PARTIAL DERIVATIVES



FINAL EXAM REVIEW

A. Compute the following limits:

1. [pic] 2. [pic]

3. Evaluate each of the following expressions.

[pic]

a) [pic] [pic] [pic] [pic]

b) [pic] [pic] [pic] [pic]

c) [pic] [pic] [pic] [pic]

B. TANGENT LINES: Find the equation of the tangent line for the following:

1. [pic] at (0, 9) 2. [pic] at (2, 8) 3. [pic] at (1, 0)

C. MAX, MIN, INFLECTIONS, INCREASING/DECREASING, CONCAVITY.

1. Find the relative min and max, and inflection points for [pic] State the intervals of

increase, decrease for the function. Find the intervals where the function is concave up and concave down.

2. Find the relative min and max, and inflection points [pic] State the intervals of

increase, decrease for the function. Find the intervals where the function is concave up and concave down.

3. A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 90 ft, find the dimensions of the window so that the greatest amount of light is admitted.

D. EXPONENTIAL MODELS

The population growth of a small city can be modeled by the model [pic]

where t is time in years.

1. How long will it take for the population double?

2. What is the instantaneous rate of change of the population?

E. DIFFERENTIATION – power rule, product rule, quotient rule, chain rule.

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

F. INTEGRATION

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] where [pic] [pic]

7. Copper from a mine is extracted at a rate [pic] tons per year.

a. How much copper is removed in the first 3 years?

b. How much copper is removed in the 3rd year?

G. AREA BETWEEN THE CURVES

1. Find the area between the curves [pic] and [pic] for [pic]

2. Find the area between the curves [pic] and [pic].

H. PARTIAL DERIVATIVES

1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic]

Compute 1 thru 6 for each of the following functions:

A. [pic] C. [pic]

B. [pic] D. [pic]

E. [pic] F. [pic]

G. [pic] H. [pic]

7. The productivity of a computer manufacturer is given by the Cobb-Douglass function

[pic] where x stands for the utilization of labor and y is units of capital.

a. Compute the marginal productivity of labor at 4000 units of labor and 2500 units of

capital, i.e. find[pic].

b. Compute the marginal productivity of capital at 4000 units of labor and 2500 units

of capital, i.e. find[pic]

8. Given the function [pic]find all the values of x

and y that will make [pic] and [pic] simultaneously.

9. Poiseuille’s Law states that the resistance R for blood flowing in a blood vessel varies

directly as the length L of the vessel and inversely as the fourth power of its radius r.

Stated as an equation, [pic] where k is a constant. Find [pic] and [pic]

ANSWERS

A. Compute the following limits:

1. [pic]

2. [pic]

3. a) –2, –2, –2, –1 b) 3, 4, DNE, 4 c) –2, –2, –2, UND

B. TANGENT LINES

1. y = 33x + 9 2. y = 12x – 16 3. y = x – 1

C. MAX, MIN, INFLECTIONS, INCREASING/DECREASING, CONCAVITY.

1. Relative min: (–1, –1) and (1, –1) Relative max: (0, 1)

Inflections: [pic] and [pic]

Increasing: [pic], Decreasing: [pic]

Concave up: [pic] Concave down: [pic]

2. Relative min: (7, –145) Relative max: (–4, 227/3) Inflection: [pic]

Increasing: [pic], Decreasing: ( –4, 7)

Concave up: [pic] Concave down: [pic]

3. [pic] [pic]

[pic] [pic]

[pic] [pic]

Width = 2x = [pic] ft, Length = y = [pic] ft

D. EXPONENTIAL FUNCTIONS

1. ln(2)/0.025 = 27.7 years 2. P’(t) = 3000[pic]

E. DIFFERENTIATION

1. [pic] 2. [pic]

3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

F. INTEGRATION

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. a. [pic] tons

b. [pic]tons

G. AREA BETWEEN THE CURVES

1. [pic]

2. [pic]

H. PARTIAL DERIVATIVES

A. 1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

B. 1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic]

6. [pic]

C. 1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic]

6. [pic]

D. 1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic]

6. [pic]

E. 1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic] 6. [pic]

F. 1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic]

G 1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic]

6. [pic]

H 1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. a. [pic] [pic]

b. [pic] [pic]

8. [pic] (2, 4)

9. a. [pic] b. [pic]

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