Homework
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Homework Assignment #6
Read Section 2.7
Page 102, Exercises: 1 – 45 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
1. Is f (x) squeezed by u(x) and l(x) at x = 3? At x = 2?
y
u(x)
f(x)
1.5
l(x)
1
2
3
4
x
The function f (x) is squeezed at x = 3 and trapped at x = 2.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Use the Squeeze Theorem to evaluate the limit.
1
5. lim x cos
x 0
x
1
1
lim x cos   x  x cos  x  lim x  0
x 0
x 0
x
x
1
 lim x cos  0
x 0
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit using Theorem 2 as necessary.
sin x cos x
9. lim
x 0
x
sin x cos x
sin x
lim
 lim
lim cos x  1 1  1
x 0
x 0
x
x x 0
sin x cos x
lim
1
x 0
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit using Theorem 2 as necessary.
x2
13. lim 2
x 0 sin x
x2
1
1
11
lim 2  lim
lim

1
x 0 sin x
x 0 sin x x 0 sin x
11
x
x
x2
lim 2  1
x 0 sin x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
sin10 x
17. Let L  lim
x 0
x
(a) Show, by letting θ = 10x, that L  lim10
 0
sin 

sin10 x 10
sin10 x
L  lim
 lim10
 Let   10 x
x 0
x
10 x0
10 x
sin 
L  lim10
 0

(b) Compute L.
L  lim10
 0
sin 

 10lim
 0
sin 

 10
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit.
sin 6h
21. lim
h 0
6h
sin 6h
sin 6h
sin 
lim
 Let   6h  lim
 lim
1
h 0
h 0
 0 
6h
6h
sin 6h
lim
1
h 0
6h
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit.
tan 4 x
25. lim
x 0
9x
tan 4 x 4
4 tan 4 x 4
tan 4 x
lim
 lim
 lim
x 0
9 x 4 x 0 9 4 x
9 x 0 4 x
4
sin 4 x 1
4
 lim

9 x0 4 x cos 4 x 9
tan 4 x 4 4
lim

x 0
9x 4 9
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit.
sin z
3
29. lim
z 0 sin z
1
sin z z
3 3
lim
z 0 sin z
1
z
3
sin z
sin z
sin z
3
3
3
z
z
z
1
1
3
3
3
 lim
 lim
 lim

z 0 sin z
z 0
sin z 3 z 0 sin z
3
3
z
z
z
3
sin z
1
3
lim

z 0 sin z
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit.
sin 5 x sin 2 x
33. lim
x 0 sin 3 x sin 5 x
sin 2 x
1
sin 5 x sin 2 x
sin 2 x 6 x
lim
 lim
 lim 6 x
x 0 sin 3 x sin 5 x
x 0 sin 3 x 1
x 0 sin 3 x
6x
6x
1 sin 2 x 1
2
sin 5 x sin 2 x 2
3
2
x
3
 lim
   lim

x 0 1 sin 3 x
x 0 sin 3 x sin 5 x
1 3
3
2 3x
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
Evaluate the limit.
1  cos t
37. lim
t 0
sin t
1  cos t 1  cos t
1  cos 2 t
lim
 lim
t 0
sin t 1  cos t t 0 sin t 1  cos t 
sin 2 t
sin t
 lim
 lim
0
t 0 sin t 1  cos t 
t 0 1  cos t 
1  cos t
lim
0
t 0
sin t
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
sin x
sin x
41. Calculate:  a  lim
and  b  lim
x 0
x 0
x
x


sin x
1
 a  xlim
0
x

sin x
 1
 b  xlim
0
x

Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 102
45. Plot the graphs of u  x   1  x 

2
and l  x   sin x
on the same axis. What can you say about the lim f  x  if
x
f  x  is squeezed by u  x  and l  x  at x 

2
2
?
y
lim f  x   1
x


2
x




Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 2: Limits
Section 2.7: Intermediate Value Theorem
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
If f (x) is a continuous function on some interval [a, b],
f (x) must equal every value between f (a) and f (b) at least
once on the interval [a, b].
Consider an airplane that is climbing to 20,000 feet after
take-off. It must pass through every altitude between
ground level and 20,000 ft. It can not just “jump” from
10,000 ft to 12,000 ft without passing through 11,000 ft.
This is the basis of the Intermediate Value Theorem.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The graph on the left illustrates the climb of the aircraft we
just mentioned, while the graph on the right is of the greatest
integer function. The greatest integer function is not
continuous on [1, 2], or between any other two integers, so
the Intermediate Value Theorem does not apply to it when
considering closed intervals between integers. Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The graph below shows that there is a value of x on [0, π/2] for
which sin x = 0.3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
y

B

A

C

x










E

D


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Find a length of 1/8
containing a zero of f (x)
in Figure 3.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 106
6. Use the IVT to find a length of ½ containing a root of
f (x) = x3 + 2x + 1.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 106
Use the IVT to prove each of the following statements.
7. c  c  1  2 for some number c.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 106
Use the IVT to prove each of the following statements.
14. tan x  x has infinitely many solutions.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 106
Use the IVT to prove each of the following statements.
16. tan 1 x  cos 1 x has a solution.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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

Homework Assignment #7
Read Section 2.8
Page 106, Exercises: 1 – 25 (EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company