The Ultimate Guide to Calculus

 Home

 Limits

 Differentiation

 Derivative Tests

 Integration

 Logarithmic Functions

 Area/Volume
Derivative Tests
[First Derivative Test and Increasing and Decreasing Functions] [Second Derivative Test and Concavity and Points of Inflection] [Curve Sketching] [Derivative Test Links]

First Derivative Test and Increasing and Decreasing Functions
It is possible to find the behavior of a function using the derivative and second derivative. The first derivative test funds the intervals where the function is increasing or decreasing.
Guidelines for applying the first derivative test
1. first take the derivative
2. find the zeros of the derivative
3. set the zeros to intervals putting the smallest number in an interval with negative infinity and the largest number with positive infinity
4. pick a number in between each of the intervals
5. test each of the numbers in the derivative
6. if the result is postive, the interval is increasing and if the result is negative, the interval is decreasing
example: find the relative extrema and determine the intervals where the function is increasing or decreasing:
In the graph of the function, the function is increasing in the intervals [-infin,0] and [4,infin] and decreasing on the interval [0,4]
The critical numbers are 0 and 4
[click here for enlarged graph]
Second Derivative Test and Concavity and Points of Inflection
Using the second derivative test, it is possible to find the concavity and minimums of a function
Guidelines for applying the second derivative test
1. first take the derivative
2. then take the derivative of the derivative (second derivative)
3. find the zeros of the second derivative (these are the probable points of inflection)
4. set the zeros to intervals putting the smallest number in an interval with negative infinity and the largest number with positive infinity
5. pick a number in between each of the intervals
6. test each of the numbers in the second derivative
7. if the result is positive, the interval is concave upward and if the result is negative, the interval is concave downward
8. if the intervals change from positive to negative or negative to positive, then the zeros are points of inflection
example: determine the concavity of the following equation:
In the graph of the function, the function is concave downward on the interval [-infin,2] and concave upward on the interval [2,infin]
With this result, it can also be concluded that x=2 is the x point of the point of inflection making the point of inflection
[click here for enlarged graph]
Curve Sketching
Using the different techniques learned in Calculus, it is possible to be able to determine fairly accurate graph of an equation with only the equation itself. The techniques that can be used include
x-intercepts and y-intercepts
symmetry
domain and range
continuity
vertical asymptotes
differentiability
relative extrema
concavity
points of inflection
horizontal asymptotes
example: analyze the graph of
first derivative:
second derivative:
y-intercept: (0,1)
Domain: all real numbers
Using this information, an accurate graph can be made
[click here for enlarged graph]
Derivative Test Links
Visual Calculus Drill: First Derivative Test Hot - first derivative test drill with answer explanations
First Derivative Test - first derivative test slide presentation
Second Derivative Test - second derivative test slide presentation
Concavity Test - in depth concavity test description
Calculus-Help: Problem of the Week #26 Hot - challenging concavity problem

 
© 2003 Calculus World: The Ultimate Guide to Calculus.