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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Jason Furney*

Jason has taught both College and High School Mathematics and holds a Master's Degree in Math Education.

This lesson develops the understanding of what a critical point is and how they are found. It explores the definition and discovery of critical points using functions and graphs as well as possible uses for them in the everyday world.

Critical points are key in calculus to find maximum and minimum values of graphs.

Let's say you bought a new dog, and went down to the local hardware store and bought a brand new fence for your yard, but alas, it doesn't come assembled. Of course, this means that you get to fence in whatever size lot you want with restrictions of how much fence you have. Wouldn't you want to maximize the amount of space your dog had to run? Critical points can tell you the exact dimensions of your fenced-in yard that will give you the maximum area!

Critical points in calculus have other uses, too. For example, they could tell you the lowest or highest point of a suspension bridge (assuming you can plot the bridge on a coordinate plane). Now we know what they can do, but how do we find them? First, let's officially define what they are.

**Definition of a Critical Point**

Let *f* be defined at *b*. If *f(b)* = 0 or if *'f*' is not differentiable at *b*, then *b* is a critical number of *f*. If this critical number has a corresponding *y* value on the function *f*, then a critical point exists at (*b, y*).

What exactly does this mean? Well, *f* just represents some function, and *b* represents the point or the number we're looking for. The second part of the definition tells us that we can set the derivative of our function equal to zero and solve for *x* to get the critical number! The third part says that critical numbers may also show up at values in which the derivative does not exist. We'll look at an example of this a bit later. Lastly, if the critical number can be plugged back into the original function, the *x* and *y* values we get will be our critical points.

**Finding Critical Points**

Now we're going to look at a graph, point out some critical points, and try to find why we set the derivative equal to zero.

The red dots on the graph represent the critical points of that particular function, f(x). It's here where you should start asking yourself a few questions:

- Is there something similar about the locations of both critical points? You should look for visual similarities.
- How does this compare to the definition from above?

If you understand the answers to these two questions, then you can understand how we find critical points.

Notice how both critical points tend to appear on a hump or curve of the graph. More specifically, they are located at the very top or bottom of these humps. Mathematically speaking, the slope changes from positive to negative (or vice versa) at these points. It's why they are so critical!

To understand how number one relates to the defection of a critical point, we have to remember what exactly a derivative tells us. The derivative of a function, *f(x)*, gives us a new function *f(x)* that represents the slopes of the tangent lines at every specific point in *f(x)*. So why do we set those derivatives equal to 0 to find critical points? Take a look at the following graph that shows different tangent lines to *f(x)*:

The green tangent lines run through our critical points. What's the difference between those and the blue ones? For one thing, they have the same slope, whereas the blue tangent lines all have different slopes. For another thing, that slope is always one very specific number. Who remembers the slope of a horizontal line? That's right! The slope of every tangent line that passes through a critical point is always 0!

*Find the critical points of the following:*

The first thing we can look at is whether or not the function is differentiable every time. Remember, a function cannot be differentiable at points where it is undefined. In this case, *x* cannot be 0.

We must take the derivative and set it equal to zero. Remember to use the quotient rule here.

Now that you've taken the derivative, simplify it and set it equal to zero.

Now you can solve for *x* to get the other critical numbers. We've already taken care of the undefined part so we just need to set the top of the fraction equal to 0 and solve.

So, we have *x* = 1, and *x* = -1 as our critical numbers, but are they points yet?

The last step is to find the corresponding *y* values by plugging our *x* values into *f(x)*.

*x* = 1 and *y* = 4 so our critical point is (1,4)

*x* = -1 and *y* = 0 so our critical point is (1,0)

Here is the graph with our critical points marked.

Let's review.

**Critical points** are points on a graph in which the slope changes sign (i.e. positive to negative). These points exist at the very top or bottom of 'humps' on a graph. We also know the slope of the tangent line at these points is always 0. We can use this to solve for the critical points.

To find these critical points you must first take the derivative of the function. Second, set that derivative equal to 0 and solve for *x*. Each *x* value you find is known as a critical number. Third, plug each critical number into the original equation to obtain your *y* values. Critical points will be (*x1, y1*), (*x2, y2*), etc.

To find the maximum and minimum values of graphs, you need to locate the critical points. Critical points are points on a graph in which the slope changes sign from positive to negative, and vice versa. Finding critical points can be essential in real-world applications such as finding out the area of a fenced-in yard or the lowest and highest points of a suspension bridge.

- The slope of the tangent line at these points is always 0
- Take the derivative of the function, set that derivative equal to 0 and solve for
*x*(*x*values known as critical numbers) - Plug each critical number into the original equation to solve for
*y*values - Critical points= (
*x1, y1*), (*x2, y2*), etc.

After reading about critical points in this lesson, you should be able to

- Define critical points
- Locate critical points on a graph
- Solve a function for critical points

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CAHSEE Math Exam: Tutoring Solution21 chapters | 211 lessons

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