1. AP CALCULUS AB FINAL REVIEW APPLICATIONS OF THE DERIVATIVE

2. ABSOLUTE EXTREMA The Extreme Value Theorem guarantees that a continuous function on a closed interval has both an absolute maximum and minimum on the interval. These absolute/global extrema occur either at the end points of the interval or at relative max/mins inside the interval. To find absolute extrema: (i) find the critical numbers of the function on the interval, (ii) plug the critical numbers and endpoints into the function to see which x-values give the largest/smallest function values. Recall a critical number is an x-value where f(x) is defined and the derivative is zero or undefined.

3. Rolle’s Theorem and The MVT Rolle’s Theorem states that under the right conditions a function is guaranteed to have a value where the derivative is zero. The Mean Value Theorem states that under the right conditions a function is guaranteed to have a value where the derivative is equal to the mean value.

4. The First Derivative If f(x) is differentiable on an interval: (i) When the derivative is positive, f(x) is increasing. (ii) When the derivative is negative, f(x) is decreasing. The First Derivative Test states that when the derivative changes sign at a critical number, f(x) has either a relative max or relative min. positive to negative → relative max at the critical number negative to positive→ relative minimum at a critical number To find the actual value of the relative extrema plug the critical number into f(x).

5. The Second Derivative If the second derivative of f(x) exists on an interval: (i) When the second derivative is positive f(x) is concave up (ii) When the second derivative is negative f(x) is concave down. A point of inflection is a point where f(x) has a tangent line and f(x) changes concavity. Graphically this means the change of concavity must be smooth, not a cusp or at an asymptote. To find points of inflection: (i) Find x-values where f(x) is defined and the second derivative is zero or undefined. (ii) Check to see if the second derivative changes sign at these possible points of inflection. A change if sign indicates a point of inflection. The Second Derivative Test: We plug the critical number x = c of f(x) into the second derivative. (Only works for critical numbers where the derivative is 0) If the result is positive  rel min at x = c If the result is negative  rel max at x = c If the result is zero  the test fails.

6. Optimization Optimization problems are basically absolute max/min problems involving real life scenarios. We must write an equation that models the real life situation using the given info. Primary Equation: Involves the quantity you want to optimize. Secondary Equation: Involves the given info. We then find the feasible domain. The absolute max/min usually falls at a relative extrema inside the feasible domain. [Ex] See Board