Chapter 4

4.1

Extreme Value Theorem (EVT): If f is continuous on a closed interval [a, b], it has both a global max and a global min on [a, b]. (Rausch’s emphasis) Finding Global Extrema: Check endpoints of the interval Check local extrema f’(x) = 0 or undefined are potential local extrema (called critical values when written with just the x-value and critical points when written as an ordered pair)

Find the extreme values on [c, d] on [d, e] on [c, e] on [a, f] on [a, b]

Find the extreme values on [a, b]

4.2

You will SAFELY and LEGALLY check the speedometer of a vehicle. You will need the help of one other person (one to drive and one to take the data) who may or may not be from class. 1. Either set the cruise control or keep the speedometer needle steady while driving on a section of Interstate. 2. As the vehicle passes a mile marker, start a stopwatch. Time the distance from one mile marker to the next. 3. Calculate your average velocity. 4. Discuss the relationship between your average velocity and your instantaneous velocities. What does this say about your speedometer? What does this have to do with the MVT?

On the AP Test (and so in here) there are only four things you are allowed to say “my calculator told me” plot the graph of a function within an arbitrary viewing window find the zeros of functions (solve equations numerically) numerically calculate the derivative of a function numerically calculate the value of a definite integral Assignment: pg. 202 (1-28, 39-42, 45-54, 56) Project – due the day after the Chapter 4 Test

4.3

First Derivative Test (for finding local extrema) Find all critical numbers (derivative is zero or undefined). Find test points for each region created by the critical numbers. Use test points to see where the function is increasing or decreasing. Look for a change of directions. A switch from increasing to decreasing indicates a local maximum. A switch from decreasing to increasing indicates a local minimum. “On the Role of Sign Charts in AP© Calculus Exams for Justifying Local or Absolute Extrema”

4.4

Optimization: As the name implies, optimization involves using the first derivative test and/or the procedure for finding global extrema in context (i.e. a story problem) to find the largest or smallest value that “optimizes” a particular situation. This is often done through the use of a primary and secondary equation. The primary equation involves the quantity that you want the maximum or the minimum of. If the primary equation has more than two variables, you will need a secondary equation to restrict one or more variables. This secondary equation will involve a quantity that you are not trying to optimize but that you have more information about.

4.5

The line tangent to the curve of f(x) at x = a does a “good” job of approximating f(x) “near” a. Tangent Line Approximation: L(x) = f(a) + f’(a)(x-a) [this is just the point-slope form of a line!] Example: Compare the tangent line approximations at (0, 1) to the actual values for f(x) = 1 + sin x at -.5, -.1, -.01, 0,.01,.1, and.5. Note: Tangent lines where the graph is concave down overestimate and where the graph is concave up underestimates.

4.6