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Trig/Math Anal
HW NO.
L-1
L-2
L-3
L-4
L-5
L-6
L-7
L-8
L-9
Name_______________________No_____
SECTIONS
ASSIGNMENT
DUE
√
Practice Set A
Practice Set B
Practice Set C
Practice Set D
Practice Set E
Practice Set F
Practice Set G
Practice Set H
Practice Set J
California (Math Analysis) Standard(s):
3.0 Give proofs of various formulas by mathematical induction.
8.0 Understand the notion of limit of a sequence and limit of a function as the independent variable approaches a
number or infinity. Determine whether certain sequences converge or diverge.
Next Test Date:
Practice Set A: Mathematical Induction
Prove by induction.
1. 1  4  7  ...   3n  2   n  3n  1 2. 2  6  10... 4n  2  2n 2
3. 2  6  18...2  3n1  3n  1
Practice Set B: Mathematical Induction
Prove by induction.
1. 21  16 ... n 21 n  nn1
2.  n3  3n 2  2n  is  3
3.  5n  1 is  4
b g
2
b g
b g
4. Expand 2 x  y
5. Write the first four terms of x  y
Practice Set C: Mathematical Induction
Prove by induction.
n n  1 2n  1
2. 3  5  7... 2n  1  n n  2
5n 1  5
1. 12  2 2 ...n 2 
3. 51  52 ...5n 
6
4
3
2n
4.  3  1 is  4
5.  n  n  is  3
5
14
b gb g
b gb g
Practice Set D: Limits
Evaluate.
1. lim x 2  4
x 1
c h
4. lim
x 4
478165373 Page 1
x 2
x4
x 2  16
x 4 x  4
2x  4
5. lim
x 
5x
2. lim
x2  x  6
x 2
x2  4
4  3x
6. lim
x  5  2 x
3. lim
y3
8. lim 1 1
y 3
y  3
x3  8
x 2 x 2  4
Prove by induction.
7. lim
10. 13  23 ...n3 
b g
n2 n  1
4
2
Practice Set E: Limits
Evaluate.
x2  1
1. lim
x 1 3x  3
1 1
4. lim 2 x
x 1 x  1
x2  9x
x 9
x 3
25  5x
5. lim
x 5 1  25
x2
2  x  6x
8. lim
x  2 x 2  10
b g
11. 12  32 ... 2n  1 
2
x3  1
10. lim
x 1 x  1
13.  n 4  2n3  n 2  is  4
y 
14. lim
y 4
bg 4x
bg
6. f x  6 x  x 2
Practice Set G: Limits and Derivatives
Find the limit.
x4  1
x2  4x  4
1. lim
2. lim 2
x 1 x  1
x 2 x  x  6
3
5x 2  7
5. lim
6. lim 2
x  x  4
x  3x  x
10. lim
h
3. f x 
bg x4x2
5x 2  2
x 
x3
x9
13. lim
x 9
x 3
h0
bg
5. f x 
9. lim
2
4
9. lim
x  5 x  1
2
x  25
5 x
b3  hg 9
6. lim
2. f x  x 3
1. f x  5x 2  3x  2
x
x 25
12.  23n  1 is  7
4 n3 n
3
Practice Set F: Derivatives
Find f '( x)
4.
3. lim
2. lim
7y
7. lim
y 0 1  1  y
Prove by induction.
bg
f bg
x 
1
1
9. lim 2  h 2
h 0
h
x 2  6x  5
x 1 x 2  3x  4
x2
7. lim 2
x  x  2 x  1
3. lim
2 y
7  6x5
x  x  3
11. lim
7  6y2
4 y
2 y
15. lim
x0
x3 3
x
b g
7. Expand x  h
4. lim
x 
6
3x  1
2x  5
3s 7  4s 5
s 
2s 7  1
5  2t 3
12. lim 2
t  t  1
8. lim 3
16. lim
x 
radians)
sin x
(use
x
x 1 x  3
bg R
S
T3x  7 x  3 , find:
x
x
x
17a. lim f bg
17b. lim f bg
17c. lim f bg
