College Mathematics: Learning Worksheets Chapter 9
Step 4
:
()()
lim lim (2 5)
Px h Px xh
+− =++
c) (2) 2(2) 5 9P′=+= means that when the second item was sold, your profit is
5. The distance of a particle from some fixed point is given by
2
() 5 2,st t t
where t is time measured in seconds.
a) Find the average velocity from t = 4 to t = 6.
b) Find the instantaneous rate of change equation using the four-step procedure.
c) Using the equation found in part b, find the instantaneous rate of change when
t = 4 and interpret the results.
Solution:
a) First find the function values at 4 and 6:
(4)16202
(4) 38
s
s
(6) 36 30 2
(6) 68
s
s
Now use the average rate of change formula. Note that 64 2.h
College Mathematics: Learning Worksheets Chapter 9
250
b) Step 1
: 2
22
()()5()2
() 2 552
st h t h t h
st h t th h t h
c) (4) 2(4) 5 9s
means that after 2 seconds the particle is
College Mathematics: Learning Worksheets Chapter 9
Name ________________________________ Date ______________ Class ____________
Goal: To find the first derivatives using the basic properties
In Problems 1–6, find the indicated derivatives.
1.
y
for 5
y
x=
y
2. dy
dx for 4
1
x
y=
Section 9-5 Basic Differentiation Properties
Notation: If (),yfxthen (), ,dy
dx
fx y
all represent the derivative of f at x.
Theorems:
1. If () ,yfxCthen () 0fx
(Constant Function Rule)
n
3. If () (),
y
fx kux then () ()
f
xkux
(Constant Multiple Property)
4. If () () (),
y
fx ux vxthen () () ()
f
xuxvx
(Sum and Difference
Property)
252
3. 2
(2 3 8)
d
du uu
430
43
du du du du
u
u
4. ( )
f
x
if 3.8 1.7
() 6 8 2fx x x x=+ −+
2.8 0.7
2.8 0.7
3.8 6(1.7) 8 0
3.8 10.2 8
xx
xx
=+ −+
=+ −
5. 4
1
(6 58)
d
dx xxx
Use theorem 4 to break the original function into pieces, then use a combination of
theorems 1–3.
12
12
5
4
51
2
3
4
658
46() 50
5
dx dx dx dx dx
xx
dd dd
dx dx dx dx
x
x
xxx
xx
6. ( )
f
x
if 742
55 5
() 3 3 8fx x x x
−
=+ − −
Use theorem 4 to break the original function into pieces, then use a combination of
theorems 1–3.
7
21
555
76
12
55 5
xx x
−−
=+ +
253
7. Given the function 2
() 7 9 2fx x x
a) Find ().
f
x
b) Find the slope of the graph of f at 3.x
c) Find the equation of the tangent line at 3.x
d) Find the value of x where the tangent is horizontal.
Solution:
College Mathematics: Learning Worksheets Chapter 9
254
In Problems 8 and 9, use the following information for both problems:
If an object moves along the y axis (marked in feet) so that its position at time x (in
seconds) is given by the indicated function, find
a) The instantaneous velocity function ‘( )vfx
b) The velocity when 0xand 4x
c) The time(s) when 0v
8. 2
() 5 3 6fx x x=−+
Solution:
a) 2
2
() 5 3 6
() 5 ( ) 3 () (6)
fx x x
fx fx fx f
=−+
′′ ′′
=−+
255
9. 32
2
() 30
f
xx x x
Solution:
a) 32
21
2
32
21
2
() 30
() ( ) ( ) 30 ()
fx x x x
f
xfx fx fx
2
2
0 3 21 30
0 3( 7 10)
03( 5)( 2)
2,5
xx
xx
xx
x
10. Find ( )
f
x
if 2
() (5 6)fx x
First simplify the function by using the FOIL to expand it.
22
( ) (5 6) (5 6)(5 6) 25 60 36fx x x x x x
Now use theorem 4 to break the original function into pieces, then use a combination
of theorems 1–3.
2
( ) 25 ( ) 60 ( ) (36)
( ) 50 60
fx fx fx f
fx x
256
11. Find ( )
f
x
if 87
() .
x
x
fx −
=
First simplify the function.
1
87 8 7
() 8 7
xx
xxx
f
xx
−
−
==−=−
Now use theorem 4 to break the original function into pieces, then use a combination
of theorems 1–3.
