Calculus 5.1: Mean Value Theorem and Critical Points

​5.1: Mean Value Theorem and Critical Points

By J Barrientos

We will determine the critical points of a function.

​Objective #2

We will use the Mean Value Theorem to determine when the derivative of a function is equal to its average rate of change.

Objective #1

Content Objectives

​Setup

In Topic 1 we discussed the Intermediate Value Theorem. The IVT allowed us to determine whether a function had a specific value over a given interval when certain conditions were met. In this lesson we are going to on the Mean Value Theorem, which will allow us to take a similar approach with the derivative.

​Discuss

Think about this scenario. You wake up in Harlingen in the morning, and you drive to Austin, where you end the day.

Which direction can you guarantee that you drove that day? Discuss with your partner how you know you can guarantee this.

​Discuss

Think about this scenario. You wake up in Harlingen in the morning, and you drive to Austin, where you end the day.

Answer: You can guarantee that you drove NORTH.

How do we know?

Austin, is due north of Harlingen. And while we don't know the exact route that was driven, you had to have gone north to have ended up north. Could you have driven in the other directions? Yes, but we can't guarantee.

​Idea

When we talk about direction, that is similar to the slope of a graph. So how can we take this concept and formalize it in calculus?

Mean Value Theorem

MVT

If a function f is continuous on over the interval [a, b] and differentiable over the interval [a, b], then there exists a point within that open interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval.

Mean Value Theorem

MVT

In other words, there must be a point where f'(x) = AROC as long as the function is continuous and differentiable on a given interval [a, b].

1. f(x) is continuous and differentiable

2. ​f'(x) = AROC

​

​MVT

MVT

​Example #1

MVT

​Example #1

MVT

​Example #1

MVT

Things to Note

If you are going to apply the MVT always ensure a function is continuous or differentiable.

Ensure there are no corners or cusps on a graph.

Math Response

Use the function $$f\left(x\right)=x^2-10x+3$$ on the interval [-1, 4] to determine when the IROC equals the AROC.

Type answer here

Correct Answer: 1.5

Deg^°

Rad

MVT (w/Calculator)

​Using the MVT, find where the IROC = AROC

MVT (w/Calculator)

​Using the MVT, find where the IROC = AROC

MVT (w/Calculator)

​Using the MVT, find where the IROC = AROC

MVT (w/Calculator)

​Using the MVT, find where the IROC = AROC

Math Response

Using the MVT, find the value of x where the IROC = AROC of the given function.

$$f\left(x\right)=x^3-2x^2+10x-1\left[2,5\right]$$ Round to two decimal places as needed.

Type answer here

Correct Answer: 3.63

Deg^°

Rad

On a given interval, you will have a y-value at each of the end points of the interval. Every y-value exists between these two y-values.

Only looks at y-values.

​IVT

The derivative must equal the average rate of change.

Only looks at derivative and slope.

MVT

MVT vs IVT

MVT vs IVT

MVT vs IVT

Critical Points

Setup

Next, we are going to focus on identify the critical points of a function. An important part of mathematics and calculus is understanding the behaviors and properties of a function.

Critical points will help us 'shape' a function by simply knowing its behaviors.

Critical Points

What is a Critical Point?

As the name implies, a critical point is an important point of a function.

Specifically it is 'A point on a function where the derivative of the function is either zero or undefined'.

Critical Points

Discuss

What happens when the slope of a graph is 0?

What happens when the slope is undefined?

Critical Points

When do they occur?

Critical Points will occur when graphs have a Horizontal Tangent Line (0 slope), at corners, and at cusps.

Critical Points

Types of Critical Points

There are two types of critical points you will encounter:
- Absolute (Global) or Relative (Local) Maximum/Minimum
- Points of Inflection (Lesson 5.4)

Why do Maximums/Minimums occur?

On a graph, critical points can be found where the derivative is equal to 0 or undefined.

Critical Points on a Graph

Critical Points

How do we find them?

1. Take the derivative of a given function.

2. Set the derivative equal to 0 and/or set the denominator equal to 0 if rational.

3. Solve for the value of x. Check the domain.

Critical Points

Graph Example

Identify the critical points on the following graph. Identify the types of critical points.

Critical Points

Graph Example

Identify the critical points on the following graph. Identify the types of critical points.

Critical Points

Graph Example

Identify the critical points on the following graph. Identify the types of critical points.

Critical Points

​Extreme Value Theorem

If a function is continuous on a closed and bounded interval [a, b], it is guaranteed to have an absolute max and minimum within that interval.

*Note that this does not tell us where the maximum or minimum is located, only that it exists in the interval.

Critical Points

From a Table

Critical Points

From a Table

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Critical Points

Function Example

Multiple Select

$$f\left(x\right)=\frac{1}{3}x^3-3x^2+8x+14$$  
Find the critical numbers of f(x).   Select ALL that apply

• x=2

• x=0

• x= -4

• x= -2

• x = 4

Critical Points

AP Exam Questions