Name:_____________________
AP Calculus Chapter Two Homework Packet – keep for the whole unit
Complete all work on separate paper. Some answers are given in the back of the packet.
NO CALCULATOR ON ENTIRE PACKET!!
Section 2.1 – complete all problems listed.
Determine the limit by substitution.
7.
y2 + 4 y + 3
11. lim
y ® -3
y2 - 3
lim 3 x (2 x - 1)
2
x® - 1
2
13. lim ( x - 6)
9. lim ( x 3 + 3 x 2 - 2 x - 17)
3
x® -2
x® 1
Determine the limit by substitution. Answers of
!
!
are considered FATAL and not allowed. TRY SOME ALGEBRA!
5 x3 + 8 x 2
21. lim
x ® 0 3 x 4 - 16 x 2
t 2 − 3t + 2
20. lim
t→ 2
t2 − 4
x -1
19. lim 2
x® 1 x - 1
2
23. lim
x→ − 4
x 2 − 16
x+4
For # 37 - 38, determine if the statements are TRUE or FALSE.
37.
y = f(x)
a) lim+ f ( x) = 1
b) lim- f ( x) = 0
c) lim f ( x) = 1
-
d) lim- f ( x) = lim+ f ( x)
e) lim f (x) exists
f) lim f ( x) = 0
g) lim f ( x) = 1
h) lim f ( x) = 1
i) lim f ( x) = 0
j) lim- f ( x) = 2
x ® -1
y
x® 0
1
x® 0
x® 0
x® 0
x® 0
x® 0
x® 0
x® 1
x
-1
1
2
x® 1
x® 2
38.
y = f(x)
y
a) lim f ( x) = 1
+
b) lim f ( x) = does not exist
c) lim f ( x) = 2
d) lim f ( x) = 2
-
e) lim+ f (x) = 1
f) lim f ( x) = does not exist
x ® -1
2
x® 2
x® 2
x® 1
x® 1
x® 1
1
g) lim- f ( x) = lim+ f ( x)
x® 0
x
-1
1
2
3
x® 0
h) lim 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎𝑡 𝑒𝑣𝑒𝑟𝑦 𝑐 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (1, 3)
"→ %
i) lim 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠 𝑎𝑡 𝑒𝑣𝑒𝑟𝑦 𝑐 𝑖𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (0, 2)
"→ %
For 39, 41, 43, use the graph to estimate the limits and value of the function, or explain why the limits do not exist.
39.
41.
a) lim- f (h)
y
y
5
a) lim- f ( x)
4
h® 0
2
x® 3
1
b) lim+ f (h)
3
b) lim+ f ( x)
2
-2
x
1
2
3
4
5
-2
6
7
8
c) lim f ( x)
43.
x® 3
y
-4
a) lim- F ( x)
x® 0
4
b) lim+ F ( x)
x® 0
x
c) lim F ( x)
x® 0
-4
c) lim f (h)
h® 0
-3
d) f(3)
-3
h® 0
-1
x® 3
1
-3 -2 -1
-1
x
d) F(0)
Keep going…
d) f(0)
For 51:
a) draw the graph of f(x)
ì 3 - x,
ï
c = 2, f ( x ) = í x
ïî 2 + 1,
b) determine lim+ f ( x) and lim- f ( x)
x®c
x®c
x<2
x>2
c) Does lim f ( x) exist? State why or why not.
x® c
For 52 and 53:
52.
ì
ï 3 - x,
ï
c = 2, f ( x ) = í 2,
ïx
ï ,
î2
x<2
x=2
⎧ 2x + 2,
x <1
⎪
⎩ x − 2x + 5,
x≥1
53. c = 1, f (x) = ⎨
x>2
3
a) determine lim+ f ( x ) and lim- f ( x )
x® c
x® c
b) Does lim f ( x) exist? State why or why not.
x® c
MULTIPLE CHOICE and No calculator:
ì 2 - x,
ï
In Exercises 67 – 70, use the following function: f ( x ) = í x
ïî 2 + 1,
x £1
x >1
67. What is the value of lim- f ( x) ?
A) 5/2
B) 3/2
C) 1
D) 0
E) DNE
68. What is the value of lim+ f ( x) ?
A) 5/2
B) 3/2
C) 1
D) 0
E) DNE
69. What is the value of lim f ( x)?
A) 5/2
B) 3/2
C) 1
D) 0
E) DNE
70. What is the value of f (1) ?
A) 5/2
B) 3/2
C) 1
D) 0
E) DNE
x® 1
x® 1
x® 1
Section 2.2 – complete all problems listed.
Without a calc, find the limits.
1.
lim
$!%&
!→# '!
2.
3.
$! &
!→# '!()
4. lim
13.
16.
*+,p %.! &
/! &
!→#
lim
lim+
1
x-2
14.
lim -
1
x+3
17.
x® 2
x ® -3
%&
!→# '!
lim
lim-
x
x-2
lim +
x
x+3
x® 2
x ® -3
! ' %1
!→0 !%0
5. lim
15.
! ' %&!%$.
!→/ ! ' %/!
lim
In 27 – 30, determine the vertical asymptotes.
