5.1 Using the Mean Value Theorem

• What does the Mean Value Theorem state?

  Relates derivative to slope
• The Mean Value Theorem states that if a function f(x)</latex> is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one point $c$ in $(a, b)$ such that f'(c) equals the average rate of change.
• If $f(x)$ is continuous on $[a, b]$, there exists a $c$ in $(a, b)$ such that $f'(c) =$$\frac{f(b) - f(a)}{b - a}$
• The Mean Value Theorem requires that $f(x)$ is differentiable on the open interval (a, b).
• What is the derivative of $f(x) =$$x^{2}$?


  $f'(x) =$$2x$
• In the example $f(x) =$$x^{2}$ on $[1, 3]$, the value of $c$ that satisfies the Mean Value Theorem is $c =$$2$
• What are the two conditions required for the Mean Value Theorem?

  Continuity and differentiability
• The Mean Value Theorem requires $f(x)$ to be continuous on the closed interval [a, b].
• For the function $f(x) =$$x^{2}$ on $[1, 3]$, both continuity and differentiability conditions are satisfied.
• What is the first step in applying the Mean Value Theorem?

  Verify conditions
• The second step in applying the Mean Value Theorem is to calculate the average rate of change using $\frac{f(b) - f(a)}{b - a}$, which represents the slope of the secant line.
• For the function f(x) = x^{2}</latex> on $[1, 3]$, the average rate of change is 4.
• What is the final step in applying the Mean Value Theorem?

  Solve for $c$
• What is the average rate of change of $f(x) =$$x^{2}$ over the interval $[1, 3]$?


  4
• The derivative off(x) = x^{2}</latex> is $f'(x) =$$2x$
• The value of $c$ in the Mean Value Theorem for $f(x) =$$x^{2}$ on $[1, 3]$ is 2
• Steps to apply the Mean Value Theorem

  1️⃣ Verify continuity and differentiability

  2️⃣ Calculate the average rate of change

  3️⃣ Find the derivative $f'(x)$

  4️⃣ Set $f'(c)$ equal to the average rate of change

  5️⃣ Solve for $c$ in $(a, b)$
• For the Mean Value Theorem to apply, $f(x)$ must be continuous on $[a, b]$ and differentiable on (a, b)</latex>
• What formula is used to calculate the average rate of change in the Mean Value Theorem?

  $\frac{f(b) - f(a)}{b - a}$
• To set up the equation in the Mean Value Theorem, $f'(c)$ is set equal to the average rate of change
• What is the function being considered in the example for the Mean Value Theorem?

  $f(x) =$$x^{2}$
• What is the first step in solving problems using the Mean Value Theorem?

  Verify continuity and differentiability
• Steps to solve problems using the Mean Value Theorem in order.

  1️⃣ Verify continuity and differentiability

  2️⃣ Calculate the average rate of change

  3️⃣ Find the derivative $f'(x)$

  4️⃣ Set $f'(c) =$$\frac{f(b) - f(a)}{b - a}$ and solve for $c$
• What do you set f'(c)</latex> equal to when solving for $c$?


  $\frac{f(b) - f(a)}{b - a}$
• The Mean Value Theorem states that if a function $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, there exists at least one point $c$ in $(a, b)$ such that $f'(c) =$$\frac{f(b) - f(a)}{b - a}$.
• The Mean Value Theorem guarantees that there exists a point $c$ in $(a, b)$ where $f'(c)$ equals the average rate of change.
• Match the condition with its result in the Mean Value Theorem.

  $f(x)$ is continuous on $[a, b]$ ↔️ There exists $c$ in $(a, b)$ such that $f'(c) =$$\frac{f(b) - f(a)}{b - a}$

  $f(x)$ is differentiable on $(a, b)$ ↔️ The derivative $f'(x)$ can be found
• What is the average rate of change for $f(x) =$$x^{2}$ on $[1, 3]$?


  4
• For $f(x) =$$x^{2}$, the derivative $f'(x)$ is 2x.
• The Mean Value Theorem requires a function to be continuous on (a, b)</latex> and differentiable on $[a, b]$.


  False
• What does the Mean Value Theorem state regarding a function $f(x)$?


  There exists $c$ such that $f'(c) =$$\frac{f(b) - f(a)}{b - a}$
• The formula for the slope $f'(c)$ in the Mean Value Theorem is \frac{f(b) - f(a)}{b - a}
• If $f(x)$ is continuous on $[a, b]$, the Mean Value Theorem guarantees that there exists a point $c$ in $(a, b)$ where $f'(c) =$$\frac{f(b) - f(a)}{b - a}$.
• What is the function used in the example to illustrate the Mean Value Theorem?

  $f(x) =$$x^{2}$
• The average rate of change for $f(x) =$$x^{2}$ on $[1, 3]$ is 4
• What is the derivative of $f(x) =$$x^{2}$?


  $f'(x) =$$2x$
• Match the condition with its description:

  Continuity ↔️ $f(x)$ is continuous on $[a, b]$

  Differentiability ↔️ $f(x)$ is differentiable on $(a, b)$
• The Mean Value Theorem requires that a function be differentiable on a closed interval $[a, b]$.


  False
• The first step to apply the Mean Value Theorem is to verify that $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$. This ensures the conditions of the theorem
• How is the average rate of change calculated in the Mean Value Theorem?

  \frac{f(b) - f(a)}{b - a}</latex>