Labware - MA35 Multivariable Calculus - Single Variable Calculus
 MA35 Labs 1 » Single Variable Calculus Contents1.3 More Derivatives 1.3.1 Second Derivatives 1.3.2 Convexity and Concavity 1.3.3 Derivative Test 1.3.4 Taylor Series 1.4 Integration Search

Convexity and Concavity

Text

The second derivative of a function is the derivative of the first derivative.

Demos

 Convexity and Concavity This demonstration is similar to 1.2.2 (critical points). As in 1.2.2, the function f(x) and its domain are defined in the control panel. This time, the graph of the function is colored magenta where it is concave downward and yellow where it is concave upward.

Exercises

• 1. Using the demo, investigate the concavity of the following functions. For which intervals is the graph concave upward? For which intervals is it concave downward? For which intervals is there no concavity?:
• f(x) = x + 0.3
• f(x) = x2
• f(x) = x2 + x + 1
• f(x) = -3x2
• f(x) = x3 + 3x2 + 3x + 1
• f(x) = cos(x)
• 2. Describe the concavity of any function of the form f (x) = ax + b.
• 3. Now consider a function of the form f(x) = ax2 + bx + c. How do the values of a, b, and c affect the concavity of f(x)?