1
−
College Mathematics: Learning Worksheets Chapter 9
257
Name ________________________________ Date ______________ Class ____________
Goal: To use differentials to solve problems
1. Given the function 3
2,
y
x=find , , and y
x
xy
given 12xand 25.x
2. Given the function 3
2,
y
x=find , , and y
x
xy
given 13xand 26.x
3
x
378
y
Δ=
Section 9-6 Differentials
Definition: Differentials
If ()yfxdefines a differentiable function, then the differential dy or df is defined as
the product of ()
f
x
and dx, where .dx x Symbolically,
where .dx x
Recall that ( ) ( )
y
fx x fx
258
3. Given the function 32
312 416,yx x x find .dy
2
2
9244
(9 24 4)
dy
dx
xx
dy x x dx
4. Given the function 4
3
9
(7 ),
x
yx=+find .dy
3
9
37
1
9
26
7
9
26
7
9
7
21
(21 )
x
dy
dx
yx x
xx
dy x x dx
=+
=+
=+
5. Given the function 23
36 5 2 ,
y
xx=− + find and dy y given 4xand 0.1.dx x
259
6. A company will sell N units of a product after spend x thousand dollars in advertising,
as given by
2
120Nxx 10 60x
Approximately what increase in sales will result by increasing the advertising budget
from $15,000 to $17,000? From $25,000 to $27,000?
7. The average pulse rate y (in beats per minute) of a healthy person x inches tall is
given approximately by
590
y
x
30 75x
Approximately how will the pulse rate change for a change in height from 49 inches
to 52 inches?
12
590 590
y
x
x
College Mathematics: Learning Worksheets Chapter 9
260
College Mathematics: Learning Worksheets Chapter 9
Name ________________________________ Date ______________ Class ____________
Goal: To solve problems involving marginal functions in business and economics
In Problems 1–10, find the indicated function if cost and revenue are given by
2
( ) 600 4 0.0006Cx x x=−+ and 2
( ) 1, 000 20
R
xxx=−
1. Marginal cost function
( ) 4 0.0012
Cx x
′=− +
2. Average cost function
() 600 4 0.0006 600
Cx xx
xxx
−+
Section 9-7 Marginal Analysis in Business and
Economics
Definition: Marginal Cost, Revenue, and Profit
If x is the number of units of a product produced in some time interval, then
total cost = ( )Cx and marginal cost = ( )Cx
Definition: Marginal Average Cost, Revenue, and Profit
If x is the number of units of a product produced in some time interval, then
Cost per unit: average cost = ()Cx
x
C and marginal average cost = () ()
d
dx
Cx Cx
R
x
x
dx
263
10. Marginal average profit function
2
() () ()
x
Px Rx Cx
′′′
=−
11. Consider the revenue (in dollars) of a stereo system given by
1000
( ) 1000 .
x
R
xx
a) Find the exact revenue from the sale of the 101st stereo.
b) Use marginal revenue to approximate the revenue from the sale of
the 101st stereo.
Solution:
a) To find the exact revenue, find the revenue from the 101st and 100th and
subtract their values:
1000
( ) 1000
x
Rx x
1000
( ) 1000
x
Rx x
b) Find the marginal revenue formula and then substitute in 100.
1000
( ) 1000
x
Rx x
2
1000
( ) 1000
x
Rx
265
13. The total profit (in dollars) from the sale of x units of a product is
2
( ) 30 0.03 200.Px x x
a) Find the exact profit from the 201st unit sold.
b) Find the marginal profit from selling the 201st unit.
Solution:
a) To find the exact profit, find the profit from the 201st and 200th and subtract
their values:
b) Find the marginal profit formula and then substitute in 200.
266
14. The total cost and revenue (in dollars) for the production and sale of x units are given,
respectively, by
( ) 32 36,000Cx x and 2
( ) 300 0.03 .
R
xxx
a) Find the profit function P(x).
b) Determine the actual cost, revenue, and profit from making and
selling 101 units.
c) Determine the marginal cost, revenue, and profit from making and
selling the 101st unit.
Solution:
a)
() () ()
Px Rx Cx
b) ( ) 32 36, 000
Cx x
( ) 32 36,000
Cx x
College Mathematics: Learning Worksheets Chapter 9
c) Marginal functions would be
( ) 32 36,000
() 32
Cx x
Cx
2
( ) 300 0.03
( ) 300 0.06
R
xxx
R
xx
( ) (300 0.06 ) (32)
( ) 0.06 268
Px x
Px x
Marginal values for the 101st unit are
() 32
Cx
( ) 300 0.06
Rx x
College Mathematics: Learning Worksheets Chapter 9
268