Math 131 Final Exam Review
Find A, B, and C in each for problems 1, 2, and 3.
1. f ( x )  A  Be Cx lim f ( x )   2 f ( 0 )  4 f (1)  1
x 
2.
f ( x )  A  B ln( x  C )
lim
x 2
3.
f ( x )  A  Be
Cx
lim
x  
4. Solve for x if
4
2x
2x

f ( x )  
f (x)  4
f ( 3)  2
f (0)  2
3
f (e  2)  8
f (ln 3 )   50
.
.
x
 17 ( 4 )  16  0 .
 4e
5. Solve for x if
e
6. Solve for x if
2 log 2
x
5 0
.
x  6  log 2 ( x  1)  4 .
7. $1500 is invested at a continuous compound interest rate for 5 years. If the amount at
the end of the 5 years is $1897.36, find the interest rate as a decimal rounded to 3 decimal
places.
8. A radioactive substance begins with 54 g. It decays at a rate proportional to its weight.
(The rate at t years is proportional to the weight at t years). After 30 years there are 36 g.
Find the rate of decay rounded to 4 decimal places.
9.
2x  2

f (x)   x2  1
 x2

x 1
a) Find any or all values of c where
1 x
not exist.
b) Find any and all x values where f is not continuous.
c) Find any and all x values where f is not differentiable.
 x  4x  3
2

x 1

10. f ( x )  
x2
 x 2  25

2
 3 x  15 x
2
x 1
1 x  5
5 x
Find all discontinuities of f and give a reason for each.
11.
f (x)  (x
2
 25 )
1 3
Find f '(x) and tell where f is not differentiable.
lim f ( x )
x c
does
12. Evaluate each limit. First identify the expression as a difference quotient for some
function.
a)
lim
ln( 2  h )  ln 2
h 0
h
13. Find

d  
ln 
dx  
 
b)
lim
h 0
4
 4)  

3
x  7 x  

x (x
e
h
1
3
c)
h
lim
2
5( x  h )  8( x  h )  3  5 x
h 0
2
 8x  3
h
2
Do it the easy way or it will not be graded.
14. Find the derivative of each function of x.
a)
f ( x )  sin
2
x
f (x)  e
b)
x 3
a(x
2
 7 x  20 )
c)
f (x)  x e
3
x
2
2
d)
f (x) 
e
e)
2 x 1
f ( x )  sec x
4
f)
f ( x )  tan
2
x
15. Find the tangent line to the function at the given value of x.
a)
b)
f ( x)  ( x  2)
2
f (x) 
e
3
x 1
at
x
x 1
e
at
x0
x
c)
f (x) 
d)
g ( x )  f ( x  1)
x 4
2
2
at
x0
at
x 1
given
f ( 2 )  5 and f ' ( 2 )   3
16. The tangent line to f(u) at u=1 is y=5u+7. The tangent line to g(x) at x=2 is y= -4x+9.
Find the tangent line to h(x)=f(g(x)) at x=2.
17. Find all local max, min and inflection point(s) of each.
2
a)
f ( x)  x ( x  2)
b)
f ( x)  x( x  2)
c)
f ( x)  x ( x  2)
d)
f (x) 
3
3
2
2
1
x
2
4
18. Use a differential or a tangent line to approximate
3
8 .5
.
19. A company can sell 200 cups per day of its super energy drink if the price is $1.50
per cup. For each decrease of $0.25 in the price, they can sell 40 more cups per day. What
price per cup will maximize their sales revenue?
20. A farmer wants to fence an area of 1000 sq.ft. One side of the area is the wall of a
building. He will partition the area into 3 parts with fencing perpendicular to this wall.
find the dimensions that will minimize the amount of fence material.
21. Re do #18 if the partitions are parallel to the wall of the building.
22. Find the left and right hand Riemann sums L 4 , and R 4 , for f ( x )  16 x on [0, 1].
Find lim L n  lim R n .
n 
23. If
n 
v (t ) 
ln( 1  t )
1 t
for t >0, t in seconds, is the velocity of an object moving in a straight
line, find:
a) a(t), the acceleration at time t.
b) the distance traveled in the first 4 seconds.
24. Evaluate each by calculator and also by showing all work.
1
a)

0
e
x
8
dx
x
b)
 (x 
3
2
9
2 )( x  4 ) dx
c)

0
x
x
2
4
1
dx
d)
x
0
3 x  1dx
25. Find F(x) given that
a) F " ( x )  24 x  16
b) F " ( x ) 
1
x
2
F (1)  6 and F (  1)  12
F (1 )  10 and F (  1)  16
26. Find the area between the graphs.
a) y  x 2  5 x and y  7 x  3 .
b)
y  x
2
 5x
and
y  7x  3
for
0 x4
.