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.
What Are Critical Points?
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!
Example with Graph
Find the critical points of the following:
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