Increasing and Decreasing Functions in Calculus
Learn about increasing and decreasing functions, important concepts in calculus. Explore how to determine if a function is increasing or decreasing on an interval and their applications in real-world scenarios such as maximizing profit and minimizing cost.
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MAT 1221 Survey of Calculus Section 3.1 Increasing and Decreasing Functions http://myhome.spu.edu/lauw
HW WebAssign HW 3.1 Additional HW listed at the end of the handout (need to finish, but no need to turn in) Please do your HW ASAP. Do not wait till Thursday to do your HW. Read 3.2 for next class.
Prelude If ?(?) is a function, then ? ? is also a function. ? ? =1 4?4 5 2?2+ ? 7 ? ? = ?3 5? + 1
Prelude ? ? = ?3 5? + 1
Prelude ? ? = ?3 5? + 1
Prelude ? ? = ?3 5? + 1
Prelude If ? 1 = 5, can you tell whether ? ? is all positive or negative on the interval?
Prelude Summary
Max/Min We are interested in max/min values Minimize the production cost Maximize the profit Maximize the sunlight exposure of plants
Preview Define increasing/Decreasing Function Increasing/Decreasing Test
Increasing/ Decreasing Functions (a) A function is increasing on an interval if for any two numbers ? > ? in the interval implies ?(?) > ?(?) ? y = (x ) f ? ? ?
Increasing/Decreasing Test (a) If on an interval, then ? is increasing on that interval. (b) If on an interval, then ? is decreasing on that interval. 0 ) ( x f x ( ) 0 f
Increasing/Decreasing Test y y = (x ) f ? is increasing on ? x ? ?
Increasing/Decreasing Test y = ? (x ) f ? is increasing on ? ? (?) > 0 on ? ? ? ?
Increasing/Decreasing Test ? ? is decreasing on ? y = (x ) f ? ? ?
Increasing/Decreasing Test ? ? is decreasing on ? ? (?) < 0 on ? y = (x ) f ? ? ?
Increasing/Decreasing Test ? Why is it important to find the intervals of increasing and decreasing? How to find the intervals of increasing and decreasing? y = (x ) f =0 <0 >0 x ? ? ?
Increasing/Decreasing Test ? Why is it important to find the intervals of increasing and decreasing? How to find the intervals of increasing and decreasing? y = (x ) f =0 <0 >0 x ? ? ?
Increasing/Decreasing Test ? Why is it important to find the intervals of increasing and decreasing? How to find the intervals of increasing and decreasing? Therefore, points where ? ? = 0 are very important y = (x ) f =0 <0 >0 x ? ? ?
Critical Number A critical number of a function ? is a number ? in the domain of ? such that either or does not exist. 0 ) ( = c f f (c )
Critical Number A critical number of a function ? is a number ? in the domain of ? such that either or does not exist. For nice functions such as polynomials, critical numbers are those with c f = (c ) ( ) 0 f c = ( ) 0 f
Increasing/Decreasing Test y To find the intervals of increasing and decreasing, we look for the values of ? such that ? (?) = 0 (The critical numbers) y = (x ) f =0 <0 >0 ? ? ? ?
Increasing/Decreasing Test The critical numbers divided the interval into subintervals. ( , ) ( c a ( c b , ) ( b a , ) , ) x ? ? ? x = x x = ( ) 0 f = ( ) 0 f ( ) 0 f
Increasing/ Decreasing Test The critical numbers divided the interval into subintervals. On each subinterval, the signs of the ? (?) are the same (Why?). ( , ) ( c a ( c b , ) ( b a , ) , ) x ? ? ? x = x x = ( ) 0 f = ( ) 0 f ( ) 0 f
Prelude ? ? = ?3 5? + 1
Example 1 Find the intervals of increasing and decreasing of = x f + 2 ( ) 6 10 x x
= + 2 ( ) 6 10 f x x x Example 1 1. Find the critical numbers
= + 2 ( ) 6 10 f x x x Example 1 2. Sketch a diagram of the subintervals formed by the critical numbers
= + 2 ( ) 6 10 f x x x Example 1 3. For each subinterval, pick a point ? and compute ? (?).
Expected Solution Steps Find the critical numbers Sketch a diagram of the subintervals formed by the critical numbers For each subinterval, pick a point ? and compute ? (?) Test ? = 5:? (5)= =-999<0 So, ? (?)<0 on (1,6). Thus, ? is decreasing on (1,6). 1. 2. 3.
