WLAC One Sided Limits Strange Behaviors Polynomials & Step Functions Exam Practice

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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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