Math 142
Week in Review
Set of Problems
Week 4
• Continuously Compounded Interest
A = Pert
1) Solve for t: 4 = e0.05t
2) Melborn Trust offers a 10 year CD that earns 3 ¼ % compounded continuously.
a. If $5000 is invested, what will its value be in 8 years?
b. How long before it grows to $12, 000?
3) I bought $10, 000 of stock in Mead Publishing and 52 months later sold it for $18, 000. If the
interest is compounded continuously, what annual nominal rate of interest did the original
investment earn?
4) A mathematical model for world population growth is given by P = Poert. How long will it
take theworld population to increase by 50% at its current continuous compound rate of 1.45%
per year?
• Derivatives of Exponentials and Logarithmic Functions
y =e
x
,
y '=e
x
,
y =b lnb
y =b
y =logb x ,
x
and
y =ln x ,
x
y =1 / x ln b
y '=1 /x
Find the derivative of each of the following in #5 − #7:
5) f(x) = -7ex + e8 − ln x6
6) g(x) = 4x − 8 log4 x
7) h(x) = ln x5 + 3 ln x
8) Write the equation of the tangent to the curve f(x) = ln x3 at x = e, in slope intercept form
(y = mx+b).
9) The estimated salvage value (in thousands of dollars) of a company car is S(t) = 30(0.5)t.
What is the rate of depreciation in dollars per year after a) 3 years, b) after 6 years?
10) A mathematical model for the average of a group of people learning to type is given by
N(t) = 8 + 5 ln t, t ≥ 1, where N(t) is the number of wpm typed after t hours of instruction and
practice. What is the rate of learning after 20 hrs of instruction and practice?
• Derivatives of Products and Quotients
y = f x g x  ,
y=
f x 
g x 
,
y '= f  x g x'= f  x g ' x f ' x g x 
 
y '=
f x 
g x  f ' x− f  x g 'x 
'=
g x 
g  x2
11) Find
y=
dy
dx
for
y =−4 x e
12) Find
y=
dy
dx
for
y =6 x ln x
13) Find
y=
dy
dx
for
y=
x
3x−1
14) Find
y=
dy
dx
for
y=
x −2x
x 2 −16
3
4
2
3
x
15) Find
y=
dy
dx
for
y=
3e
7−ex
16) Find
y=
dy
dx
for
y=
ln x
2
x 7
17) Find
y=
dy
dx
for
y =5x1x −2x10
x
2
18) Find the equation of the tangent line to the curve at x=1 for the previous problem
19) Find
y=
dy
dx
for
y =1e lnt
20) Find
y=
dy
dx
for
y=
t
ln x
2
x
e e
21) The concentration of a drug in the bloodstream t hours after the injection is modeled by
C t=
0.18t
2
2t 1
,
C(t) is measured in milligrams per milliliter.
a) Find C(2) and interpret.
b) Find C′(2) and interpret.
• The Chain Rule
a) If y = f( u (x) ) , then y' = f ' ( u (x) ) u ' (x)
u x
, then
y '=e
u x
, then
y '=b
b) If
y =e
c) If
y =b
u x
u '  x
u x
u ' x ln b
and
y =lnu x 
and
, then
y =logb u  x
y '=
, then
u ' x 
u x 
y '=
u' x
u x ln b
2
2
22) For f x = x and g x =3x −1 , find f( g(x) )
23) Write f(x) and g(x) such that
24) Find
dy
5x2 −3x14
dx
25) Find
dy 2x x
e

dx
26) Find
dy
log 2 3x2− 4x9
dx
27) Find
dy lnx 5
x
dx
e
28) Find
dy x
e 14
dx
29) Find
dy
[ ln x 3−2x 2 ]4
dx
30) Find
dy 5x −3x1
e
dx
31) Find
dy 4  x 21
dx x2
2
2
2
4
f g x =e
2
x −3x
32) Find
dy 7x 10
e 
dx
33) Find
dy e
dx x 4 −1
34) Find
dy
ln 4x−3 2.5
dx
35) Find
dy
4
ln   x 
dx
36) Find
dy
xln1ex 
dx
37) Find
dy
ln1ex 7
dx
38) Find
dy
−5x 3−73
dx
39) Find
dy
3x−12e−4x7
dx
40) Find
dy
 3x x2
dx
3
5x−1