Quinton Fischer
Assignment Homework 07 F19 due 11/12/2019 at 07:00pm EST
MA103 F19
4. (1 point)
Find the particular antiderivative that satisfies the following
conditions:
1. (1 point)
Find the most general antiderivative F of
f (x) = 3x2 + 1x + 8.
√
F 0 (x) = x4 + 4 x;
F(x) =
F(1) = −1.
F(x) =
Answer(s) submitted:
NOTE: Don’t forget the constant in your answer.
• xˆ5/5+8xˆ(3/2)/3-58/15
Answer(s) submitted:
(correct)
• xˆ3+xˆ2/2+8x+c
5. (1 point)
Find the particular antiderivative that satisfies the following
conditions:
(correct)
2. (1 point)
Find the most general antiderivative F of
H 0 (x) =
f (x) =
−6x2 − 3x + 10
.
x
5 10
− ;
x3 x5
H(1) = 0.
H(x) =
Answer(s) submitted:
F(x) =
• 5/(2xˆ4)-5/(2xˆ2)
NOTE: Don’t forget the constant in your answer.
(correct)
Answer(s) submitted:
6. (1 point) Suppose f 00 (x) = − (sin(x)), f 0 (0) = −4, and
f (0) = 5.
• 10lnx-3xˆ2-3x+c
(correct)
3. (1 point)
Find the most general antiderivative y of
Find f (π/2).
f (π/2) =
Answer(s) submitted:
• 6-5pi/2
dy
= 8ex + 3.
dx
(correct)
Z 29
7. (1 point) Estimate I =
y=
8
(7x − 7x2 )dx using a Riemann
sum, midpoints, and n = 3 subintervals.
NOTE: Don’t forget the constant in your answer.
I≈
Answer(s) submitted:
Answer(s) submitted:
• 8eˆx+3x+C
• -52393.25
(correct)
(correct)
1
8. (1 point)
Z 10
Estimate I =
∆x =
(3x2 + 4x + 5) dx using a Riemann sum,
4
n = 3 subintervals, and
xi =
(a) Left endpoints.
(b) Using the definition, evaluate the integral.
I≈
(b) Right endpoints.
Z 5
(4 + 3x) dx =
−1
I≈
Answer(s) submitted:
Answer(s) submitted:
• 6/n
• -1+6i/n
• 60
• 870
• 1422
(correct)
(correct)
9. (1 point)
Z b
11. (1 point) Suppose that the integral I =
Z π/2
Estimate I =
f (x) dx can
0
x sin(x)dx
0
π π π π
using the partition {0, , , , } and
6 4 3 2
be represented by the following Riemann sum using right endpoints and n subintervals:
s
(a) Left endpoints.
s
s
2
2
2
4
8
4n
4
4
4
16 −
· + 16 −
· + . . . + 16 −
·
n
n
n
n
n
n
I≈
(a) Determine b.
b=
(b) Right endpoints.
(b) Determine f (x).
I≈
f (x) =
Answer(s) submitted:
(c) Evaluate the integral by interpreting it in terms of areas.
• 0.68878
• 1.342364
I=
(correct)
Answer(s) submitted:
10. (1 point)
• 4
• sqrt(16-xˆ2)
• 4pi
Recall the definition of a definite integral:
(correct)
n
Z b
12. (1 point)
Consider the function
f (x)dx = lim
∑ f (xi )∆x
n→∞
a
i=1
where xi are right endpoints.
(a) Determine ∆x and xi in the definition for
f (x) =
Z 5
(4 + 3x) dx.
−1
2
−5, if 1 ≤ x < 8
4, if 8 ≤ x ≤ 15
Evaluate the following definite integral using properties of
definite integrals.
Z 5
(b)
9 f (x) − 10 dx =
6
Answer(s) submitted:
• -7
• 73
Z 15
f (x) dx =
1
(correct)
15. (1 point) The sum:
Answer(s) submitted:
• -7
Z 2
(correct)
Z 5
f (x) dx +
−2
13. (1 point)
Given that 4 ≤ f (x) ≤ 6 for −6 ≤ x ≤ 1, use a property
of definite integrals to find an interval in which the value of
f (x) dx −
can be written as a single integral in the form:
f (x)dx lies.
−6
≤
Answer:
Z b
f (x) dx.
f (x)dx ≤
a
−6
Answer(s) submitted:
• 28
• 42
Determine a and b.
(correct)
Z 7
Z 5
f (x) dx = 9,
14. (1 point) Suppose that
4
Z 7
a=
f (x) dx = 10,
4
f (x) dx = 6.
and
b=
6
Answer(s) submitted:
• -1
• 5
Evaluate the following definite integrals:
Z 6
f (x) dx =
(a)
f (x) dx
−2
2
Z 1
Z 1
Z −1
(correct)
5
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