Section 2.2.
1. Find the derivative of the following function.
f (x)  x4  x6
Use the product rule.
d 4 6
(x  x )  x4  6x5  x6  4x3  6x9  4x9  10x9
dx
2. Find the derivative of the following function. f (x) = x 2 (x 3 + 3)
Use the product rule.
f '(x)  x2  3x2  2x(x3  3)  3x4  2x4  6x  5x4  6x
3. Find the derivative of the following function. f (x) = √x (6x + 2)
Use the product rule.
f (x)  x1 2 (6x  2)
1 1 2
f '(x)  x  6  x (6x  2)  6x1 2  3x1 2  x  1 2  9x1 2  x 1 2
2
12
4. Find the derivative of the following function. f (x) = (x 2 + x) (3x + 1)
Use the product rule.
f '(x)  (x2  x)  3  (2x  1)  (3x  1)
 3x  3x  6x  5x  1  9x  8x  1
2
2
2
5. Find the derivative of the following function. f (x) = (2x 2 + 1) (1 - x)
Use the product rule.
f '(x)  (2x  1)  (  1)  (1  x)  (4x)
2
  2x2  1  4x  4x2   6x2  4x  1
6. Find the derivative of the following function.
4
3
f (t)  6t (3t
2
3
1)
Use the product rule.

4
3
f '(t)  6t  2t
1
3
 36t  8 t
1
3
1
3
2
3
 8 t (3t  1)  12 t  24t  8 t
1
3
7. Find the derivative of the following function. f (x) = (x 4 + x 2 + 1) (x 3 - 3)
Use the product rule.
f '(x)  (x  x  1)  3x  (x  3)  (4x  2x)
4
2
2
3
3
 3x6  3x4  3x2  4x6  2x4  12x3  6x
 7x  5x  12x  3x  6x
6
4
3
2
8. Find the derivative of the following function.
Use the quotient rule.
x8
y 
x2
d  x8  x2  8x7  x8  2x 8x9  2x9
6x9
5




6x
 2
dx  x 
((x2 ))2
x4
x4
9. Find the derivative of the following function.
x4  1
f(x)  3
x
Use the quotient rule.
x3  4x3  3x2 (x4  1) 4x6  3x6  3x2
f '(x) 

3 2
((x ))
x6
x6  3x 2
x4  3


6
x
x4
10. Find the derivative of the following function.
Use the quotient rule.

3x 1
f (x) 
2 x
11. Find the derivative of the following function.
Use the quotient rule.

s3 1
f (s) 
s 1
12. Find the derivative of the following function.
Use the quotient rule.

x 4  x 2 1
f (x) 
x 2 1

13. Economics: Marginal Average Revenue Use the Quotient Rule to find a general
expression for the marginal average revenue. That is calculate
d R(x)
[
] and simplify your answer.
dx x
14. Environmental Science: Water Purification If the cost (in cents) of purifying a gallon of
water to a purify of x percent is
for ( 50  x 100)
100
C( x) 
100  x
a. Find the instantaneous rate of change of the cost with respect to purity.
b. Evaluate this rate of change for a purity of 95% and interpret your answer.
c. Evaluate this rate
of change for a purity of 98% and interpret your answer
0  (100  x)  (100)  (  1)
100
a. C'(x) 

2
(100  x)
(100  x)2
100
100
b. C'(95) 

4
2
2
(100  95)
(5)
It will cost 4 cents per gallon to increase the purity from 95% to 96% pure.
100
100
c. C'(99) 

