Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Survey

Document related concepts

Transcript

4.2 Critical Points Mon Oct 19 Do Now Find the derivative of each 1) f (x) = x - 3x + 2 2) f (x) = sin x cos x 4 3 HW Review Critical points • A number a in the domain of a given function f(x) is called a critical point of f(x) if f '(a) = 0 or f ’(x) is undefined at x = a. • To find a critical point, we find the 1st derivative and set it equal to 0 • Example 1: Find the critical point(s) of the polynomial function f given by f(x) = x 3 - 3x + 5 Solution • Solution to Example 1. – The first derivative f ' is given by f '(x) = 3 x 2 - 3 – f '(x) is defined for all real numbers. Let us now solve f '(x) = 0 • 3 x 2 - 3 = 0 = 3(x-1)(x+1) =0 • x = 1 or x = -1 – Since x = 1 and x = -1 are in the domain of f they are both critical points. • Example 2: Find the critical point(s) of the rational function f defined by f(x) = (x 2 + 7 ) / (x + 3) • Solution to Example 2. – Note that the domain of f is the set of all real numbers except -3. – The first derivative of f is given by f '(x) = [ 2x (x + 3) - (x 2 + 7 )(1) ] / (x + 3) 2 – Simplify to obtain f '(x) = [ x 2 + 6 x - 7 ] / (x + 3) 2 – Solving f '(x) = 0 • x2 + 6 x - 7 = 0 (x + 7)(x - 1) = 0 x = -7 or x = 1 • f '(x) is undefined at x = -3 however x = -3 is not included in the domain of f and cannot be a critical point. • The only criticalpoints of f are x = -7 and x = 1. • Example 3: Find the critical point(s) of function f defined by f(x) = (x - 2) 2/3 + 3 • Solution to Example 3. – Note that the domain of f is the set of all real numbers. – f '(x) = (2/3)(x - 2) -1/3 = 2 / [ 3(x - 2) 1/3] – f ’(x) is undefined at x = 2 and since x = 2 is in the domain of f it is a critical point. You try: Find the critical points • • • • a) f(x) = 2x 3 - 6 x - 13 b) f(x) = (x - 3) 3 - 5 c) f(x) = x 1/3 + 2 d) f(x) = x / (x + 4) answers • • • • A) 1, -1 B) 3 C) 0 D) none Extreme Values • Extreme values refer to the minimum or maximum value of a function • There are two types of extreme values: – Absolute extrema: the min or max value of the entire function or interval – Local extrema: the min or max value of a piece of a function Absolute vs Local (pics) • Absolute extrema may or may not exist • Local extrema always exist How to find absolute extrema • 1) Find all critical points in an interval. • 2) Test all critical points and endpoints into the original function • 3) The biggest is the absolute max The smallest is the absolute min Ex • Find the extrema of the function on [0,6] f (x) = 2x -15x + 24 x + 7 3 2 Ex 2 • Find the max of the function on [-1, 2] f (x) =1 - (x -1) 2/3 Ex 3 • Find the extreme values of the function 2 on [1, 4] f (x) = x - 8ln x Ex 4 • Find the min and max of the function on 2 [0, 2pi] f (x) = sin x + cos x You try • Find the extrema of the given function on the indicated interval • 1) f (x) = x 4 - 8x 2 + 2, [-1,3] • 2) f (x) = x , [-1,3] 2/3 Rolle’s Theorem • Assume that f(x) is continuous on [a,b] and differentiable on (a,b). If f(a) = f(b), then there exists a number c between a and b such that f’(c) = 0 Ex • Verify Rolle’s Theorem on [-2, 2] f (x) = x - x 4 2 Closure • Find the min and max of the function on given interval f (x) = 2x + 4x + 5, [0,2] 2 • HW: p.222 #1 5 7 21 35 47 55 65 78 83