x . If given, find the slope of the tangent line at the given point.
Find f ' bg
x  3x ; (3, 27)
x  x  x ; (2, 2)
x  2 x  1; (-1, -1)
18. f bg
19. f bg
20. f bg
1
1
x  ax  b
23. f bg
21. f bg
x  x 1 ; (8, 3) 22. f bg
x 
x 
24. f bg
x
x 1
If f x 
x  3
x  3
2
x 3
2
3
2
2
478165373 Page 2
Practice Set H: Review
1. Prove by induction:  32 n  1 is  8
Find:
5 x  5
x
2a. lim
x 0
x 0
2 x 3x  2 I
F
G
Hx  1  x J
K
2c. lim 1
2e. lim
2 x
x2  4
2f. lim
x 
b g (use a table of
2d. lim 1  x
2b. lim
1
x
x2
values)
Find f '( x) :
bg
Practice Set J: Review
Find:
x 3  3x 2  x  3
10  x 2  3
1. lim
2. lim
x 3
x 1
x3
1 x
2 x
x 3
5. lim
6. lim 2
x 2 4  x 2
x 9 x  9 x
2 x  3 if x  2
Given: f x  2
, find:
x
if x  2
9a.
x2 
bg
13. f x  2 x 3
bg
9b. lim f x
x2
Prove by induction.
10.  n 4  2n3  n 2  is  4
Find f '( x)
x3  1
x 1 x  1
3b. f x  6  3x  4 x 2
bg x 2 1
3a. f x 
bg R
S
T
lim f bg
x
x0
3x
1
4  4 x
11.  22 n  1 is  3
bg
14. f x  4 x  1
3. lim
x0
x
1
1
x 6  6
2  x  6x 2
x  2 x 2  10
7. lim
bg
9c. lim f x
x 2
8x 3  1
x 2 2 x  1
4. lim1
cx  1hb3x  4g
lim
2
8.
x 
1  2 x  3x 2
9d. Is it continuous?
bg bg b g nbn  1g6b2n  7g
12. 1 3  2 4 ...n n  2 
bg 1x
15. f x 
bg
16. f x  x 1
Practice Set K: Power Series
1. Write infinite series in expanded form for cos 4 and sin   4  .
2. Write infinite series in expanded form for e and e 1 .
3a. Substitute x  1 in the series for ln(1  x) to get an infinite series for ln 2 .
b. Why can’t you substitute x  2 to get an infinite series for ln 3 ?
4. Substitute x  1 in the series for Tan 1 x to get an infinite series for a number involving  .
Find a power series for each function. State the interval of convergence in each case.
2
5. e  x
7. Tan 1 2 x
6. e x
8. ln(1  x)
10. cos 2x
9. sin x 2
sin x
 1.
11. Use the sine series to show that lim
x 0
x
1  cos x
.
12. Use the cosine series to find lim
x 0
x2
ln(1  x)
.
13. Use series (5) to find lim
x 0
x
478165373 Page 3
14a. Use series (1), (2), and (3) to prove that ei  cos   i sin  .
b. Use part (a) to show that ei  1 and e2i  1
ei  ei
c. Use part (a) to show that cos  
2
i
e  ei
d. Use part (a) to show that sin  
2i
e. Show that the expressions for sin  and cos given in part (c) and (d) satisfy the equation
(sin  )2  (cos  )2  1 .
Prove by induction.
n
n(n  1)(n  2)
16. n(n2  5) is  by 6
15.  i (i  1) 
3
i 1
Determine whether each function is continuous. If it is discontinuous, state where any
discontinuities occur.
2  x 2 if x  1
 x 2 +1 if x  0
17. f ( x)  
18. f ( x)   2
 x if x  0
 x if x  1
x 1
 x2  4
20. f ( x) 
if x  2

