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epsilon - delta definition of limits - Homework
In exercises 1-4, find the limit 7 and find ) ] 0 such that f ( x ) ! L % ( whenever 0 % x ! c % )
1. lim(2 x ! 4 )
( # .01
3. lim ( 3 x + 5)
(#
x *3
x *!2
2. lim(6 x + 1)
( # .05
,
x/
4. lim. 4 ! 1
x *420
(#
x *5
1
5
1
20
In problems 5 - 10, do the following:
a. plot the graph on your calculator
b. graphically find the limit of the function as x approaches the given value
c. calculate the maximum value of ) that can be used for the given value of ( at the point.
1
x = 3, ( = 0.8
6. f ( x ) # 5 ! 2 sin( x ! 3)
7. f ( x ) # ( x ! 2) + 3
x = 2, ( = 0.5
8. f ( x ) # 1 + 33 7 ! x x = 6, ( = 0.7
9. f ( x ) # 1 + 2 4 ! x
x = 4, ( = 0.8
10. f ( x ) # 6 ! 2( x ! 3)
5. f ( x ) # ( x ! 2) 3 + 3
3
MasterMathMentor.com
-8-
1
3
x = 3, ( = 0.5
x = 3, ( = 0.4
Stu Schwartz
Newton’s Method of Roots - Homework
In the following exercises, use Newton’s Method by hand to find the first iteration in approximating the zeros and
continue the process with the calculator until 2 successive approximations differ by less than .001.
1.
f ( x ) # x 3 + x + 3 (Initial Guess = 1)
2.
f ( x ) # x 5 + x + 3 (Initial Guess = 1)
3.
f ( x) # x 2 !
1
(Initial Guess = 3)
x !1
4.
f ( x ) # x 4 ! 10 x 2 ! 7 (Initial Guess = 3)
5.
4
f ( x ) # 2 x + sin( x + 1) (Initial Guess = ! )
N
6.
f ( x ) # x 3 ! cos x + 2 (Initial Guess = 3)
7.
f ( x) # 3)8 x has a root at x = 0. Suppose your initial guess is 1.3. Show that Newton’s method does not work
here and formulate a reason why it does not. Look at the picture visually in order to understand what is
happening.
8) Newton’s method can be used to determine square roots. For x # a , use the equation f ( x ) # x % ! a .
Use this method to find
7 and
214 . Use this method to find
5
5
9. Show that Newton’s method fails to converge for the function f ( x) # x! + using x! # #!
MasterMathMentor.com
- 22 -
Stu Schwartz
Riemann Sums - Homework
For each problem, approximate the area under the given function using the specified number of rectangles/
trapezoids. You are to do all 4 methods to approximate the areas.
v
Function
Interval Number Left
Rectangles
q1,4r
6
1
f ( x ) # x 2 ! 3x + 4
2
f ( x) # x
q2,6r
8
3
f ( x) # 2x
q0,1r
5
4
f ( x ) # sin x
q0,#r
8
Answers are below:
v Left Rectangles
1 9.125
2 7.650
3 1.345
4 1.974
Right Rectangles
12.125
8.168
1.545
1.974
Right
Rectangles
Midpoint Rectangles
10.438
7.914
1.442
2.013
Midpoint
Rectangles
Trapezoids
Trapezoids
10.625
7.909
1.445
1.974
5. Roger decides to run a marathon. Roger’s friend Jeff rides behind him on a bicycle and clocks his pace every 15
minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. The data
Jeff collected is summarized below. Assuming that Roger’s speed is always decreasing, estimate the distance
that Roger ran in a) the first half hour and b) the entire race. (Trapezoids)
Time spent running (min)
Speed (mph)
0
12
15
11
30
10
45
10
60
8
75
7
90
0
6. Coal gas is produced at a gasworks. Pollutants in the air are removed by scrubbers, which become less and less
efficient as time goes on. Measurements are made at the start of each month (although some months were
neglected) showing the rate at which pollutants in the gas are as follows. Use trapezoids to estimate the total
number of tons of coal removed over 9 months.
Time (months)
Rate pollutants are escaping
(tons/month)
0
5
1
7
3
8
4
10
6
13
7
16
9
20
1
, where t is in hours and v is
1+ t
in meters/hr. Find the distance that the bug crawls during this hour using 10 minute increments.
7. For 0 $ t $ 1, a bug is crawling at a velocity v, determined by the formula v #
8. An object has zero initial velocity and a constant acceleration of 32 ft sec 2 . Complete the chart to find the
velocity at these specified times. Then determine the distance traveled in 4 seconds.
t (sec)
v (ft sec)
MasterMathMentor.com
0
.5
1
1.5
2
- 41 -
2.5
3
3.5
4
Stu Schwartz
Arc Length - Homework
Find the exact length of the curves between the indicated x values. No calculators.
1)
2) y # 4 x
(0,2)
y # 10 x ! 1
3
2
(0,4)
x3 1
+ from x = 1 to x = 2. No calculators. The algebra is
12 x
intricate but straightforward. Take your time and do it neatly.
3) Find the exact length of the curve y #
Find the length of the curves between the x-values. Calculators allowed.
4) y # x 2 ! 5 x + 3
5) y # x 3 ! 9 x 2 + 5 x + 50
(1,6)
6) y # sec x
5 48
760, 4 :9
MasterMathMentor.com
7) y # (ln x )
- 59 -
2
(-1,9)
(0.1, e)
Stu Schwartz
dx
1.
# 1+ 4 x
3.
# 4+
2
4 $ x2
dx
2
( x $ 1)
4.
x
dx
4
x + 16
6.
#
8.
# 9+e
5.
#
7.
# 1 + cos
sin x
3 2
9.
Inverse Trig Functions Integration - Homework
dx
2. #
#
0
2
x
dx
1
dx
1+ 4 x2
MasterMathMentor.com
#
t
1$ t4
sin$1 x
1$ x2
e 2x
4x
dt
dx
dx
Find the area of the region bounded by the curves
10.
1
, y = 0, x = 0, x = 1
y=
4 $ x2
- 144 -
Stu Schwartz
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