Math Help

1. (4 points) Use the graph of y = f(x) below to evaluate the following.

lim x→6−

f(x) = lim x→6+

f(x) = lim x→6

f(x) = f(6) =

2. (4 points) Consider the following function:

f(x) =

  x + a if x < 4

20 if x = 4

b−x if x > 4

What values for a and b make f(x) continuous at x = 4?

3. (4 points) Consider the table below.

x f(x) g(x) f ′(x) g′(x)

7 50 2 −1 3

(a) Find u′(7) if u(x) = f(x) ·g(x).

(b) Find q′(7) if q(x) = f(x)/g(x).

4. (4 points) Let f(x) = sin(4g(x)). Evaluate f ′(x).

5. (2 points) What is the 100th derivative of 5x99? Explain your answer.

6. (a) (4 points) Find the equation of the line that is tangent to y = 8−x4 at x = 1.

(b) (2 points) Use your answer from the previous part to approximate 8−(1.01)4

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7. (6 points) Determine the positive real numbers x and y such that

• x + y = 4 and

• 2x2 + y2 is as small as possible.

8. (4 points) Suppose that f(x) is a function such that f ′(x) = x3 − 6x2 + 9x

x2 + 5 . Find

the interval(s) on which f(x) is increasing.

9. (6 points) Consider the following information about the graph of a continuous function m(x):

• limx→−∞ m(x) = −2 and limx→∞ m(x) = 2

• The table below summarizes when m′ and m′′ are positive and negative, but with some information unknown:

x < −2 −2 < x < −1 −1 < x < 0 0 < x < 1 1 < x < 2 2 < x m′(x) − − + + − − m′′(x) ? + + − − ?

Copy or print the axes below and sketch a graph of m(x) with −5 < x < 5 that matches the information.

10. (4 points) The rocketship Magnificence takes off from its launch pad with velocity function v(t) = 12+80e−t. Its height at time t = 0 is h(0) = 150. Find the rocket’s height function h(t).

11. (4 points) The picture below shows the graph of y = 4 + sin x. Using a Riemann

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sum with n = 5 and using right endpoints as your sample points, estimate the area below y = 4 + sin x and above the interval [4, 14]. Round your answer to the nearest tenth. (Use a scientific calculator in radians mode.)

12. (4 points) The following is a graph of y = f(t):

Let g(x) =

∫ x 0

f(t) dt and use the graph to answer the following. Show your

reasoning.

(a) g′(4) =

(b) g′′(4) =

5

13. (4 points) The following is a graph of y = f(x).

Use the graph to evaluate

∫ 10 4

f(x) dx.

14. (4 points) Evaluate the following definite integral using the Fundamental Theorem

of Calculus:

∫ 5 1

( 5 +

1

x2

) dx

15. (4 points) Evaluate the following indefinite integral using Substitution:

∫ x √

4x2 + 4 dx

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