To understand important points on a graph, we can use a few simple techniques:
Find Critical Points: Check where the first derivative, written as , equals zero or is not defined. These points might show the highest or lowest points on the graph, or where the line is flat.
First Derivative Test: Look at the sign of before and after the critical points. If it changes from positive to negative, then you have a local maximum (the highest point). If it changes from negative to positive, then you have a local minimum (the lowest point).
Second Derivative Test: Use to see how the graph curves. If , the critical point at is a local minimum. If , it’s a local maximum.
Graphical Analysis: Draw the graph of and its derivatives. This helps you see how the graph behaves around the critical points, like where the peaks and valleys are located.
To understand important points on a graph, we can use a few simple techniques:
Find Critical Points: Check where the first derivative, written as , equals zero or is not defined. These points might show the highest or lowest points on the graph, or where the line is flat.
First Derivative Test: Look at the sign of before and after the critical points. If it changes from positive to negative, then you have a local maximum (the highest point). If it changes from negative to positive, then you have a local minimum (the lowest point).
Second Derivative Test: Use to see how the graph curves. If , the critical point at is a local minimum. If , it’s a local maximum.
Graphical Analysis: Draw the graph of and its derivatives. This helps you see how the graph behaves around the critical points, like where the peaks and valleys are located.