worksheet

Q1 Review
– round 2
Name_______________
Free Response
2
2

 x  a x if x  2
2

4  2 x if x  2
1) Let f ( x)  
(a)
Find lim f ( x).
(b)
Find lim f ( x).

x 2
x 2
(c) Find all values of
a that make f
continuous at 2. Justify your answer.
f ( x)  2 x  x 2 .
2) Let
(a)
Find f (4)
(b)
Find f (4  h)
(c) Find
f (4  h)  f (4)
h
(d) Find the instantaneous rate of change of f at
3) Let
f ( x)  x 4  4 x 2 .
(a)
(b)
(c)
4)
x  4.
lim
x →2
Find all the points where f has horizontal tangents.
Find an equation of the tangent line at x = 1.
Find and equation of the normal line at x = 1. (Normal lines are perpendicular to tangent lines)
when

 x 2  3x  6 when x  2
= 2

 x  3x  2 when x  2
5) Find a c such that
is continuous on the entire real line.
2
 x when x  4

= c
 x when x  4
6) l i m
x →3
x 1 2
x 3
1
1

7) l i m x  3 3
x
x 0
8) The graph of a function
is shown below.
6
f
4
2
-10
-5
5
10
-2
-4
Answer the following questions about function .
a. f (5) 
b. f (2) 
c. f (4) 
d. lim f ( x) 
e. lim f ( x) 
f. lim f ( x) 
g. lim f ( x) 
h. lim f ( x) 
i. lim f ( x) 
j. lim f ( x) 
k. lim f ( x) 
l. lim f ( x) 
m. lim f ( x) 
n. lim f ( x) 
x 7
x 4
x 5
x 0
x  0
x 4
x 
x 2
x  0
x 4
x 
9) Use the definition of a continuous function at an -value to answer the following.
a.
is not continuous at
because:____________________________
b.
is not continuous at
because:_____________________________
c.
is not continuous at
because:______________________________
10) For the following problems, sketch a graph of a function that has the indicated features and write an
equation for the function that has these features. The function may be a piecewise.
a) The function is continuous at x = 3, but has a
cusp there.
b) The function has a limit as x approaches 3 but fails to
be continuous there because f(3) is undefined.
c) The function has a limit as x approaches -1,
has a value for f(-1), but still is not continuous
there.
d) The function has no limit as x approaches 0, but
f(0)=3.
e) The function has a limit of 2 as x approaches 0
from the right, but has no limit as x approaches 0
from the left.
f) The function has a step (or jump) discontinuity at
x = 1, and f(1) = 6.
g) The function has a limit as x approaches 2 of 5
but f(2) = 4.
h) The function has a right-hand limit of -2 and a lefthand limit of 2 as x approaches -1.
Multiple Choice
x 3
is
x  2x  3
11) lim
2
x 3
(A) 0
12) lim
x 0
(B) 1
lim
x 7
(B) nonexistent
(B)

(E) none of these
(C) 1
(D) -1
(E) none of these
7
(C) 0
(D) 2 7
(C) 1
(D)
e
1
2a 2
(E) nonexistent
x
is
x 1 ln x
lim
(A) 0
(B)
15) If a  0 , then lim
xa
16)
(D)
x  7 is
x 7
(A) 2 7
14)
1
4
x
is
x
(A) 0
13)
(C)
1
e
x2  a2
is
x4  a4
(A)
1
a2
(B)
(C)
1
4
(D) 0
(E) nonexistent
(C)
1
6a 2
(D) 0
x3  2 x 2  3x  4

x  4 x 3  3 x 2  2 x  1
lim
(A)
(B) 1
4
(E) 1
17) Let f ( x)  4  3x . Which of the following is equal to f (1) ?
(A) -7
(B) 7
(C) -3
(D) 3
(E) nonexistent
18) Let f be the function given by f ( x)  x . Which of the following statements about f are true?
I.
II.
III.
f is continuous at x = 0.
f is differentiable at x = 0.
f has an absolute minimum at x = 0.
19) True or False (with reason):
If is undefined at
, then the limit of
20)
as
approaches does not exist.
True of False (with reason): If the l i m f(x)  L then f(c) = L.
x c
(E) DNE

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