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.”
Page 1 of 4
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.)
Page 8 of 4