# MAT135H1 Lecture 1: 4.1 Increasing and Decreasing Functions

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4.1 Increasing and Decreasing Functions

• By reading a graph of a function from left to right, we can determine if a function is increasing or decreasing

on an interval

•

()fx

is increasing (rises) on an interval if,

! for any value of

12

xx< on the interval ,

12

() ( )fx fx<

• ()fx is decreasing (falls) on an interval if,

! for any value of

12

xx< on the interval ,

12

() ()fx fx>

Example 1: Given

2

()fx x= , determine the intervals where ()fx is increasing and decreasing

• for a function ()fx that is continuous and differentiable on an interval,

o ()fx is increasing if '( ) 0fx

>

for all values of x on the interval

! the slope of the tangents are positive

o ()fx is decreasing if '( ) 0fx

<

for all values of x on the interval

! the slope of the tangents are negative

increasing when X0

decreasing when xso

i

slopeof

tangent

is