27. f ( x ) =
1
2
x -4
28. f ( x ) =
x2 - 1
2x + 4
29. f ( x ) =
x2 - 2 x
x +1
30. f ( x ) =
1- x
2 x - 5x - 3
2
In 31, 32: determine the vertical asymptotes on [0,2p ) w/o a calculator. (think about precal and the reciprocal…)
In 39, 41, 43: determine the horizontal asymptotes:
39.
f ( x) = 3x2 - 2 x + 1
41. f ( x ) =
x-2
2
2 x + 3x - 5
43. f ( x ) =
4 x3 - 2 x + 1
x-2
In 55 & 56, sketch a graph of a function, f(x), that satisfies the stated conditions. (there are many correct answers)
f (-2) = 0
f (2) = 0
f (0) is undefined
lim f (x) = 4
55.
x® 0
f (2) = 6
lim f (x) = 3
x® 2
56.
lim f (x) = 0
x® -2
lim f (x) = does not exist
x® 2
73. Without a calculator, determine lim f (x) and lim f (x)
x→ ∞
⎧2x + 2,
a) f (x) = ⎨
x→ − ∞
x <1
b)
⎪ x − 2x + 5, x ≥ 1
⎩
3
⎧x + 1
⎪⎪ x 2 , x < 2
f (x) = ⎨ 2
⎪ x ,x>2
⎪⎩ 3x 2 + 1
Section 2.3 – complete all problems listed.
y
2
y = f(x)
x
-1
In 11 – 14 answer with YES or NO. Use the function f graphed above to answer the questions.
11. a) Does f(0) exist?
b) Does
lim 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡?
!→2)
12. a) Does f (1) exist?
b) Does lim+ f ( x) exist?
x® 1
c) Does
lim 𝑓(𝑥) = 𝑓(0)?
!→2(
d) Is f continuous at x = 0?
c) Does lim- f ( x) exist?
x® 1
d) Is f continuous at x = 1?
13. On what interval(s) of x is f continuous? Use proper interval notation.
1
2
3
19. My answer for showing if f(x) is continuous is shown to the right. What, if anything, is wrong with my answer?
f (2) = 1
at x = 2:
lim f ( x ) =1
ì3 - x , x £ 2
ï
f ( x) = í x
ïî 2 + 1, x > 2
x ® 2-
lim f ( x ) = 2
x ® 2+
Since lim f ( x ) ¹ f (2), then f ( x ) is not continuous
x® 2
For 20 – 21, use the continuity test to show whether the function f(x) is continuous. Be DETAILED!
20. at x = 2
ì
ï3 - x,
ï
f ( x ) = í2,
ïx
ï ,
î2
x< 2
x=2
21. at x = 1
x>2
⎧2x + 2,
x<1
f (x) = ⎨ 3
⎪ x − 2x + 5, x ≥ 1
⎩
23. Use the full continuity test to show whether the
y
function f(x) is continuous at x = 0 and x = 1.
1
x
-1
1
Find the one-sided limits WITHOUT a calc:
Use your 2.2 lesson and plot points
approaching the VA
In 41 – 42, sketch a possible graph for a function f that has the stated properties.
41. f (3) exists but lim f ( x) does not.
x® 3
42. f (-2) exists, lim + f ( x ) = f ( -2 ) , but lim f ( x ) does not exist .
x® -2
x® -2
2
ì x 2 - 1,
f ( x) = í
î2ax,
47. Find a value for a so that the function
49. Find a value of a so that the function
2
ì
ï4 - x ,
f ( x) = í 2
ï
îax - 1,
x<3
x³3
x < -1
x ³ -1
is continuous at x = 3.
is continuous at x = -1.
ì 2 x,
ï
58. Identify all of the following statements about the function f ( x ) = í 1,
ï - x + 3,
î
A) f (1) does not exist
C) lim f ( x) exists
-
B) lim f ( x) exists
+
D) lim f ( x) exists
x® 0
x® 2
0 < x <1
x =1
1< x < 2
which are not true.
E) lim f ( x) = f (1)
x® 1
x® 1
Section 2.4 – complete all problems listed.
7. The number of butterflies spotted in Amelie’s garden each afternoon is modeled by the increasing function h
for 0 ≤ ℎ ≤ 14, where t is measured in days.
a) Find the average rate of change of butterflies from day 0 to day 12. I could also say day 0 to day 12 as (0, 12).
b) Find the average rate of change of butterflies on the interval (7, 14).
c) Any idea what the units would be for (a) and (b)?
9. For y = x 2 at x = - 2
11. For y = 2 x2 - 6 at x = 5
a) find the slope equation using lim
h ®0
f ( x + h) - f ( x )
h
a) find the slope equation using lim
h ®0
b) the slope at the given value
b) the slope at the given value
c) the equation of the tangent line
c) the equation of the tangent line
f ( x + h) - f ( x )
h
In Exercises 13 and 14, write the piece-wise equation for f(x) and then find the slope of the curve at the indicated point.
13. f (x) = x at
(a) x = 2
14. f ( x ) = x - 2
at x = 1
(b) x = - 3
27. The equation for free fall at the surface of Mars is 𝑓(x) = 1.86t * meters with t in seconds. Assume a rock is dropped
from the top of a large cliff. Use lim
h ®0
f ( x + h) - f ( x )
to find the speed of the rock at t = 1 sec.
h
SHOW YOUR
COMPUTATIONS !!
You are penalized if
you don’t.
SOME ANSWERS…..USE THEM WISELY
Exercises 2.1:
7. -3/2
9. -15
11. 0
13. 4
c) dne
d) 1
19. ½
20. ¼
21. -1/2
43. a) 4
b) -3
37. 5 of them are TRUE
39. a) 3
b) -2
41. a) -4
c) -4
Exercises 2.2:
1. 2/7
4. -1/2
27. x = -2, x = 2
5. NOT DNE or 0/0
13. ∞
16. - ∞
15. 11/8
29. x = -1
43. No HA, This is a Parabola
Exercises 2.3:
11. One Yes
37. ∞
47. a = 4/3
7a)
+,-+
+*-!
39. ∞
58. Two of these are not true
9a) 2x
b) - 4
c) y- 4 = - 4(x+2)
11 b)
20
23. -8