R s OMQaJdqey zw5i8tShp QIMn8f6iTn4i0t2ev pCBaSlTcXuml4uPsh.D. Worksheet by Kuta Software LLC. Kuta Software - Infinite
Kuta Software - Infinite Calculus
Name___________________________________
Mean Value Theorem
Date________________ Period____
For each problem, find the values of c that satisfy the Mean Value Theorem. 1) y = − x 2 + 8 x − 17; [3, 6]
2) y = x 3 − 9 x 2 + 24 x − 18; [2, 4]
y
−8
−6
3) y = −
−4
y
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
x2 1 + x − ; [−2, 1] 2 2
4) y =
4
6
x2 − 2 x − 1; [−1, 1] 2
5) y = x 3 + 3 x 2 − 2; [−2, 0]
6) y = − x 3 + 4 x 2 − 3; [0, 4]
x2 − 9 7) y = ; [1, 4] 3x
x2 8) y = ; [−4, 1] 2x − 4
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8 x
Worksheet by Kuta Software LLC
1 2
9) y = −(−2 x + 6) ; [−2, 3]
1 2
10) y = −(−5 x + 25) ; [3, 5]
For each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not. 11) y = −
x2 ; [−3, −1] 4x + 8
2 3
13) y = −(6 x + 24) ; [−4, −1]
12) y =
−x2 + 9 ; [1, 3] 4x
2 3
14) y = ( x − 3) ; [1, 4]
Critical thinking question: 15) Use the Mean Value Theorem to prove that sin a − sin b ≤ a − b for all real values of a and b where a ≠ b.
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Worksheet by Kuta Software LLC
Kuta Software - Infinite Calculus
Name___________________________________
Mean Value Theorem
Date________________ Period____
For each problem, find the values of c that satisfy the Mean Value Theorem. 1) y = − x 2 + 8 x − 17; [3, 6]
2) y = x 3 − 9 x 2 + 24 x − 18; [2, 4]
y
−8
−6
−4
y
8
8
6
6
4
4
2
2
−2
2
4
6
8 x
−8
−4
−6
−6
−8
−8
{
x2 1 + x − ; [−2, 1] 2 2
9+ 3
4) y =
3 9− ,
4
3 3
6
8 x
}
x2 − 2 x − 1; [−1, 1] 2
{0 }
1 2
5) y = x 3 + 3 x 2 − 2; [−2, 0] −3 + 3
2
−4
{ }
{
−2 −2
9 2
−
−4
−2
{} 3) y = −
−6
3 −3 − , 3
3
6) y = − x 3 + 4 x 2 − 3; [0, 4]
}
x2 − 9 7) y = ; [1, 4] 3x
{} 8 3
x2 8) y = ; [−4, 1] 2x − 4
{2 −
{2 }
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6}
Worksheet by Kuta Software LLC
1 2
9) y = −(−2 x + 6) ; [−2, 3]
1 2
10) y = −(−5 x + 25) ; [3, 5]
{}
{}
7 4
9 2
For each problem, determine if the Mean Value Theorem can be applied. If it can, find all values of c that satisfy the theorem. If it cannot, explain why not. 11) y = −
x2 ; [−3, −1] 4x + 8
12) y =
{
The function is not continuous on [−3, −1]
2 3
13) y = −(6 x + 24) ; [−4, −1]
3}
2 3
14) y = ( x − 3) ; [1, 4]
{ } −
−x2 + 9 ; [1, 3] 4x
The function is not differentiable on (1, 4)
28 9
Critical thinking question: 15) Use the Mean Value Theorem to prove that sin a − sin b ≤ a − b for all real values of a and b where a ≠ b. Let f ( x) = sin x. Use the interval [a,b]. By the MVT, we know that there is at least one c sin b − sin a sin b − sin a = cos c. We know cos c ≤ 1 for all c. Therefore, ≤ 1, such that b−a b−a sin a − sin b ≤ 1, and sin a − sin b ≤ a − b . a−b
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Worksheet by Kuta Software LLC
