Math 151 In Class Exam 3 Review
Sections 4.3-4.6, 4.8, 5.1-5.5
1. In January of 2005, $10,000 dollars was invested at annual interest rate r compounded continuously.
By January of 2012, the investment had grown to $12,500. Find the interest rate.
2. A substance decays so the rate of change in the weight at t years is proportional to the weight at t
years. Initially it weighed 20 g and after 100 years it weighed 15 g.
a) Find a formula for the weight at t years from the time it weighed 20 g.
b) Find the rate of change at t years and the continuous decay rate.
3. Find the derivative of each function.

a ) f ( x )  ln 

3
x (x  2) 

2
(x  4)

b) f (x) 
 x
c ) f ( x )  arctan   and simplify.
 3
ln( x )
ln( x )
d ) f (x) 
x
4. Simplify each expression.
2
a)
sin(arccos
t )
d)
cos(arctan
x)
b)
tan(arccos
x)
c)
sec(arcsin
5. Evaluate each limit.
a)
lim
x 0
d)
lim
x 0

x
ln x
b)
lim
x
1 x
c)
x 
x  ln( 1  x )
x
2
e)
lim
x 0
lim
x  2
x  2
tan x

x
sin x
x)
6. Find all pertinent info of the graph of the function.
a)
f (x) 
x
2
( x  1)
3
b)
3
f ( x )  x ln | x |
7. Solve for x(y) and use it to find the inflection point of
f (x) 
L
1 e
 x
.
8. Find any local extreme points and inflection points of f(x) if the derivative of f(x) is
2
f '(x)  (x  2) x
.
9. The graph of the derivative of f(x) is shown.
Where does f have a local max?, local min?, estimate where f has inflection points?
f '(x)
10. What is the conclusion, if any, of the 2nd derivative test in each case?
x
1
2
3
4
f ‘(x)
0
1
0
0
f “(x)
0
-2
-3
1
11. what does the 2nd derivative test say about g at x=a?
g (x)  e
x
f (x)
12. A boat uses $0.5 v
f (a )  1
2
f '(a )  1
f " (a )  2
dollars per hour for fuel when running at velocity v.
Other expenses to operate the boat are $200 per hour. At what velocity should the boat run to minimize
the total cost of a D mile trip. Use the 2nd derivative test to show the critical value actually minimizes the
cost.