First and second derivative tests:
Problem 1 and solution: (1st derivative test; relative extrema): FIND THE RELATIVE EXTREMA
Problem 2 and solution: (1st derivative test; relative extrema): FIND THE RELATIVE EXTREMA
Problem 1 and solution: (1st derivative test; absolute extrema): FInd the absolute extrema
Problem 2 and solution: (1st derivative test; absolute extrema) At what
Problem 1 and solution: (2nd derivative test);
Determine all points of inflection, intervals of concave up and concave down, relative extrema, and intervals of increasing and decreasing. Problem 2 and solution: (2nd derivative test);
Determine all points of inflection, intervals of concave up and concave down, relative extrema, and intervals of increasing and decreasing. |
We know what derivatives are, but what can we use them for besides getting an A in AP calc? We can use them to help us more accurately draw a graph, as well as showing all the activity of a function in a given interval.
First derivative test: (for relative extrema) 1. Find the derivative 2. Find the critical numbers (where f'(x)=0) 3. Test the critical numbers in their respective intervals. 4. If f(x) is greater than 0 in the interval, the function is increasing and vice-versa. 5. Where the intervals of increasing/decreasing changes is where f(x) has relative extrema: If it changes from increasing to decreasing there is a relative max, if it changes from decreasing to increasing there is a relative min. First derivative test: (for absolute extrema) 1. Find the derivative 2. Find the critical numbers 3. Test the critical numbers in the function 4. Test the end points in the function if they are included on the closed interval 5. The greatest of these values is the absolute maximum and the smallest of these values is the absolute minimum. Second derivative test:
1. Find the derivative of the first derivative 2. Find the "critcal numbers," (possible points of inflection) where f''(x)=0 3. Test the ppi's in their respective intervals 4. If f''(x) is positive in the interval than the function is concave up, and vice versa. 5. Points where concavity changes is where the points of inflection are. |