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2 hours final “practice exam” start in 1 hour and 30 min from now calculus 1 7chapters: 1-function 2-limits 3-derivatives 4-applications of the derivative 5-integration 6-applications of integration 7-logaritm and exponential function

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62

CHAPTER 2

.

LIMITS

72. One-sided limits Let

55x 15 if x < 4
8(x)
Vox + 1 if x 2 4.
Compute the following limits or state that they do not exist.
a. lim 8(x) b. lim 8(x)
c. lim 8(x)
---
4+
14
73. One-sided limits Let
Tx? + 1
if x < -1
f(x) =
(Vx+ 1 if x 2 -1.
Compute the following limits or state that they do not exist.
lim_ f(x) b. lim f(x) c. lim f(x)
a.
x-1
x-1+
74. One-sided limits Let
{
a.
*-5-
c.
--5+
if x < -5
f(x) = { V25 – x2 if –5 3.
Compute lim f(x) and lim f(x). Then explain why lim f(x)
-+3
does not exist
P(x)
17. Suppose p and q are polynomials. If lim = 10 and
9(0) = 2, find p(0)
18. Suppose lim f(x) = limh(x) = 5. Find lim 8(x), where
f(x) = g(x) s h(x), for all x.
13
V10x - 9 - 1
2
51. lim
52. lim
*-1
?
(5 + k) - 25
53. lim
0 h
w2 + 5kw + 4K?
54. Jim
,ko
w2 + kw
x-1
X-1
55. lim
56. lim
I VX-1
- 4x + 5 - 3
3(x-4) Vx+5
57. lim
3- Vx+ 5
58. lim Vax +
- Ver +1-70
Practice Exercises
19-70. Evaluating limits Find the following limits or state that they do
not exist. Assume a, b, c, and k are fixed real numbers.
19. lim (3.x-7)
20. lim (-2x + 5)
4
1
21. lim 5x
22. lim 4
1-9
23. lim (2x - 3x2 + 4x + 5) 24. lim (t? + St + 7)
60. lim
sin 2
0 sin
59. lim x COS X
00
1 - COS X
61. B
Pence - 3 cos x + 2
cos X - 1
62. lim
10 cos2x - 1
w - 31
64. lim
-3 w2 - 7w + 12
1
-2
X-X
63.lim
0 1.
26. lim V-10
3
121 - 4
65. lim
2 2-4
5x2 + 6x +1
25. lim
8x - 4
3p
27. lim
p-2 V4p + 1 - 1
-5x
29. lim
- V4x - 3
28. lim (x - *)
66. lim 8(x), where g(x) =
if x < -1
I + 1
-2
ifr 2-1
31. lim (Sx-6)*2
|x - 51
68. lim
67. lim
X - 31
2
5r? - 25
x2 - 1
33. lim
--
30. lim
- V16 + 3h + 4
100
32. lim
- (10h - 1)" + 2
x² - 2x - 3
34. lim
-3
312 - 71 + 2
36. lim
2 - 1
(x + b)? + (x + b) to
38. lim
--- 4(x + b)
*- 3
14
2
x2 - 16
35. lim
4 - X
(- 6) sox+b
37. lim
x-6
1
5 + 1
40. lim
-0
1
5
X-1
69. lim
70. lim
'Vx-T
VP-1
71. Explain why or why not Determine whether the following state-
ments are true and give an explanation or counterexample.
Assume a and L are finite numbers.
a. If lim f(x) = L, then f(a) - L.
b. If lim $(x) = 4, then lim f(x) = L.
c. If lim f(x) = L and lim g(x) = L, then f(a) = g(a).
$(x)
d. The limit lim does not exist if s(a) = 0.
8)
e. If lim VFX) - lim f(x). it follows that
lim VF(x) = V lim f(x).
(2x - 1)² - 9
39. lim
*+1
Vs-3
41. lim
19
42. lim
x-9
43. lim
1
61 - 5)
6(+5)
A 415
44. lim
45. Tim
46. lim
a> 0

div>

Viva

Scanne

*2 – 25

23. f(x) =

* – 5

Q = 5

17. Finding limits from a graph Use the graph off in the figure to

find the following values or state that they do not exist. If a limit

does not exist, explain why.

