SecondOrder Derivatives and Concavity
Graphs a function with first and seconddegree 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) = x^{3}3x
f '(x) = 3x^{2}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 seconddegree
approximations are shown in red.
The tangent line and seconddegree approximation correspond to first
and seconddegree Taylor polynomials:
 P_{1}(x) = f (c)
+ f '(c)(xc)
 P_{2}(x) = f (c)
+ f '(c)(xc)
+ (f ''(c)/2)(xc)^{2}
(the center point c is the value given as x in the applet)
If the seconddegree 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.