Math 151 Exam 3 Review
1. Solve for x. a) 500 e
c) log
3
x  4 + log
.04 x
3
= 750
 x2
 x 3
b) ln 
 = 1 + ln 

 x 1 
 x 1 
x+ 4 =1
2. Find the derivative of each function. f(x)=
 ( 2x + 4 ) 2 ( 3x 2 + 12
a) f(x)  ln 
3
x +7

2
x +1
b) f ( x ) 
2
3
x +7

2

2
( 2x + 4 ) ( 3x + 12 )
d) f ( x )  log 10 (x + 1 )
c) f ( x )  log 2 (x + 7 )
e) f ( x )  ln x +
)



2

2
2
f) ln (x + y) = x ln y Find y'.
h) f ( x )  ( x  1)
2
g) f ( x )  ln | sec x |
x
3. When making chocolate fudge, the chocolate mixture must be heated to 234 degrees
Fahrenheit and subsequently cooled to 110 degrees F. If the cooling takes 30 min in a
room with constant temperature 70 degrees F, what is the temperature t minutes after the
mixture is removed from the heat? Use Boyle’s Law of cooling. ( y  T ) '  k ( y  T )
Where T is the temperature of the surroundings, 70 degrees F in this case.
4. a) At what interest rate, compounded continuously, will $1500 grow to $2000 in 4
years?
b) A culture grows from 12 grams to 60 grams in 4 hours. Find the weight after t hours.
5. Find y'(x) and express it explicitly.
a) sin y = x
b) tan y = x
6. Simplify each expression.
a) cos( arctan 4/5 )
b) sin( 2arccos(1/3))
d) sec(arctan x )
e) sin  arccos x 
7. Find the derivative of each function.
a) y = xarctan x
b) f(x)=arcsin( x )
c) cos( 2arccos(3/4))
8. Evaluate each limit.
a) lim (cot x )
x 0
d) lim
sin x

b) lim (tan x )
x
1
f) lim (x + 2 x) x
x 
x 
2 + x arctan x
e) lim ln( 5 x  2 x )  2 ln( x )
x
2

5x
c) lim
x 0
arcsin x
x 0
sin x
1
2

x
g) lim  
x 
x
9. If the function f(x) has a local max of 4 at x=2 and f(3)=7, what can you say about f '
on the interval [2, 3]?
10. Find c in [0, 2] so the conclusion of the Mean Value Theorem holds for
3
f ( x)  x  2 x .
11. Find the absolute max and min of f ( x )  2 x 3  3 x 2  12 x on the given interval:
a) [-5, 7]
b) [0, 4]
c) [-3, 0]
12. For f ( x )  2 x 3  3 x 2  12 x , find the inflection point of f and graph f and f'.
2
13. f ' ( x )  e  x , f(x) has horizontal asymptotes y=C as x approaches infinity and y= B as
x approaches minus infinity and f(0)=(C+B)/2.. Graph f(x).
14. The distance a car has traveled after t hours is given by f ( t )  400  400 e
Find the time at which the velocity of the car is greatest and find the max velocity.
 0 . 1t
2
.
15. A right circular cylinder at time t=0 minutes has a radius of 20 cm and a height of 30
cm. The radius is increasing at the constant rate of 2 cm/min. The height is decreasing at
the constant rate of 3 cm/min. When will the volume be greatest? Show this is a max. and
not a min.volume.
16. Given the table of values for f' and f", determine what the 2nd derivative test says
about f(x) for each x-value.
x
1
2
3
4
f '(x)
2
f "(x) -7
0
1
0
0
0
-1
17. Find any asymptotes, local max and min, and inflection points of each.
a) f ( x )  x 3 e x
b) g ( x )  x 4 e x
For 18 and 19, use the 2nd derivative test to show you found the appropriate type of
extreme point.
18. A fence is to be built to enclose a rectangular area. If one wall of fence is $12 per
linear foot and the other three walls are $8 per foot, what is the maximum area that can be
enclosed if the total cost of the fence is $2400?
19. If the fence of problem 18 is to enclose an area of 2500 square feet, but the cost is not
set, what dimensions will minimize the cost of the fence.
20. A cylindrical can with top and bottom has a total surface area of 54π cubic inches.
Find the dimensions that will maximize the volume.
More max-min word problems are in section 5.5