 25
2
2
(100  98)
(2)
It will cost 25 cents per gallon to increase the purity from 98% to 99% pure.
15. Environmental Science: Water Purification (14 continued)
a. Use a graphing calculator to graph the cost function C(x) from exercise
14 on the window [50,100] by [0,20]. TRACE along the curve to see how rapidly
costs increase for purity (x-coordinate) increasing from 50 to near 100.
b. To check your answer to 14, use the “dy/dx” or SLOPE feature of your calculator
to find the slope of the cost curve at x = 95 and x = 98, The resulting rates of change
of the cost should agree with your answer to Exercise 14(b) and (c). Note that
further purification becomes increasingly expensive at higher purity levels.
16. Business: Marginal Average Cost A company can produce LCD digital alarm clocks
at a cost of $6 each while fixed costs are $45. Therefore, the company’s cost function
C(x) = 6x+45.
a. Find the average cost function .
b. Find the marginal average cost function.
c. Evaluate marginal average cost function at x =30 and interpret your answer.
If you produce one more alarm clock, the 31st, the average cost will
decrease by 5 cents.
17. General: Body Temperature If a person;s temperature after x hours of strenuous
exercise is T (x) = x 3 (4 – x 2) + 98.6 degrees Fahrenheit for (0  x 2), find the rate of
change of the temperature after 1 hour.
18. General: Body Temperature (17 continued)
a. Graph the temperature function T(x) given in 17, on the window [0,2] by [90, 110].
TRACE along the temperature curve to see how the temperature rises and falls as time
increases
b. To check you answer to 17, use the “dy/dx” or SLOPE feature of your calculator to find
the slope (rate of change) of the curve at x =1. Your answer should agree with your
answer in 17.
c. Find the the maximum temperature.
19. Find the first four derivatives of f (x) = x 4 - 2x 3 – 3x2 + 5x - 7
20. Find the first four derivatives of f ( x) 
x5
21. Find the first and second derivatives of
and evaluate the second derivative at x = 3.

f ( x) 
x 1
x
22. Find the first and second derivatives of
f (x) 
and evaluate the second derivative at x = 3.

x 1
2x
23. Find the first and second derivatives of

f (x)  (x 2  2)(x 2  3)
24. Find the first and second derivatives of f (x) 
x
x1
25. Find the first and second derivatives of f (x) = r 2.
26. After t hours a freight train is s (t) = 18t 2 – 2t 3 miles due north of its starting
point (for 0 ≤ t ≤ 9).
a. Find its velocity at time t = 3 hours.
b. Find its velocity at time t = 7 hours.
c. Find its acceleration at time t = 1 hour.
a. Velocity = s’ (t) = 36 t – 6 t
2
b. Velocity = s’ (t) = 36 t – 6 t
. And s‘ (3) = 36 (3) – 6 (9) = 54 miles per hour.
2
. And s‘ (7) = 36 (7) – 6 (49) = - 42 miles per hour.
2.5 #33
c. acceleration = s” (t) = 36 – 12 t . And s” (1) = 36 – 12 = 24 miles per hour per hour.
27. If a steel ball is dropped from the top of the Taipei 101, the tallest building in the
world, its height above the ground t seconds after it is dropped will be
s (t) = 1667 – 16t 2 feet.
a. How long will it take to reach the ground?
b. Use your answer in part a to find the velocity at which it will strike
the ground.
c. Find the acceleration at any time t.
a. To find when the steel ball will reach the ground, we
need to determine what value of t produces s(t)=0. Thus,
set s(t)=0 and solve the equation by find the x-intercepts
(zeros) using our calculator.
The steel ball will reach the ground after 10.2 seconds.
b. The velocity is the derivative of the distance function
in part a. Use your calculator to find the derivative at x =
10.2 from part a. The velocity will be 326.4 feet per
second. About 222.5 mph.
c. The acceleration is the second derivative of the given
distance formula or s’ (t) = -32t and s” (t) = - 32 feet per
second per second.
28. ECONOMICS: National Debt The national debt of a South American country t
years from now is predicted to be D (t) = 65 + 9 t 4/3 billion dollars.
Find D’ (8) and D” (8) and interpret your answers.
D’ (t) = 12 t 1/3 and D’ (8) = 24 billion dollars per year. This is the amount the
national debt is expected to increase between the 8th and 9th year.
D” (t) = 4 t - 2/3 and D” (8) = 1 billion dollars per year. The rate of growth in the
national debt is expected to increase by 1 billion dollars per year after the 8th year.
29. GENERAL: Wind-chill Index the wind-chill index for a temperature of 32
degrees Fahrenheit and a wind speed of x miles per hour is
W (x) = 55.628 – 22.07 x 0.16.
a. Graph the wind-chill and find the wind-chill index for wind speeds of x = 15
and x = 30 miles per hour..
b. Notice from your graph that the wind-chill index has a first derivative that is
negative and a second derivative that is positive. What does this mean about
how successive 1-mph increases in wind speed will affect the wind-chill
index?
a.
W (15) = 21.6
W (30) = 17.6
b. Each 1-mph increase in wind speed lowers the wind-chill index. As wind speed
increases, the rate with which the wind-chill index decreases slows.