Math 18 Exam #2 Ch. 2.3 - 4.2 Spring 2011

NAME ____________________________________________

Show important algebraic steps. Partial credit can be given only if work is CLEARLY & CORRECTLY shown. If writing a sentence, use complete, correct English sentences. 1. Given r = f(w),

( 2 pts. each)

a) write the derivative r′ = f ′(w) in Leibniz notation.

1.a) _________________

b) write the second derivative r′′ = f ′′(w) in Leibniz notation. 2.

The average weight W of a oak tree in kilograms that is t meters tall is given by the function W = f(t). What are the units of

3.

b) _________________

dW ? dt

___________________________

(2 pts.)

The number N of acres harvested t years after farming began in the region is given by N = f(t) = 120 a)

Find f(9)

t .

(2 pts.)

2.a) __________ b) Interpret in the context of this problem the meaning of your answer in part a) above. Include units. (4 pts.)

c) Find f ′(9).

(4 pts.)

2.c) __________ c)

Interpret in the context of this problem the meaning of your answer in part c) above. Include units. (4 pts.)

4. Sketch the graph of a function that satisfies the following conditions.

(4 pts.)

“Second derivative everywhere negative and first derivative everywhere positive.”

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Math 18 Exam #2(a) Ch. 2.3 - 4.2 Spring 2011

5.

NAME __________________________________________

Use the given graph to answer the following questions . (3 pts. each) a)

At which of the labeled point(s), if any, is y′ positive & y′′ negative? ____________________

b) At which of the labeled point(s), if any, is y′ = 0? __________________ c) At which of the labeled point(s), if any, y′′< 0? _____________ 6. The revenue R and profit P functions for the production of q table saws are given by:

1 2 R(q) = 200q q 30 a)

−1 2 P(q) = q + 140q − 72000 30

(3 pts. each)

Find the marginal revenue function MR.

6.a) _____________________________ b) Using the specific results given, interpret the following in the context of this problem. Include units! R′(1500) = 100 means: ___________________________________________________________________ ___________________________________________________________________ P′(1500) = – 60 means: ___________________________________________________________________ ___________________________________________________________________ 7. The temperature H, in degrees Celsius of a bottle of water put into the refrigerator for t minutes is given by -0.02t H = 4 + 16e . a) Find the rate at which the temperature of the bottle of water is changing (in °C/minute) at any time t. (4 pts.)

7.a) _________________ b) How fast is the water cooling initially (t = 0) Include units?

(2 pts.)

b) _____________________

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8.

t In 2009, the population of Hungary was approximated by: P = 9.906(0.997) where P is in millions and t is in years since 2009. a) Find

dP , the rate of change of the population with respect to time t. dt

(5 pts.)

4.a) ___________________________________ b) What is the predicted rate of change of the population in the year 2019 (t = 10)? Use a calculator and round to 2 decimal places. Include appropriate units and indicate if the population is increasing or decreasing. (3 pts.)

b) ____________________________________ 9. Find the first derivative of the following functions. (4 pts.) a) y =

3 −7 x x11

9.a) ____________________ (5 pts.) b) y = ln(2x -9)

b) ____________________ (6 pts.) c) y =

e x ⋅ x 2 (Use the product rule)

c) ____________________

(6 pts.) d) y =

2x − 5 3x + 1

(Use the quotient rule)

d) ____________________ Page 3 of 4

f (x) = 2(2x −1)3 and f ′(x) = 12(2x −1)2 . a) Find f (1) and f ′(1) . (1 pt. each) 10.a) f (1) = _______

10. Given

f ′(1) = _______

b) Write the equation of the tangent line to f at the point where x = 1. (5 pts.)

b) ________________________________ 11. Fill in the blank with a word or words to make the following statement true. (2 pts.) “Graphically, at a critical point p the line tangent to the graph at p is

_______________________. “

12 . A function and its first and second derivatives are given below: 1 3 2 2 f(x) = − x + 3x - 5x; f ′(x) = − x + 6x - 5; f ′′(x) = −2x + 6. 3 a) x =1 is a critical point. Use the First Derivative Test to identify it as a local maximum, local minimum or neither. CLEARLY show work and justify your answer! (5 pts.) sign of y′ f inc. or dec.

__________________|_________________ x=1

Thus, f has a __________________________ at x = 1. local max or local min or either

b) x = 5 is also a critical point. Use the Second Derivative Test to identify it as a local maximum, local minimum or neither. CLEARLY show work and justify your answer! (5 pts.)

Thus, f has a __________________________ at x = 5. local max or local min or either

2 3 2 13. Given f(x) = x + 3x - 4 and f ′(x) = 3x + 6x = 3x(x + 2). So f ′(x) = 0 if x = 0 or x = – 2 Use either the First or Second Derivative Test to find where f has a local maximum or local minimum. Clearly indicate which test you are using, test BOTH critical points, and clearly show work. (4 pts.)

