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 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 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 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: • Examples: Differentials: dy = f’ dx Separation of variables
4.5 cont’d (Newton/Euler) CHART: Orig. Pt. dx dy/dx dy New pt. given 0.1 Continue……… Do an example with the chart……..
4.6 Related Rates 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