5.1 Using the Mean Value Theorem

• The average rate of change of a function over the interval [a, b] is equal to the instantaneous rate of change at some point c
• What are the two conditions required for the Mean Value Theorem to hold true for a function f(x) over an interval [a, b]?

  Continuity and differentiability
• Order the steps to apply the Mean Value Theorem:

  1️⃣ Verify continuity on [a, b]

  2️⃣ Verify differentiability on (a, b)

  3️⃣ Calculate the average rate of change (f(b) - f(a))/(b - a)

  4️⃣ Find the point c such that f'(c) = (f(b) - f(a))/(b - a)
• What are the two conditions required for the Mean Value Theorem (MVT) to hold true for a function f(x) over an interval [a, b]?

  Continuity and differentiability
• Match the condition with its requirement for the Mean Value Theorem:

  Continuity ↔️ f(x) must be continuous on [a, b]

  Differentiability ↔️ f(x) must be differentiable on (a, b)
• In the Mean Value Theorem formula, f(b) - f(a) represents the change in the function value over the interval [a, b].


  True
• Match the component of the Mean Value Theorem formula with its description:

  f(b) - f(a) ↔️ Change in function value

  b - a ↔️ Interval length

  (f(b) - f(a)) / (b - a) ↔️ Average rate of change

  f'(c) ↔️ Instantaneous rate of change at point c
• In Example 1, the Mean Value Theorem guarantees the existence of a point c in (1, 5) where f'(c) = 3.


  True
• For the quadratic function, the value of c guaranteed by the Mean Value Theorem is 1.5.

  True
• For the function $f(x) =$$x^{2} - 2x +$$1$, the value of $c$ that satisfies the MVT in the interval $( - 1, 2)$ is $1.5$.


  True
• What does the Mean Value Theorem (MVT) state if a function f(x) is continuous on [a, b] and differentiable on (a, b)?

  f'(c) = (f(b) - f(a))/(b - a)
• Match the theorem with its key idea:

  Mean Value Theorem (MVT) ↔️ The average rate of change equals the instantaneous rate of change

  Intermediate Value Theorem (IVT) ↔️ A continuous function must pass through zero if it changes sign
• What does differentiability on the open interval (a, b) imply for the function f(x)?

  The derivative exists
• What is the second condition required for the Mean Value Theorem, and on which interval must it hold?

  Differentiability on (a, b)
• The Mean Value Theorem requires f(x) to be differentiable on the open interval (a, b).


  True
• The formula for the Mean Value Theorem is \frac{f(b) - f(a)}{b - a}
• The Mean Value Theorem states that there exists a point c in the interval (a, b) where f'(c) is equal to the average rate of change
• The average rate of change for the linear function f(x) = 2x + 3 over [1, 5] is 3
• The average rate of change for the quadratic function over [-1, 2] is 1
• By the Mean Value Theorem, there exists a point c in the open interval $( - 1, 2)$ such that $f'(c) =$$1$.
• The Mean Value Theorem states that if a function $f(x)$ is continuous on [a, b], it must also be differentiable on $(a, b)$.
• What does the Intermediate Value Theorem state about continuous functions?

  They must pass through zero
• The function $f(x)$ must be differentiable on the open interval $(a, b)$ to satisfy the Mean Value Theorem.


  True
• The term $b - a$ in the MVT formula represents the length of the interval.
• In the MVT formula, $f'(c)$ represents the instantaneous rate of change at the point c.
• For the quadratic function $f(x) =$$x^{2} - 4x +$$5$, the value of $c$ that satisfies the MVT in the interval $(1, 3)$ is 2.
• Rolle's Theorem states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point c in (a, b) such that f'(c) = 0
• For Rolle's Theorem to hold, the function must start and end at the same y-value
• What is the average rate of change for f(x) = x^3 - 3x + 2 on the interval [-2, 2]?

  1
• Both values of c ≈ ±1.15 for f(x) = x^3 - 3x + 2 are within the interval (-2, 2).


  True
• The value of c ≈ 0.69 for f(x) = cos(x) is within the interval (0, π).


  True
• What is the formula for the Mean Value Theorem?

  \frac{f(b) - f(a)}{b - a} = f'(c)</latex>
• Order the key ideas of the Mean Value Theorem and Intermediate Value Theorem:

  1️⃣ MVT: Average rate of change equals instantaneous rate of change at some point

  2️⃣ IVT: A continuous function must pass through zero if it changes sign
• Match the condition for the MVT with its requirement:

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

  Differentiability ↔️ $f(x)$ must be differentiable on $(a, b)$
• The MVT asserts that there exists a point $c$ in the interval $(a, b)$ where the instantaneous rate of change equals the average rate of change.


  True
• The Mean Value Theorem guarantees that the instantaneous rate of change at some point $c$ is equal to the average rate of change over the interval $[a, b]$.


  True
• Rolle's Theorem is a special case of the Mean Value Theorem where $f(a) =$$f(b)$.


  True
• Rolle's Theorem is a special case of the Mean Value Theorem.

  True
• What is the first step to apply the Mean Value Theorem (MVT)?

  Verify the conditions
• The derivative of f(x) = x^3 - 3x + 2 is 3x^2 - 3