Name_____KEY_____________
Chapter 2 Review
Limits and continuity
Per.______
For the following functions
a. Find the intervals of continuity of each function
b. Label each discontinuity as removable or non-removable
c. IF the discontinuity is removable then write the extended function that is continuous
at that domain value.
4
x
a. (, 0)  (0, )
b. f has an infinite discontinuity at x = 0.
c. x = 0 is not a removable discontinuity
1) f ( x)  3 
2) g (t ) 
t 3  2t 2  t
t 2 1
a. (, 1)  (1,1)  (1, )
b. f has an infinite discontinuity at x = -1 and a removable discontinuity at x = 1
t  t  1
c. Extended function for g(t): f (t ) 
t 1
Rates of change:
3
do the following.
n2
Name the domain of p(n):
Compute the average rate of change of p(n) on the interval [3, 4].
Compute the instantaneous rate of change function of p(n) for n = 1.
Write an equation for the tangent line to the curve p(n) when n = 1.
Write an equation for the normal line to the curve p(n) when n = 1.
3. Given p (n) 
a.
b.
c.
d.
e.
a. Domain:
 , 2   2, 
b. Average rate of change:
p (4)  p (3) 3

43
2
c. Instantaneous rate of change at n = 1: lim
h 0
d. y  3  3  x 1
y3
1
 x  1
3
p(1  h)  p(1)
 3
h
A 10 ft3 wooden box with a square base of t feet by t feet is going to made of three
different woods. The cost for the wood on the top is $1.15 per square foot. The sides are
made of wood that is $0.50 per square foot. And the bottom material costs $0.85 per
square foot.
a. Write a volume equation for the box.
b. Write a cost equation that is a function of only t. That is C(t).
c. What is a reasonable domain of the cost function?
d. Draw a sketch of C(t) on the domain from part c.
e. What are the dimensions of the box that costs the least to produce? (t and height)
C (tm  h)  C (tm )
 0 where tm is the t value from part e.
f. Explain why: lim
h 0
h
10
a) V (t )  t 2 h where h  2
t
 10 
C (t )  (1.15)t 2  4(0.50)    (0.85)t 2
 t 
b)
20
 2t 2 
t
c) Reasonable domain:  0,  
4.
d)
t  1.71 ft, h  3.42 ft, cost  $17.54
5) Use the graph of f(x) on the right to evaluate the
following limits.
a. lim f ( x)  
x 
b. lim f ( x)  9
x 6
c. lim f ( x)  DNE
x 0
d. lim f ( x)  2
x 0
e. lim f ( x)  4
x 
f. lim
h 0
f (1  h)  f (1)
3
h
g. lim
h 0
f (6  h)  f (6)
0
h
lim
6) What does the limit
3 4  h  2  3 4  2
h 0
h
tell you about the graph of the function
f ( x)  3x  2
f ( x)  3x  2
The limit tells you that the slope of the curve
at x = 4.
7) Find a right and a left end behavior model for the functions below.
f ( x)  x  e x
g ( x)  ln x  sin x
a.
b.
y  ex
y  ln x
R.E.B.M:
R.E.B.M:
yx
L.E.B.M.
L.E.B.M.
y  ln   x 
8) An object is dropped from the top of a 200 meter tower. Its height above ground after t
seconds is given by the equation
h(t )  200  4.9t 2
a.
What is the average speed of the object over the first two seconds of its fall?
9.8 m/
b. When does the object hit the ground?
 6.38877 seconds
c.
What is the object’s instantaneous speed three seconds after it begins to fall?
 29.4 m/sec
9) Consider the function f ( x) 
3
.
x  5x  6
2
a. Is f(x) a continuous function? Explain your answer.
Yes. f(x) is a rational function. The function is continuous on its domain which means
that it is a continuous function. The only points of discontinuity f(x) has are x = -2 and x = -3.
Both lie outside the domain of the function.
b. Is f(x) continuous on the interval from [-10,10]?
No. f(x) is discontinuous at x = -2 and x = -3 both of which are in the interval [-10,10]