2. f(1)

b. lim f(x)

lim f(x)

1

d. lim f() e. f(3)

f. lim f(x)

X-3

h. lim

1. f(2)

3. lim f(x) k. lim f(x)

1. lim f(x)

12

– 100

a = 100

1 24. f(x) =

Vx – 10

x2 + x-2

25. f(x) =

-a

-1

1 –

26. f(x) =

3

*? – a=1

YA

y = f(x)

27-32. Estimating limits graphically and numerically Use a graph

off to estimate lim f(x) or to show that the limit does not exist. Eval-

wate f(x) near x = a to support your conjecture.

ia=0

X-2

27. f(x) =

; a = 2

sin (x – 2)

tan? (sin x)

28. f(x)

1 – Cox

1 – cos (2x – 2)

29. f(x)

;a=

(x – 1)

3 sin x – 2 cos x + 2

30. f(x)

a=0

3

S

18. One-sided and two-sided limits Use the graph of g in the figure

to find the following values or state that they do not exist. If a

limit does not exist, explain why.

a. 8(2)

b. lim 8(x) C. lim 8(x)

-2

1-2

d. lim 8(x)

e. g(3)

1. lim g{x)

g. lim 8(x)

h. g(4)

i. lim g(x)

31. f(x)

sin (x + 1)

; a = -1

-3

→3

YA

5

4

y=80)

3

2

1

0

1

3

2 – 4x + 3x

32. f(x) =

:a = 3

1x – 31

33. Explain why or why not Determine whether the following state-

ments are true and give an explanation or counterexample.

-9

a. The value of lim does not exist.

3 3

b. The value of lim f(x) is always found by computing f(a).

c. The value of lim f(x) does not exist if f(a) is undefined

d. lim Vx = 0. (Hint: Graph y = V.)

0

e. lim cotx = 0. (Hint: Graph y = cotx.)

**/2

34. The Heaviside function The Heaviside function is used in engi-

neering applications to model flipping a switch. It is defined as

ſo if x < 0
H(x) =
11ifx 20.
a. Sketch a graph of H on the interval [-1.1).
b. Does lim (*) exist?
0
35. Postage rates Assume postage for sending a first-class letter in
the United States is $0.47 for the first ounce (up to and including
1 oz) plus $0.21 for each additional ounce (up to and including
cach additional ounce).
. Graph the function p = f(w) that gives the postage p for
sending a letter that weighs w ounces, for 0 -1
if x < 2
20. (x) =
1x -
VE if x < 4
- 21. f(x) = 3
if. 4; a 4
lat1 x 4
22. 56) - x + 2] + 2; a = -2
- 19. 1(x) = {
ift > 212 = 2

Scanned wie

50

CHAPTER 2

LIMITS

T9.

a

5. Use the graph of f in the figure to find the following values or

state that they do not exist.

a. f(1) b. lim f(x)

c. f(0)

d.

YA

b.

3

T 10.

LE

y=f(x)

a.

с.

11. Ex

6.

12. Ex

Use the graph of f in the figure to find the following values or

state that they do not exist.

a. f(2)

b. lim f(x)

c. lim f(x)

x5

YA

d. lim f(x)

>4

13. If

.

rea

6

T 14. Let

5

y = f(x)

4

3-

b.

2

1

0

5

1. Calculate the limits. Specify + or -o if appropriate. You cannot use L’Hopital’s

rule.

3.02 – 3

(a) (6 pts) lim

(Show work.)

x=+00.22 – 3x + 2

Math 150 A Final.

3.2 – 3

(b) (6 pts) lim

31.x2 -3.0 + 2

(Show work.)

CSU Northridge.

6.2

(c) (6 pts) lim

2-0 sin(3.c)

(Show work.)

