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AP Calculus
Name __________________________________hr___
Chapter 2 Review
Date____/____/____
Find the limits if they exist. If the limit does not exist, explain why not.
1.
lim
x  2
( x 3  2 x 2  1)
1
1
lim 2  x  2
3.
x0
x
5.
7.
lim x csc x  1
x  0 x csc x
lim
x
e  x cos x
2.
lim
x4
1 2x
lim
x4  x3
4.
x   12 x 3  128
6.
8.
lim
x  7 / 2
int(2 x  1)
lim sin 2 x
x  0 4x
Determine whether the limit exists on the basis of the graph of y = f(x). The domain of f is the set of real
numbers.
9.
lim
xd
11.
f ( x)
lim
x b
10.
lim
x  c
f ( x)
f ( x)
Determine whether the function f used in 8-11 is continuous at the indicated point.
12. x = a
13. x = c
14. Determine:
a)
lim
x  3
g( x)
b) g (3)
c) whether g(x) is continuous at x = 3
d) the points of discontinuity of g(x).
e) whether any points of discontinuity are removable. If so, describe the new function. If not, explain why not.
15. Find the vertical asymptotes of the graph of the function f ( x ) 
x3
and describe the behavior of f(x) to
x2
the left and right of any vertical asymptotes.
1,
−𝑥
16. Given 𝑓(𝑥) =
1
−𝑥
{1
𝑥 ≤ −1
−1 <𝑥 <0
𝑥=0
0 <𝑥 <1
𝑥 ≥1
a) Find the right-hand and left-hand limits of f at x = -1, 0, and 1.
b) Does f have a limit as x approaches -1? 0? 1? If so, what is it? If not, why not?
c) Is f continuous at x = -1? 0? 1? Explain.
17. Find all points of discontinuity of the function.
x 1
f ( x) 
4  x2
In exercises 18 and 19 find a) a power function end behavior model and b)any horizontal asymptotes.
2x  1
18. f ( x )  2
x  2x  1
x 3  4 x 2  3x  3
19. f ( x ) 
x3
20. Find a) a right end behavior model and b) a left end behavior model for the function f ( x)  x  e x .
What value should be assigned to k to make f a continuous function?
𝑥 2 +2𝑥−15
21. 𝑓(𝑥) = {
𝑥−3
𝑘
𝑥≠3
𝑥=3
Sketch a graph of a function f that satisfies the given conditions.
lim
lim
22.
f ( x)  3 ,
f ( x)  
x
x  
23. Find the average rate of change of f ( x )  1  sin x over the interval 0,  / 2 .
24. Find the instantaneous rate of change of the surface area S  6x 2 of a cube with respect to the edge length x
at x = a.
25. Let f ( x )  x 2  3x and P= (1, f (1)). Find a) the slope of the curve y = f(x) at P, b) an equation of the
tangent at P, and c) an equation of the normal at P.
26. The number of bears in a federal wildlife reserve is given by the population equation
200
p( t ) 
1  7e 0.1t
where t is in years.
(a) Find p(0). Give a possible interpretation of this number.
(b) Find
lim
x
p( t )
27.
a) Let x = 0 represent 1990, x = 1 represent 1991,
and so forth. Make a scatter plot for the data.
b) Let P represent the point corresponding to 2003,
Q1 the point corresponding to 1998, Q2
the point corresponding to 1999,…, and Q5 the point
corresponding to 2002. Find the slope of the secant the PQi
for i =1, 2, 3, 4, 5.
c) Predict the rate of change of population in 2003.
d) Find a linear regression equation for the data, and use it to
calculate the rate of the population in 2003.
Population of Florida
Year
1998
1999
2000
2001
2002
2003
Population (in
thousands)
15,487
15,759
15,983
16,355
16,692
17,019