1 x
19. f ( x)   x  2
4 if x  2

Determine values for a and b so that each function is continuous.
 x 2 if x  1
1  2 x if x  2


22. f ( x)  ax  b if  2  x  1
21. f ( x)  ax  b if  1  x  1
3x  2 if x  1
 x 2 if x  1


ANSWERS
Practice Set B
4. 32 x5  80 x 4 y  80 x3 y 2  40 x 2 y3  10 xy 4  y 5 5. x14  14 x13 y  91x12 y 2  364 x11 y 3  ...
Practice Set D
1. 5
2. -8
3. 1.25
4. .25
5. .4
6. 1.5
7. -3
8. -9
9. -.25
Practice Set E
2. 54
3. -10
4. ½
5. -12.5
6. 6
7. -14
8. -3
9. 0
10. 3
1. 23
Practice Set F
1. 10x  3
4
1
4.
2
x
2 x
6
5
4 2
3 3
2 4
5
5
7. x  6 x h  15 x h  20 x h  15 x h  6 xh  h
Practice Set G
1. 4
2. 0
3. -4/5
4. 3/2
5. 0
6. 5/3
7. 0
11. 
12. 
13. 6
14. 4
15.
17c. 2
22. x23
18. 6x; 18
23. 2ax
19. 2x-1; 3
24. bx 11g2
2. 3x 2
10. 1 6
9. 5
17a. 2
17b. 2
1
21. 2 x1 ; 1/6
478165373 Page 4
3.
6. 6  2x
5. bx82 g2
1
2 3
20. 6 x 2 ; 6
8. 3 3 2
16. 0
Practice Set H
2b. -1
2a. 105
2d. 
2c. 48
2e. -.25
Practice Set J
1. 8
2. 1/3
3. -36
4. 3
9a. 1 9b. 4
9c. limit does not exist
5. ¼
9d. no
3a. bx21g2
2f. 3
3b. 3  8x
6. 1/54
7. -3
2
14. 4 15.  x12
13. 6x
8. 
16. 2
1
x1
Practice Set K
2
4
6
3
5
7
1. cos 4  1  42 2!  44 4!  46 6!  ...;sin   4    41!  43 3!  45 5!  47 7!  ...
2. e  1  1!1  2!1  3!1  ...; e1  1  1!1  2!1  3!1  ...
3b. x  2 is not in the interval of convergence
3a. 1  12  13  14  ...
4.

4
 1  13  15  71  ...
5. 1  1!x  x2!  x3!  ..., all real x
2
7. 2 x  
6. 1  x1!  x2!  x3!  ..., all real x
2
4
6
8.  x  x2  x3  x4  ..., 1  x  1
2
3
4
9. x 2 
12. ½
10. 1   2!   4!   6!  ..., all real x
17. cont. 18. disc. at x=0
19. cont.
2x
2
2x
4
2x
6
Power Series

(1) e x  1  1!x  x2!  x3!  ...   xn! , for all real x
3
n
n 0

(2) cos x  1  x2!  x4!  x6!  ...   ( (21) n )!x , for all real x
2
4
6
n
2n
n 0

(3) sin x  x  x3!  x5!  x7!  ...   ( (21)n x1)! , for all real x
3
5
7
n
2 n1
n 0

(4) Tan 1 x  x  x3  x5  x7  ...   ( 1)2 n x1 ,  1  x  1
3
5
7
n
2 n1
n 0

(5) ln(1  x)  x  x2  x3  x4  ...   ( 1)n
2
3
4
n 1
478165373 Page 5
3
x 
2 3
3!
20. disc. at x=1
The Derivative of a Function
f ( x  h)  f ( x )
f '( x)  lim
h 0
h
2
2 x
3
n1 n
x
, 1  x  1
3


2 x
5
5
x 
2 5
5!


2 x
7
7
x 
2 7
7!
 ...,  12  x  12
 ..., all real x
13. 1
21. a=-1, b=0
22. a=0,b=5