Multiple Choice If f(x)=cos x, then the Mean Value Theorem guarantees that somewher

Name: Problem 53, Chapter 4: Calculus: Graphical, Numerical, Algebraic
Uploaded: 2020-03-08T16:03:41-08:00
Duration: 1 min 45 s
Channel: Linh Vu
Description: Multiple Choice If f(x)=cos x, then the Mean Value Theorem guarantees that somewhere between 0 and π / 3, f^'(x)= (A) -(3)/(2 π) (B) -(√(3))/(2) (𝐂)-(1)/(2) (D) 0

step 1 So by mean value theorem, we need to find the interval between 0 and pi over 3. There exists f p

Question

$$ \begin{array}{l}{\text { Multiple Choice If } f(x)=\cos x, \text { then the Mean Value }} \\ {\text { Theorem guarantees that somewhere between } 0 \text { and } \pi / 3, f^{\prime}(x)=} \\ {\text { (A) }-\frac{3}{2 \pi} \quad \text { (B) }-\frac{\sqrt{3}}{2} \quad(\mathbf{C})-\frac{1}{2} \quad \text { (D) } 0}\end{array} $$

   $$
\begin{array}{l}{\text { Multiple Choice If } f(x)=\cos x, \text { then the Mean Value }} \\ {\text { Theorem guarantees that somewhere between } 0 \text { and } \pi / 3, f^{\prime}(x)=} \\ {\text { (A) }-\frac{3}{2 \pi} \quad \text { (B) }-\frac{\sqrt{3}}{2} \quad(\mathbf{C})-\frac{1}{2} \quad \text { (D) } 0}\end{array}
$$

Calculus: Graphical, Numerical, Algebraic

Ross L. Finney,… 3rd Edition

Chapter 4, Problem 53

Instant Answer

Solved by Expert Linh Vu

03/08/2020

Step 1

Step 1: The Mean Value Theorem states that if a function is continuous on the interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in the interval (a, b) such that the derivative of the function at c is equal to the average Show more…

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$$ \begin{array}{l}{\text { Multiple Choice If } f(x)=\cos x, \text { then the Mean Value }} \\ {\text { Theorem guarantees that somewhere between } 0 \text { and } \pi / 3, f^{\prime}(x)=} \\ {\text { (A) }-\frac{3}{2 \pi} \quad \text { (B) }-\frac{\sqrt{3}}{2} \quad(\mathbf{C})-\frac{1}{2} \quad \text { (D) } 0}\end{array} $$

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Key Concepts

Mean Value Theorem

The Mean Value Theorem is a fundamental theorem in calculus stating that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point in the interval at which the derivative (instantaneous rate of change) equals the average rate of change over the interval.

Derivative

The derivative of a function measures its instantaneous rate of change at any given point. It is defined as the limit of the difference quotient, and it provides insight into the behavior of the function, such as increasing or decreasing trends.

Average Rate of Change

The average rate of change of a function over an interval is calculated by taking the difference in the function’s values at the endpoints of the interval divided by the length of the interval. The Mean Value Theorem ensures that there is some point where the instantaneous rate of change (the derivative) equals this average rate.

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