Chapter 4 AP Calculus BC

4.1 Extreme Values of Functions Absolute Maximums/Minimums Local Maximums/Minimums Graphs Theorem 1 – If f is cts. on a closed interval [a,b], then f has both a max. and a min. on the interval. Theorem 2 – Local Extreme values – If a function, f, has a local max. or local min. at an interior pt., c, of its domain, and if f’ exists at c, then f’(c) = 0. Critical Points are where f’ = 0 or DNE Examples:

4.2 Mean Value Theorem Examples: Examples: Y=f(x) is cts. on[a,b] and differentiable on (a,b) then Examples: Corollary 1: f is cts. on [a,b] and differentiable on (a,b) f’>0 on (a,b) then f increases [a,b] f’<0 on (a,b) then f decreases [a,b] Examples: Corollary 2: If f’(x)=0 at each pt in interval then f(x)=C. Corollary 3: Functions with the same derivative differ by a constant.

4.2 cont’d Do your homework!!!!!!!! Examples: Antiderivatives – Reverse of derivatives Position, velocity, and acceleration Do your homework!!!!!!!!

4.3 Connecting f’ and f” with the graph of f Thm. 4 – 1st derivative test Critical points where f’=0 or DNE f” = 0 or DNE possible points of inflection “concavity changes” 1. f’ goes + to - Local Max 2. f’ goes – to + Local Min 3. Left end pt + Local Min - Local Max Concavity: y= f(x) Concave up if f”>0 Concave down if f”<0 Right end pt - Local Min + Local Max Find extreme values Find concavity: Examples:

4.3 cont’d. Graph examples Given f’ Given f WHY???? Theorem 5 – 2nd derivative test for local extrema If f’(c)=0 and f”(c)<0 then f has a local max at c. If f’(c)=0 and f”(c)>0 then f has a local min at c. WHY????

4.4 Modeling and Optimization Strategy p. 219……. Examples: 1. Two numbers sum is 20. Find the product to be as large as possible. 2. A rectangle inscribed under one arch of the sine curve, largest area ? 3. Open top box out of 20 by 25 foot sheet, cut squares out of corners, largest volume? Thm. 6 – Maximum Profit is where R’ = C’ Thm. 7 – Min. Avg. Cost is where avg. cost = marginal cost.

4.5 Linearization/Newton’s(Euler’s) If f is differentiable at x = a, then the equation of the tangent line: L(x)=f(a) + f’(a)(x-a) defines the linearization of f at a. The approx. f(x)~L(x) is the standard linear approx. of f at a. The point x = a is the center of the approximation. Examples: Differentials: dy = f’ dx Separation of variables Examples:

4.5 cont’d (Newton/Euler) CHART: Orig. Pt. dx dy/dx dy New pt. 0.1 given Continue……… Do an example with the chart……..

4.6 Related Rates EXAMPLES: Multiple Variables changing with respect to time, t. Derivatives of each individual variable with respect to t. Take a derivative with respect to t for each. Hot Air Balloon EXAMPLES: Highway Chase Ladder Problem