AP Calculus Unit 5 Analytical Applications of Differentiation

University:
Whitman College
Course:
MATH-125 | Calculus I
Academic year:

2022
Views:

406

Pages:

2
Author:

Nesbyenvkk

Unit 5: Analytical Applications of Differentiation ● Intervals of Increasing/Decreasing Functions ○ 1) Take the derivative ○ 2) Find the critical points of the function - the points where f(x) equals zero. ○ 3) Make a Wiggle Chart ■ Make a number line with all the critical points labeled. ■ Use a test point that is between each number and on the ends. ■ Plug that point into the derivative and determine if it is positive or negative. ● A positive value means that the original function (not the derivative) has a positive slope between those two points. ● A negative value means that the original function has a negative slope between those two points. ● Extrema ○ Extreme Value Theorem: a continuous function must have a minimum and maximum ○ Minimums and Maximums are called Extrema or Extreme Values ○ If continuity is broken, there may not be extrema on the interval ○ Where can it occur? ■ Endpoints of an interval ■ Rounded off point on a graph ■ Derivative equals zero ■ Slope changed from positive to negative or negative to positive ■ A sharp point ○ Absolute extrema: The highest or lowest extreme value on an interval ○ Relative extrema: Every Minimum or Maximum on an interval ● Concavity: explains if a function faces up or down ○ 1) Take the second derivative of the function ○ 2) Make a second derivative chart ■ Find where all the relative extrema is - the points where the f’(x) = 0. ■ Create a number line with the relative extrema points labeled. ■ Use test points between each number and plug them into the original equation. ● Positive means concave up, negative means concave down ○ Concave up: This means that the first derivative (slope) is increasing ○ Concave down: The slope is decreasing ○ Inflection Point: where the graph changes from concave up to down or down to up ● Mean Value Theorem: spot on the function where the average rate of change is the same as the slope of a tangent line. o The function must be continuous and differentiable on the interval ● Rolle's Theorem: guarantees the existence of a minimum or maximum on an interval o F(x) is continuous and differentiable on the interval [a,b] o F(a) = F(b) o There exists a point where the derivative is zero - there’s a minimum or maximum ● Optimization: applied max or min ○ 1) Determine an equation for what you are trying to maximize or minimize using geometric formulas. ○ 2) Use a secondary equation that you will substitute into the primary equation ■ One variable per side o 3) Determine the implicit domain o 4) Take the derivative of both sides and set them equal to zero o 5) Solve