5/15/21

Math 150A Final

(d) (8 pts) For the function f(c) graphed below:

i) lim f(x) = ii) lim f(c) = iii) lim f(x) =

2+2

2+2+

iv) lim f(x) =

23+

4

3

o

2

1

Page 1

2

-1

5

6

Page 1

2. Find the derivative of f(c). Show work.

sin(x)

(a) (6 pts) f(x)

1 + cos(x)

Page 2

(b) (6 pts) f(x) = xV1 +4x.

Math 150A Final

5/15/21

(c) (6 pts) f(x) = ln(1+x).

CSU Northridge.

(d) (8 pts) f(x) =

= [71

V1 + 3 dt. (Do not integrate.)

Math 150 A Final.

Page 2

3.

f(x) = (3x +1.

(a) (12 pts) Use the limit definition of derivative to find f'(2).

Math 150A Final.

CSU Northridge.

5/15/21

(b) (4 pts) Find the equation of the tangent line to f(2) at x = 5, in the form

y = mx +b.

Math 150A Final

Page 3

Page 3

5. Suppose f(x) is continuous, and the derivatives satisfy the following sign charts:

+

1

+

f”(x)

2

Math 150A Final.

(a) (2 pts) List the x values of any local maxima of f.

(b) (2 pts) List the r values of any local minima of f.

CSU Northridge.

(c) (2 pts) List the x values of any inflection points of f.

(d) (10 pts) Sketch a graph of f(x). (Assume the graph does not leave this region):

5/15/21

5

-f(x)

4

3

Math 150A Final

2

1

1

2

3

4

5

6

Page 5

Page 5

6. Fencing costs $5/ft, so that the cost of the following fence is 5x + 10y.

The area of the fence is to be cy = 200 ft.

(a) (3 pts) Find a function f(x) that gives the cost of the fence.

Y

200 ft2

Page 6.

.

(b) (9 pts) Find a critical point of f(), where a local minimum could occur.

Math 150A Final

5/15/21

(c) (5 pts) Using either the first or second derivative test, explain why your critical

point is a local minimum.

CSU Northridge.

Math 150A Final.

(d) (3 pts) What is the minimum cost of the fence?

Page 6

4. (a) (12 pts) Find out at the point (-1,3):

sin(2y – -6)

= ry +3.

Page 4.

Math 150A Final

5/15/21

(b) (16 pts) A balloon is launched 8 meters from the point P and is rising at a rate of

2 m/sec. Find out when the balloon is at height h = 6 meters.

2 m/sec

CSU Northridge.

>

P

8 m

Math 150A Final.

Page 4

8. Let R be the region bounded by y = (3x +1, y = 22, and x = 0). See picture.

(a) (10 pts) Find the volume of the solid given by revolving R around the x-axis.

Page 8.

R

C

= 2

Math 150A Final

5/15/21

(b) (5 pts) Set up an integral for the volume given by revolving R around the y-axis.

Do not evaluate.

CSU Northridge.

(c) (5 pts) Set up an integral for the volume given by revolving R around the line

= 2. Do not evaluate.

Math 150A Final.

Page 8

7. Evaluate each integral. Show work.

**? [()u}sos ] (syd 8) (8)

Math 150A Final.

(b) (8 pts)

l a terme det.

CSU Northridge.

5/15/21

(c) (8 pts)

[l– 21 de.

Math 150 A Final

Page 7

Page 7

9. (12 pts) A ball on a spring requires 12 Joules (=Newton-meters) of work to be moved

from equilibrium to x = 2 meters, where x is the distance from equilibrium. Find the

amount of work that it takes to move the ball from x=2 meters to r= 3 meters.

Math 150A Final.

CSU Northridge.

10. A function f(x) is graphed below.

-f(2)

6

5

4

3

2

1

5/15/21

1

2

3

4

5

6

7

8

(a) (6 pts) Estimate So* f(x) dx by a left Riemann sum, with N = 4.

Math 150A Final

(b) (6 pts) Estimate $$ $(x) dx by a right Riemann sum, with N = 4.

Page 9.

Page 9

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Tags:

real numbers

One sided limits

Strange Behaviors

Step Functions

polynomials

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