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Problem of the Day An equation of the line tangent to the graph of y = x + cos x at the point (0, 1) is a) y = 2x + 1 d) y = x - 1 b) y = x + 1 e) y = 0 c) y = x Problem of the Day An equation of the line tangent to the graph of y = x + cos x at the point (0, 1) is a) y = 2x + 1 d) y = x - 1 b) y = x + 1 e) y = 0 c) y = x Finding increasing and decreasing intervals 1. Increasing function - graph moves up as x moves to the right. 2. Decreasing function - graph moves down as x moves to the right. What do you know about the slope of an increasing function? a decreasing function? a constant function? Finding increasing and decreasing intervals function - graph moves up as x 1. Increasing moves to the right. 2. Decreasing function - graph moves down as x moves to the right. What do you know about the slope of an increasing function? a decreasing function? a constant function? In Calculus what gives you the slope of the curve? Finding increasing and decreasing intervals Theorem 3.5 - Derivative tests If f '(x) > 0 f '(x) < 0 f '(x) = 0 f is increasing f is decreasing f is constant Additional derivative test information f '(x) goes from - to + + (relative minimum) + - f '(x) goes from + to (relative maximum) + f '(x) goes from + to + + (neither*) f '(x) goes from - to (neither*) (*strictly monotone - neither on the interval) - Example for find the increasing/decreasing intervals 4 3 f(x) = x - 2x Example for find the increasing/decreasing intervals 4 3 f(x) = x - 2x 1) need derivative Example for find the increasing/decreasing intervals 4 3 f(x) = x - 2x 3 2 1) need derivative f '(x) = 4x - 6x 2) find critical numbers Example find the increasing/decreasing intervals for 3 2 4 3 1) need derivative f '(x) = 4x - 6x f(x) = x - 2x 2) find critical numbers 3 2 0 = 4x - 6x 2 0 = 2x (2x - 3) 2 2x = 0 or 2x - 3 = 0 x = 0 or 3/2 Example find the increasing/decreasing intervals for 3 2 4 3 1) need derivative f '(x) = 4x - 6x f(x) = x - 2x 2) find critical numbers x = 0 or 3/2 3) describe interval behavior Example find the increasing/decreasing intervals for 3 2 4 3 1) need derivative f '(x) = 4x - 6x f(x) = x - 2x 2) find critical numbers x = 0 or 3/2 3) describe interval behavior if x < 0 if x > 0 and < 3/2 if x > 3/2 3 2 f '(-1) = 4(-1) - 6(-1) = -10 3 2 f '(½) = 4(½) - 6(½) = -1 3 2 f '(2) = 4(2) - 6(2) = 8 Example for find the increasing/decreasing intervals 4 3 3 2 f(x) = x 2x 1) need derivative f '(x) = 4x - 6x 2) find critical numbers x = 0 or 3/2 3) describe interval behavior f '(x) - 0 x 0 0 + 3/2 3 2 3 2 f '(-1) = 4(-1) - 6(-1) = -10 f '(½) = 4(½) - 6(½) = -1 3 2 f '(2) = 4(2) - 6(2) = 8 Example find the increasing/decreasing intervals for 4 3 3) describe interval f(x) = x - 2x - 0 + behavior f '(x) - 0 3/2 x 0 so what does the graph look like? Example find the increasing/decreasing intervals for 4 3 3) describe interval f(x) = x - 2x - 0 + behavior f '(x) - 0 3/2 x 0 so what does the graph look like? Example for find the increasing/decreasing intervals f(x) = x domain x+1 Note: -1 is not in Example for find the increasing/decreasing intervals f(x) = domain 1) need derivative x x -1+ 1 Note: -1 is not in f(x) = x(x + 1) f -2 -1 '(x) = x (-1)(x + 1) + 1(x + 1) -2 = (x + 1) (-x + x + 1) = 1 2 (x + 1) Example for find the increasing/decreasing intervals f(x) = domain 1) need derivative x Note: -1 is not in f x'(x) + 1= 2) find critical numbers 0= 1 2 (x + 1) 1 2 (x + 1) only critical number is -1 which is not in domain Example for find the increasing/decreasing intervals f(x) = 1) need derivative domain f '(x) = 1 (x + 1)2 x Note: -1 is not in 2) find critical numbers x+1 0= 1 (x + 1)2 only critical number is -1 which is not in domain 3) describe interval behavior f '(x) + undef + x -1 f '(-2) = 1 f '(0) = 1 Example for find the increasing/decreasing intervals f(x) = 1) need derivative domain f '(x) = 1 (x + 1)2 x Note: -1 is not in 2) find critical numbers x+1 0= 1 (x + 1)2 only critical number is -1 which is not in domain 3) describe interval behavior f '(x) + undef + x -1 What does it look like? f '(-2) = 1 f '(0) = 1 Example for find the increasing/decreasing intervals f(x) = x + 3 domain x2 Note: 0 is not in Example for find the increasing/decreasing intervals f(x) = x + 3 -1 domain Note: 0 is not in -2 f(x) = x + 3x x -3 f -2 '(x) = -x + 3(-2)x -3 = -x (x + 6) 2 f '(x) x - 0 -6 critical numbers are 0 and f '(-8) = -.003 + undef6 f '(1) = -7 0 f '(-1) = 5 Attachments