Download Homework, Page 292

Homework
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Homework Assignment #31
Review Section 4.9
Page 292, Exercises: 1 – 41(EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Find the general antiderivative of f (x) and check your answer by
differentiating.
1. f  x   12 x
f  x   12 x  F  x   6 x 2  C
F   x   6  2 x   12 x  f  x   checks
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Find the general antiderivative of f (x) and check your answer by
differentiating.
5. f  x   8 x 4
8
 1

f  x   8 x 4  F  x   8  x 3    x 3
3
 3

8

F  x     3x 4   8 x 4  checks
3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Match the function with its antiderivative (a – d).
 a  F  x   cos 1  x 
 b  F  x    cos x
c
1
F  x    cos  x 2 
2
d
F  x   sin x  cos x
9. f  x   sin x
f  x   sin x matches  b  , since F   x      sin x   sin x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
13.   x  1 dx
x2
  x  1 dx   xdx   dx  2  x  C
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
9
17.  t 5 dt
t
9
5
5 14 5
dt   t
C
14
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
21.   5t  9  dt
52
  5t  9  dt   5tdt  9 dt  2t  9t  C
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
2
25.   x  3 dx
  x  3
2
dx    x  3  C
1
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
29.  x  x  1 dx

2 52 2 32
x  x  1 dx   x dx   x dx  x  x  C
5
3
3
2
1
2
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
33.   4sin x  3cos x  dx
  4sin x  3cos x  dx  4 sin xdx  3 cos xdx
 4 cos x  3sin x  C
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
37.  cos  3  4t  dt
1
 cos  3  4t  dt   4 sin 3  4t   C
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 292
Evaluate the indefinite integral.
41.  25e5 x dx
5x
5x
5x
25
e
dx

5
5
e
dx

5
e
C


Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Differential Equations
A differential equation is an equation in the form
dy
 f  x.
dx
For differential equations, we are able to find a specific
antiderivative solution because we are provided an initial
condition. That is, we are told y  x0   y0 for some x0 .
The following examples will demonstrate this process.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 292
Solve the differential equation with initial conditions.
dy
48.
 x3 , y  0   2
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 292
Solve the differential equation with initial conditions.
dy
50.
 0, y  3  5
dt
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 292
Solve the differential equation with initial conditions.
dy
52.
 8 x3  3 x 2  3, y 1  1
dx
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 292
Solve the differential equation with initial conditions.
dy
 
56.
 sin 2 z, y    4
dz
4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #32
Review Section 4.9
Page 292, Exercises: 47 – 69(EOO)
Quiz next time
Work on Practice Final Exam
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company