College Mathematics: Learning Worksheets Chapter 10
Name ________________________________ Date ______________ Class ____________
Goal: To work with the constant eand solve continuous compound problems
1. Use a calculator to evaluate A to the nearest cent.
0.05
700 t
Ae= for 3, 6, 9t
0.05(3)
0.15
700
813.28
Ae
A
=
=
0.05(6)
0.3
700
944.90
Ae
A
=
=
0.05(9)
0.45
700
1097.82
Ae
A
=
=
Section 10-1 The Constant e and Continuous
Compound Interest
Definition: The number e
1
lim 1
n
n
en
or
1
lim 1
s
s
es
Theorem 1: Continuous Compound Interest Formula
If a principal P is invested at an annual rate r (expressed as a decimal) compounded
270
2. Solve for t to two decimal places.
0.07
4t
e
0.07
4
t
e
3. Solve for r to two decimal places.
3
7r
e=
3
7
r
e
=
4. Bank A offers a 3-year CD that earns 2.05% compounded continuously.
a) If $5,000 is invested in this CD, how much will it be worth in 3 years?
(0.0205)(3)
5000
rt
APe
Ae
=
=
College Mathematics: Learning Worksheets Chapter 10
271
b) How long will it take for the account to be worth $10,000?
0.0205
0.0205
10,000 5000
2
t
t
e
e
=
=
5. A bond will be worth $25,000 at maturity 10 years from now. How much should you
be willing to pay for the bond now if money is worth 2.6% compounded continuously?
(0.026)(10)
0.26
25, 000
25, 000
rt
APe
Pe
Pe
6. A woman invests $18,000 in an account that pays 2.9% compounded continuously. If
the woman needs $21,000, how long will it take to accumulate that much money?
(0.038)
20,000 15, 000
rt
t
APe
e
It will take approximately 7.57 years for the woman to have the money she needs.
College Mathematics: Learning Worksheets Chapter 10
272
College Mathematics: Learning Worksheets Chapter 10
273
Name ________________________________ Date ______________ Class ____________
Goal: To find the first derivative of functions that are exponential or logarithmic in nature.
In Problems 1–5, find the first derivative for the given function.
1. 5
() 7 4 3 21
x
fx e x x=+−+
2. () 5 7ln 13 4
x
fx e x x=− − − +
Section 10-2 Derivatives of Exponential and
Logarithmic Functions
Definition: Derivatives of Exponential and Logarithmic Functions
For
0, 1,bb
x
x
dee
dx ln
xx
dbbb
dx
274
3. 32 5
() 4 2 5 ln
x
f
xxxxxe
x
x
4. 3
10 5
x
yex
5. 5
7
log 3 4
x
yxx
College Mathematics: Learning Worksheets Chapter 10
275
6. Given the function () 12 5,
x
fx e=+find the equation of the line tangent to the graph
of the function at 0.x
First find the first derivative and substitute the given x value. This will give the slope
of the tangent line to the graph of the function.
Now find the y coordinate of the point by substituting the given value into the original
function:
Finally, use the slope and point to find the equation of the tangent line.
276
7. Given the function 5
() ln 2,fx xfind the equation of the line tangent to the graph
of the function at .
x
e
First, find the first derivative and substitute the given x value. This will give the slope
Now find the y coordinate of the point by substituting the given value into the original
function:
Finally, use the slope and point to find the equation of the tangent line.
College Mathematics: Learning Worksheets Chapter 10
277
Name ________________________________ Date ______________ Class ____________
Goal: To find the first derivative of functions that are written as products or quotients.
In Problems 1–10, find the first derivative and simplify.
1. 63
( ) 7 (3 14 10)fx x x x=+−
Since the problem is a product, find the functions and derivatives and then follow the
product rule above.
6
5
() 7
() 42
Fx x
Fx x
=
′=
3
2
() 3 14 10
() 9 14
Sx x x
Sx x
=+−
′=+
Section 10-3 Derivatives of Products and Quotients
Theorem 1: Product Rule
If () ()()
y
fx FxSx and if ()Fx
and ()Sx
exist, then
() () () () ().
f
xFxSxSxFx
Theorem 2: Quotient Rule
If ()
() ()
Tx
yfx
B
x
and if ()Tx
and ()
B
x
exist and () 0Bx , then
2. 22
() (4 7)(2 9)fx x x
Since the problem is a product, find the functions and derivatives and then follow the
product rule above.
