Kuta Software - Infinite Calculus
Name___________________________________
Mean Value Theorem for Integrals
Date________________ Period____
For each problem, find the average value of the function over the given interval.
1) f ( x) = − x 2 − 2 x + 5; [−4, 0]
2) f ( x) = − x 4 + 2 x 2 + 4; [−2, 1]
f(x)
−8
−6
−4
f(x)
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
For each problem, find the values of c that satisfy the Mean Value Theorem for Integrals.
2
3
+ x + ; [−3, 1]
2
2
x
f ( x) = −
3)
4) f ( x) =
4
; [−4, −2]
x2
f(x)
−8
−6
−4
f(x)
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
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4
6
8 x
Worksheet by Kuta Software LLC
For each problem, find the average value of the function over the given interval.
5) f ( x) = − x 3 + 7 x 2 − 11 x + 3; [1, 5]
6) f ( x) = − x 5 + 3 x 3 ; [0, 1]
8) f ( x) = x 5 − 2 x 3 + x; [−1, 0]
1
2
7) f ( x) = 4 x ; [0, 3]
9)
1
f ( x) = ;
x
10) f ( x) = x 5 − 4 x 3 + 2 x − 1; [−2, 2]
[2, 3]
11) f ( x) = − x 5 + 4 x 3 − 5 x − 3; [−2, 0]
12) f ( x) = x 5 − 2 x 3 − 2; [−1, 1]
For each problem, find the average value of the function over the given interval. Then, find the values of c
that satisfy the Mean Value Theorem for Integrals.
13) f ( x) = − x + 2; [−2, 2]
1
2
15) f ( x) = −3(2 x − 6) ; [3, 5]
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14) f ( x) = − x 2 − 8 x − 17; [−6, −3]
16) f ( x) =
-2-
4
(2 x + 6) 2
; [−6, −5]
Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus
Name___________________________________
Mean Value Theorem for Integrals
Date________________ Period____
For each problem, find the average value of the function over the given interval.
1) f ( x) = − x 2 − 2 x + 5; [−4, 0]
2) f ( x) = − x 4 + 2 x 2 + 4; [−2, 1]
f(x)
−8
−6
−4
f(x)
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
4
6
8 x
19
= 3.8
5
11
≈ 3.667
3
For each problem, find the values of c that satisfy the Mean Value Theorem for Integrals.
2
3
+ x + ; [−3, 1]
2
2
x
f ( x) = −
3)
4) f ( x) =
4
; [−4, −2]
x2
f(x)
−8
−6
−4
f(x)
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
6
8 x
−2 2 ≈ −2.828
3−4 3
≈ −1.309
3
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4
-1-
Worksheet by Kuta Software LLC
For each problem, find the average value of the function over the given interval.
5) f ( x) = − x 3 + 7 x 2 − 11 x + 3; [1, 5]
6) f ( x) = − x 5 + 3 x 3 ; [0, 1]
10
≈ 3.333
3
7
≈ 0.583
12
8) f ( x) = x 5 − 2 x 3 + x; [−1, 0]
1
2
7) f ( x) = 4 x ; [0, 3]
−
8 3
≈ 4.619
3
9)
1
f ( x) = ;
x
1
≈ −0.167
6
10) f ( x) = x 5 − 4 x 3 + 2 x − 1; [−2, 2]
[2, 3]
−1
ln 3 − ln 2 ≈ 0.405
11) f ( x) = − x 5 + 4 x 3 − 5 x − 3; [−2, 0]
−
12) f ( x) = x 5 − 2 x 3 − 2; [−1, 1]
−2
2
≈ −0.667
3
For each problem, find the average value of the function over the given interval. Then, find the values of c
that satisfy the Mean Value Theorem for Integrals.
13) f ( x) = − x + 2; [−2, 2]
14) f ( x) = − x 2 − 8 x − 17; [−6, −3]
Average value of function: 2
Values that satisfy MVT: 0
1
2
15) f ( x) = −3(2 x − 6) ; [3, 5]
Average value of function: −2
Values that satisfy MVT: −5, −3
16) f ( x) =
Average value of function: −4
35
Values that satisfy MVT:
≈ 3.889
9
4
(2 x + 6) 2
; [−6, −5]
1
≈ 0.167
6
Values that satisfy MVT: −3 − 6 ≈ −5.449
Average value of function:
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Worksheet by Kuta Software LLC