justify that your answer is correct.

(a)

lim x® +¥  1-3x-4x2 11x2-7

(b)

lim x® +¥  12x3+11   _______ Ö7x6+5x

5. (15 pts.) Given:

f(x) = 18x4-134x3+175x2+632x+852,

f¢(x) = 72x3-402x2+350x+632,

f¢¢(x) = 216x2-804x+350.

Use a graphing calculator to estimate these answers to two decimal places.

(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.

6. (10 pts.) A piece of wire 20 meters long is cut into two pieces. One piece is bent into a square, and the other is bent into a rectangle whose length is twice its width. How should the wire be cut so that the total area enclosed is (a) a maximum? (b)  a  minimum?

7. (10 pts.) Use Newton's method to find the root of the given equation to the best accuracy your calculator will display. Use the specified initial approximation  x₁, and show your values for x₂, x₃, x₄, and so on, until the algorithm stabilizes.

2x3+x2-x+1=0,       x1=-1.2

Recall that

x2=x1- f(x1) f¢(x1)

and

xn+1=xn- f(xn) f¢(xn) .

8. (10 pts.) Find the most general antiderivative of the given function.

(a)

f(x)=sec2 x

(b)

f(x)= 25x2+8 x5

9. (10 pts.) Find f(x), given the following.

f¢¢(x)=36x2-12x+10,       f(2)=5,     f¢(2)=4