Spring 2006

100 points possible.

The figures for Problems 1, 2, and 3 are on a supplemental page.

1. (4 pts.) For the function g whose graph is given on the supplemental page, state the following.

lim x® 1+  g(x)

A. 1

B. 2

C. 2.6

D. ¥

E. Does not exist.

2. (4 pts.) The graph of a function f is given on the supplemental page. Select the true statement from the choices listed below the graph.

A. f(0)=5, f¢(0)=-1, f¢(1)=3, f¢(2)=-5

B. f(0)=5, f¢(0)=-1, f¢(1)=-3, f¢(2)=-5

C. f(0)=5, f¢(0)=-1, f¢(1)=-3, f¢(2)=5

D. f(0)=5, f¢(0)=-1, f¢(1)=3, f¢(2)=5

E. f(0)=3, f¢(0)=1, f¢(1)=4, f¢(2)=6

3. (4 pts.) Using the graph of h on the supplemental page, estimate the value of the derivative at the point x=0.

A. 0

B. 4

C. -4

D. 2

E. -2

4. (4 pts.) A rectangle has perimeter 10 m. Express the area of the rectangle as a function A(l) of the length l of one of its sides.

A. A(l)=5l+l²

B. A(l)=10l-l²

C. A(l)=5-l

D. A(l)=10l+l²

E. A(l)=5l-l²

5. (4 pts.) The monthly cost of driving a car depends on the number of miles driven. Lynn found that in October it cost her $500 to drive 600 mi, and in July it cost her $650 to drive 900 mi. Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model.

A. C=0.5d-200

B. C=200d+0.5

C. C=200d-0.5

D. C=0.5d+200

E. C=0.5d-100

6. (4 pts.) Determine an appropriate viewing rectangle for the given function.

f(x)=sin æ ç è x 70 ö ÷ ø

A. [0,1] by [0,70]

B. [-490,490] by [-1.5,1.5]

C. [-1,0] by [-70,0]

D. [-0.14,0.14] by [-1.5,1.5]

E. [-70,70] by [-70,70]

7. (4 pts.) If an arrow is shot upward on the moon with a velocity of 55 m/s, its height in meters after t seconds is given by h=55t-0.04t². Find the average velocity over the interval [1,1.04].

A. 54.9194

B. 55.0284

C. 54.8174

D. 54.9184

E. 54.9084

F. 54.9284

8. (4 pts.) Evaluate the limit, given that lim_{x® 7} f(x)=-3 and lim_{x® 7} g(x)=5

lim x® 7  f(x) g(x)

A. 1

B. 3/5

C. -3/5

D. -7/5

E. -5/7

9. (4 pts.) Choose an equation from the following that expresses the fact that a function f is continuous at the number 6.

A. lim_{x® ¥} f(x)=6

B. lim_{x® 6} f(x)=f(6)

C. lim_{x® ¥} f(x)=f(6)

D. lim_{x® 6} f(x)=0

E. lim_{x® 6} f(x)=¥

10. (4 pts.) A curve has equation y=h(x). Write an expression for the slope of the secant line through the points P(8,h(8)) and Q(x,h(x)).

11. (12 pts.) Let f(x)=x³+1 and g(x)=1/x.

(a) Find the function f°g and simplify.

(b) Find the function f°f and simplify.

12. (12 pts.) Evaluate the limit, if it exists. Your work must completely justify your answer algebraically. Please be neat.

lim x® 2  x2-x-2 x2-4x+4

13. (12 pts.) Evaluate the limit, if it exists. Your work must completely justify your answer algebraically. Please be neat.

lim h® 0    ____ Ö2+5h   -Ö2 h

14. (12 pts.) Use the limit definition of derivative to show that the derivative of

f(x) = 3x2-2x+5

is f¢(x)=6x-2. Your work must completely justify your answer algebraically. Please be neat.

15. (12 pts.) Use the limit definition of derivative to find the derivative of

f(x) = x 3-x .

Your work must completely justify your answer algebraically. Please be neat.

Optional Bonus Problem. (8 optional bonus points possible) Suppose that f(x)=x³+3x-2, prove that there is at least one real number c such that f(c)=0.