Thus, f has a local __________ at x =____ and a local __________ at x =____ Page 4 of 4

VERSION 2(a) Show important algebraic steps. Partial credit can be given only if work is CLEARLY & CORRECTLY shown. If writing a sentence, use complete, correct English sentences. 1. Given w = f(r),

( 2 pts. each)

a) write the derivative w′ = f ′(r) in Leibniz notation.

1.a) _________________

b) write the second derivative w′′ = f ′′(r) in Leibniz notation. 2.

The average weight W of a oak tree in pounds that is t yards tall is given by the function W = f(t). What are the units of

3.

b) _________________

dW ? dt

___________________________

(2 pts.)

The number N of acres harvested t years after farming began in the region is given by N = f(t) = 120 d) Find f(16)

t .

(2 pts.)

2.a) __________ e)

Interpret in the context of this problem the meaning of your answer in part a) above. Include units. (4 pts.)

c) Find f ′(16).

(4 pts.)

2.c) __________ f)

Interpret in the context of this problem the meaning of your answer in part c) above. Include units. (4 pts.)

4. Sketch the graph of a function that satisfies the following conditions.

(4 pts.)

“Second derivative everywhere positive and first derivative everywhere negative.”

Page 5 of 4

5.

Use the given graph to answer the following questions . (3 pts. each) a)

At which of the labeled point(s), if any, is y′ negative & y′′ positive? ____________________

b) At which of the labeled point(s), if any, y′′< 0? _____________ c) At which of the labeled point(s), if any, is y′ = 0? __________________ 6. The revenue R and profit P functions for the production of q table saws are given by:

1 2 R(q) = 200q q 30

−1 2 P(q) = q + 140q − 72000 30

(3 pts. each)

b) Find the marginal revenue function MR.

6.a) _____________________________ b) Using the specific results given, interpret the following in the context of this problem. Include units! R′(1500) = 100 means: ___________________________________________________________________ ___________________________________________________________________ P′(1500) = – 60 means: ___________________________________________________________________ ___________________________________________________________________ 7. The temperature H, in degrees Celsius of a bottle of water put into the refrigerator for t minutes is given by -0.02t H = 4 + 16e . a) Find the rate at which the temperature of the bottle of water is changing (in °C/minute) at any time t. (4 pts.)

7.a) _________________ b) How fast is the water cooling initially (t = 0)? Include units.

(2 pts.)

b) _____________________

Page 6 of 4

8.

t In 2009, the population of Hungary was approximated by: P = 9.906(0.997) where P is in millions and t is in years since 2009. a) Find

dP , the rate of change of the population with respect to time t. dt

(5 pts.)

4.a) ___________________________________ b) What is the predicted rate of change of the population in the year 2020 (t = 11)? Use a calculator and round to 2 decimal places. Include appropriate units and indicate if the population is increasing or decreasing. (3 pts.)

b) ____________________________________ 9. Find the first derivative of the following functions. (4 pts.) a) y =

3 −7 x x11

9.a) ____________________ (5 pts.) b) y = ln(3x -7)

b) ____________________ (6 pts.) c) y =

e x ⋅ x 2 (Use the product rule)

c) ____________________

(6 pts.) d) y =

2x − 5 3x + 1

(Use the quotient rule)

d) ____________________

Page 7 of 4

f (x) = 2(2x −1)3 and f ′(x) = 12(2x −1)2 . a) Find f (1) and f ′(1) . (1 pt. each) 10.a) f (1) = _______

10. Given

f ′(1) = _______

b) Write the equation of the tangent line to f at the point where x = 1. (5 pts.)

b) ________________________________ 11. Fill in the blank with a word or words to make the following statement true. (2 pts.) “Graphically, at a critical point p the line tangent to the graph at p is

_______________________. “

12 . A function and its first and second derivatives are given below: 1 3 2 2 f(x) = − x + 3x - 5x; f ′(x) = − x + 6x - 5; f ′′(x) = −2x + 6. 3 a) x =1 is a critical point. Use the First Derivative Test to identify it as a local maximum, local minimum or neither. CLEARLY show work and justify your answer! (5 pts.) sign of y′ f inc. or dec.

__________________|_________________ x=1

Thus, f has a __________________________ at x = 1. local max or local min or either

b) x = 5 is also a critical point. Use the Second Derivative Test to identify it as a local maximum, local minimum or neither. CLEARLY show work and justify your answer! (5 pts.)

Thus, f has a __________________________ at x = 5. local max or local min or either

2 3 2 13. Given f(x) = x + 3x - 4 and f ′(x) = 3x + 6x = 3x(x + 2). So f ′(x) = 0 if x = 0 or x = – 2 Use either the First or Second Derivative Test to find where f has a local maximum or local minimum. Clearly indicate which test you are using, test BOTH critical points, and clearly show work. (4 pts.)

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