Mathlets: Second-Order Derivatives and Concavity

Second-Order Derivatives and Concavity

Graphs a function with first- and second-degree polynomial approximations to the function at a given point, showing how the first and second derivatives are used to determine concavity and other information.
How to use || Examples || Other Notes

Try the Derivative Calculator.

How to use

Enter a function f (x) in the text input field marked "f(x)=" (Example: f (x)=sin x)
Click the "Graph" button (this button also refreshes the graph)
Enter the derivatives f '(x) and f ''(x) in the text input fields marked "f '(x)=" and "f ''(x)=" (Example: f '(x)=cos x, and f ''(x)= - sin x)
Select an x value (Example: x=1)
This can be done in either of two ways:
- Click and drag the mouse on the graph -- value of x corresponds to horizontal mouse position on graph
- Enter the value in the text input field marked "x=" and click the "Graph" button to refresh
To erase the graph and all input fields, click the "Clear" button

The text input fields marked "f (x)=" and "f '(x)=" can accept a wide variety of expressions to represent functions. The text input field marked "x=" can accept a real number in decimal notation. The buttons under the graph allow various manipulations of the graph coordinates.

For assistance computing the derivatives f '(x) and f ''(x), try the Derivative Calculator.

Examples

Polynomial:
f (x) = x³-3x
f '(x) = 3x²-3
f ''(x) = 6x
x = 0.8

Exponential:
f (x) = e^x
f '(x) = e^x
f ''(x) = e^x
x = 1

Trigonometric:
f (x) = sin x
f '(x) = cos x
f ''(x) = -sin x
x = 0.5

Other Notes
Tangent lines are shown in blue in the graph, and second-degree approximations are shown in red. The tangent line and second-degree approximation correspond to first- and second-degree Taylor polynomials:

P₁(x) = f (c) + f '(c)(x-c)
P₂(x) = f (c) + f '(c)(x-c) + (f ''(c)/2)(x-c)²

(the center point c is the value given as x in the applet)

If the second-degree approximation is above the tangent line, then the graph of f (x) is concave up; if below the tangent line, then concave down. Labels under the text input fields show values for f (x), f '(x), and f ''(x) for the given value of x, then use these values to indicate whether the graph is increasing or decreasing (from the sign of the first derivative), as well as whether the graph is concave up or concave down (from the sign of the second derivative), at x.