First&amp;Second Derivative Notes - PC vers

# First&amp;Second Derivative Notes - PC vers - What...

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What derivatives tell you about the graph…. Let’s look at a couple functions. We’ll take the first derivative. Then take the derivative of the first derivative to get the second derivative. Let’s look at a cubic function first: Let’s take both the 1 st derivative: And the 2 nd derivative: Now let’s set the first derivative to 0. If the first derivative equals 0 for a value of x, then at that value you COULD have a local maximum, a local minimum, or an inflection point. A local maxima (or local maximum) happens wherever the curve in concave down. At that point the value of the derivative has the HIGHEST value in that area of the curve. Remember that the “top” of the bowl has a horizontal tangent line and that horizontal lines have a slope of 0. Remember also that the first derivative gives us the slope of the tangent line at an instant in time. IF THAT POINT IS A LOCAL MAXIMA, then setting the first derivative to 0 will allow us to find that time. Solving for “x” gives us the x- coordinate of the local maxima, plugging into the original function gives us the y- coordinate. A local minima (or local minimum) happens wherever the curve in concave up. At that point the value of the derivative has the LOWEST value in that area of the curve.

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## This note was uploaded on 06/15/2011 for the course MATH 122 taught by Professor Kustin during the Spring '08 term at South Carolina.

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First&amp;Second Derivative Notes - PC vers - What...

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