2
() 4 7
() 8
Fx x
Fx x
2
() 2 9
() 4
Sx x
Sx x
3. 2
17
() 513
x
fx
x
=+
Since the problem is a quotient, find the functions and derivatives and then follow the
quotient rule above.
() 17
() 17
Tx x
Tx
=
′=
2
() 5 13
() 10
Bx x
Bx x
=+
′=
College Mathematics: Learning Worksheets Chapter 10
279
4. 2
25
() 35
x
fx
x
Since the problem is a quotient, find the functions and derivatives and then follow the
quotient rule above.
() 2 5
() 2
Tx x
Tx
2
() 3 5
() 6
Bx x
Bx x
5.
()
()
() 7 7ln
x
f
xe x=
Since the problem is a product, find the functions and derivatives and then follow the
product rule above.
() 7
() 7
x
x
Fx e
Fx e
=
′=
() 7ln
7
()
Sx x
Sx x
=
′=
x
280
6. 322
(2 5 )( 2 )yxxxx
Since the problem is a product, find the functions and derivatives and then follow the
product rule above.
32
2
() 2 5
() 6 10
Fx x x
Fx x x
2
() 2
() 2 2
Sx x x
Sx x
7.
2
2
34
43
xx
y
x
Since the problem is a quotient, find the functions and derivatives and then follow the
quotient rule above.
() () () ()
()
BxT x TxB x
fx
8. 5
() 7log
x
f
xx=
Since the problem is a product, find the functions and derivatives and then follow the
product rule above.
() 7
() 7ln7
x
x
Fx
Fx
=
′=
5
() log
11
() ln 5
Sx x
Sx
x
=
⎛⎞
′=⎜⎟
⎝⎠
9.
4
2
8
() 4
x
fx
x
Since the problem is a quotient, find the functions and derivatives and then follow the
quotient rule above.
14
34
() 8
() 2
Tx x
Tx x
2
() 4
() 2
Bx x
Bx x
College Mathematics: Learning Worksheets Chapter 10
282
10.
32
2
(2 5)(3 1)
() 7
xx
fx
x
The main problem is a quotient, but in the numerator, there is a product. To find the
first derivative of the numerator, follow the product rule.
3
2
() 2 5
() 6
Fx x
Fx x
2
() 3 1
() 6
Sx x
Sx x
Now that we have the derivative for the numerator, follow the quotient rule.
College Mathematics: Learning Worksheets Chapter 10
283
Name ________________________________ Date ______________ Class ____________
Goal: To find the first derivative of functions that are written as composite functions.
1. Find
() ()mx f gxif 4
()
f
uu=and 2
( ) 2 8 15.gx x x=−+
[]
() ()
mx f gx
=
Section 10-4 The Chain Rule
Definition: Composite Function
A function m is a composite of functions f and g if
The domain of m is the set of all numbers x such that x is in the domain g and ()gxis
in the domain of f.
Theorem 1: General Power Rule
If
()uxis a differentiable function, n is any real number, and
() () ,
n
yfx ux then
Theorem 2: Chain Rule
If ( )
y
fuand ( )ugxdefine the composite function
() (),ymx fgx then
284
2. Find
() ()mx f gxif () ln
f
uuand 2
( ) 4 7 35.gx x x
3. Given 52 5
(7221),yx x x=−+− write this composite function in the form
()
y
fuand ( ).ugx
4. Given 3
5423
,
xx
ye
write this composite function in the form ( )
y
fuand
().ugx
In Problems 5–10, find the first derivative for the given functions.
5. 45
() (3 5 13)fx x x=−+
We must break the function into y and g(x) and then find each derivative.
Now, use the chain rule.
()
dy du
fx du dx
′=
College Mathematics: Learning Worksheets Chapter 10
285
6. ( ) 7 ln(12 5)fx x=+
We must break the function into y and g(x) and then find each derivative.
Now, use the chain rule.
()
dy du
fx du dx
′=
7. 23
() 7
x
fx e −
=
We must break the function into y and g(x) and then find each derivative.
7
u
y
e
=
() 2 3
ugx x
==−
Now, use the chain rule.