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Lecture 8. Increasing and Decreasing Functions. Graphs. decreasing. increasing. Sometimes increasing, sometimes decreasing. Increasing/decreasing on an interval. y = f(x). f is decreasing on [a,b], increasing on [c,d]. f is neither decreasing nor increasing on [a,d], [b,c], [b,d].
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Lecture 8 Increasing and Decreasing Functions
Graphs decreasing increasing Sometimes increasing, sometimes decreasing
Increasing/decreasing on an interval y = f(x) f is decreasing on [a,b], increasing on [c,d] f is neither decreasing nor increasing on [a,d], [b,c], [b,d]
Formal Definitions (important) • The function f is (strictly) increasing on the interval (a,b) if for any a <x<y<b it is true that f(x)<f(y) • The function f is (strictly) decreasing onthe interval (a,b) if for any a <x<y<b it is true that f(x)>f(y)
Maybe not strictly • The function f is nondecreasing on the interval (a,b) if for any a <x<y<b it is true that) • The function f is (strictly) nonincreasing onthe interval (a,b) if for any a <x<y<b it is true that
Idea of Increasing and Decreasing Does Not Depend on Derivatives There are functions that increase on ( ) but cannot be differentiated anywhere. Less exotic ones are simply not differentiable in a lot of places
Relationship Between increasing (decreasing) and calculus • f(x) = m x+b is increasing exactly when m > 0 • f(x) = m x+b is decreasing exactly when m > 0
Idea: If f has a tangent line then f is increasing when the tangent line is increasing (i.e. f ‘ (x) > 0)
Theorem: If f ‘ (x) > 0 for every x in the interval (a,b) then f is increasing on (a,b) If there are x, y between a and b with x < y and f(x) > f(y) then there is a point c between x and y where the tangent has negative slope. But then f ‘ (c) < 0
Example: Find the intervals on which f(x) is increasing (decreasing) if Solution: Want x such that f ‘ (x) >0 for increasing, f ‘ (x) < 0 for decreasing = Roots = -3, 5
Fundamental Fact • Definition: A critical number of a function f is a point c at which f ‘(c) = 0 or f ‘ (c) does not exist • FACT: The sign of the derivative of f does not change between critical numbers of f.This gives the following procedure for finding the intervals of increase and decrease of a function f: • a. Find all critical numbers. • Write down the intervals into which they divide the domain of the function • Check the sign of the derivative at any point in each interval – that will be the sign of the derivative at every point in the interval
Example = f ‘ (x) = 0 when x = -3 and c = 5. These are the only critical numbers since f ‘ exists for all x. The critical numbers divide the domain of f into the intervals ( ) ( ) and ( ) Chose -10 in the first interval, f ‘ (-10) = 3 (-7)(-15) > 0 so f is increasing on this interval Chose 0 in the first interval, f ‘ (0) = -45 < 0 so f is decreasing on this interval Chose 10 in the first interval, f ‘ (10) = 3(13)(8) > 0 so f is increasing on this interval
The graph at right is that off ‘(x). What are the critical numbers of f and on which intervals is f increasing and decreasing. Critical numbers= {-1,2,3} Intervals: (-infinity,-1), f ‘< 0 so f is decreasing(-1,1), f ‘ > 0 so f is increasing(1,3), f ‘< 0 so f is decreasing(3, infinity), f ‘ > 0 so f is increasing
Example: A particle moves along the x-axis so that its position at time t is given by What are the times at which the particle is moving left and right? It is moving to the right when x(t) is increasing and to the left when x(t) is decreasing. Find critical numbers: x ‘ (t) = 0, t = 5/8 x ‘ (t) = The critical numbers divide the domain into (-infinity, 5/8) and (5/8, infinity). We choose a convenient member of each. Take 0 in(-infinity, 5/8), f ‘ (0) = -5 so x ‘ is decreasing on (-infinity, 5/8) so the particle is moving to the left. Similarly if we take 2 in (5/8, infinity) then f ‘ (2) >0 So x(t) is increasing on (5/8,infinity) and the particle is moving to the right.
