36 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
Plug in the value of the derivative:
2 D h
0
.2/
D 0:6.2/
2
2 C C
1
D 0:4 CC
1
1:6 D C
1
h
0
.x/ D 0:6x
2
x C1:6
Now we move on to the function itself:
h.x/ D
Z
.0:6x
2
x C1:6/dx
D 0:2x
3
0:5x
2
C 1:6x CC
2
4 D h.0/
4 D 0 C0 C 0 CC
4 D C
h.x/ D 0:2x
3
0:5x
2
C 1:6x C4
And we have our answer. Notice that, since two unknown constants appeared, we gave them
different names: C
1
and C
2
. Also notice that it is much easier to deal with added information
or initial conditions that happen at time zero.
˙
1.3. INITIAL VALUE PROBLEMS 37
PROBLEMS
Problem1.55 Assume we are describing the upward motion of a projectile. Given the constant
acceleration, initial velocity, and initial position, determine the greatest height of the projectile
and the time it has that height.
1. a D 5 m/s
2
, v
0
D 20 m/sec, and s
0
D 5 m
2. a D 5 m/s
2
, v
0
D 10 m/sec, and s
0
D 4 m
3. a D 5 m/s
2
, v
0
D 12 m/sec, and s
0
D 3 m
4. a D 8 m/s
2
, v
0
D 30 m/sec, and s
0
D 4:1 m
5. a D 2 m/s
2
, v
0
D 11 m/sec, and s
0
D 1:4 m
6. a D 14 m/s
2
, v
0
D 14:2 m/sec, and s
0
D 7:33 m
Problem 1.56 Find s.t/ given the following rates of acceleration and initial velocity and posi-
tion.
1. a.t/ D 3:2 C 1:1 t m/s
2
, v
0
D 3 m/s, s
0
D 0 m
2. a.t/ D
p
t m/s
2
, v
0
D 1 m/s, s
0
D 5 m
3. a.t/ D t
1:5
m/s
2
, v
0
D 4 m/s, s
0
D 2 m
4. a.t/ D t
2
t m/s
2
, v
0
D 2 m/s, s
0
D 0:4 m
5. a.t/ D cos.t/ m/s
2
, v
0
D 1 m/s, s
0
D 0 m
6. a.t/ D
1
t
m/s
2
, v
0
D 1 m/s, s
0
D 0 m
Problem 1.57 Solve the following initial value problems.
1. f .x/ D
Z
x
2
x dx when f .0/ D 3
2. g.x/ D
Z
sin.x/ dx when g.0/ D 0:7071
3. h.x/D
Z
0:04x
3
2x C4
dx when h.1/ D 2:4
38 1. INTEGRATION, AREA, AND INITIAL VALUE PROBLEMS
4. r.x/ D
Z
0:08x C
2:3
x
dx when r.2/ D 0:4
5. s.x/ D
Z
dx
x
2
C 1
when s.1/ D 1:2
6. q.x/ D
Z
x
2
C 1
x
dx when q.1/ D 8:2
Problem 1.58 Suppose that
f
00
.x/ D 2x C 1
If f .1/ D 3 and f
0
.1/ D 0:5, find an expression for f .x/ with no unknown constants.
Problem 1.59 Suppose that
g
00
.x/ D 0:03x
2
4x C 1
If g.2/ D 7 and g
0
.1/ D 3, find an expression for g.x/ with no unknown constants.
Problem 1.60 Suppose that
h
00
.x/ D 1:4x C5:6
If h.1/ D 2 and h.2/ D 4, find an expression for h.x/ with no unknown constants. Notice that
you will have to integrate an unknown constant before you can solve.
Problem 1.61 Suppose that
r
00
.x/ D 2e
x
If r.0/ D 2 and r
0
.0/ D 8, find an expression for r.x/ with no unknown constants.
Problem 1.62 Suppose that
s
00
.x/ D e
x
If s.0/ D 1:2 and s.1/ D 5, find an expression for s.x/ with no unknown constants. Notice that
you will have to integrate an unknown constant before you can solve.
Problem 1.63 Suppose that
q
00
.x/ D sin.x/ C cos.x/
If q.0/ D 1:0 and q.=2/ D 0:5, find an expression for q.x/ with no unknown constants. Notice
that you will have to integrate an unknown constant before you can solve.
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