W-up 3-27 When is the function increasing? For the interval(s) that it is increasing, what do all the derivatives have in common? When is the function decreasing? For the interval(s) that it is decreasing, what do all the derivatives have in common? State local max and min What do you know about the derivative at the local max and mins?

14.2 Increasing/Decreasing functions; The First Derivative Test Determine where a function is increasing and where it is decreasing Use first derivative test Graph functions

Increasing/ Decreasing Find the derivative Set up a table that solves f ‘(x) >0 and f ‘(x) < 0 It’s increasing on the interval (a,b) if f ‘(x) >0 It’s decreasing on the interval (a,b) if f ‘(x) < 0

Ex: Determine when f(x) = x3 – 6x2 +9x – 2 is increasing or decreasing Find derivative: f’(x) = 3x2 – 12x + 9 Set derivative = 0 and solve 3x2 – 12x + 9 = 0 3(x2 – 4x + 3) = 0 {factored out a GCF} 3(x – 1)(x – 3) = 0 x = 1 and x = 3 Set up table to determine when positive/negative 1 3 Interval f’(x) Inc/dec (-∞,1) (1,3) (3,∞) Pick a point less than 1 f’(0) = 9 positive Pick a point btw 1 & 3 f’(2) = negative Pick a point greater than 3 f’(4) = positive Decreasing Increasing Increasing

1st derivative test: - take the first derivative and… A - MAX If f is increasing to the left of a point A on the graph f and is decreasing to the right of A, there is a local MAXIMUM If f is decreasing to the left of a point B on the graph f and is increasing to the right of A, there is a local MINIMUM increasing decreasing increasing decreasing B - MIN

steps to graph a function Find domain Located intercepts (if reasonable to find) Locate all points where tangent line is vertical or horizontal f’(x) = 0 Determine where graph is increasing and decreasing Find local max/min using 1st derivative test Determine end behavior, locate any asymptotes

Ex: follow 6 steps graph f(x) = x3 – 12x Domain – all real numbers x- intercepts - factor and solve x(x2 – 12)=0 x = 0 x2 – 12 = 0 x = ± 𝟏𝟐 =±𝟐 𝟑 3-5)Increasing/decreasing/1st derivative test/ horizontal tangent line, no vertical tangent lines f’(x) = 3x2 – 12 0 = 3x2 – 12 3(x2 – 4) =0 x =±𝟐 f(2) = 23 – 12(2) = -16 f(-2) = -23 – 12(-2) = 16 horizontal tangent lines at (2,-16) and (-2,16)

f(x) = x3 – 12x f’(x) = 3x2 – 12 MAX Plot points of the horizontal tangent line(s) Make a table to determine increasing/decreasing Plot x-intercepts End behavior – acts like x3 MIN Pick a point less than -2 say -3 f’(-3) = positive f(x) is INCREASING (-∞,−2) Pick a point between - 2& 2 say 0 f’(0) = negative f(x) is DECREASING (-2,2) Pick a point greater than 2 say 3 f’(3) = positive f(x) is INCREASING (2,∞)

14.2 hw # 1-8 all #9-21 odds (lets do #17 in class) #43-46 (read the notes on homework paper)