Suppose that a frog is about tojump 3 steps up in the stairs a dayand the following day the frog willjump 2 steps down the stairs. Now,how many days will the frog reachthe top of the stair if it is 6 steps upto reach the top?

-1-2-3-4-5-6 1 2 3 4 5 6

Increasing and Decreasing Functions

Definition:

If the graph of a function rises from left to

right over an interval I, it is said to be increasing

over an interval I. If the graph drops from left to

right over an interval I, it is said to be decreasing

over I.

Mathematically,

A function is increasing over I, if for every 𝑥1and 𝑥2 in I,

𝑥1 < 𝑥2, f(𝑥1) < f(𝑥2)

A function is decreasing on an interval I, if for every 𝑥1 and 𝑥2 in I,

𝑥1 < 𝑥2, f(𝑥1) > f(𝑥2)

Increasing Functions

Decreasing Functions

Derivatives can be used to determinewhether a function is increasing ordecreasing:

if 𝑓′ 𝑥 > 0, for all 𝑥 in an interval I, then f is increasing on I.

if 𝑓′ 𝑥 < 0, for all 𝑥 in an interval I, then f is decreasing on I.

if 𝑓′ 𝑥 = 0, for all 𝑥 in an interval I, then f is constant on I.

Checkout this graph.

You've got an ant climbing on the graph.Not just any ant...

He's kind of a

pathetic math

superhero.

Pierre the Mountain Climbing Ant!

For this, the rule is that Pierre only crawls from left to right…..

If Pierre is climbing uphill, then the graph is increasing:

So, our graph is

increasing on:

If Pierre is going downhill, then the graph is decreasing:

So, our graph is decreasing on:

When Pierre is standing right ON x = -3 ...

He's not going uphill or downhill. He's just standing there!

What if Pierre is walking on ?

That line is horizontal (slope of 0). He's not going uphill or downhill,so the graph is not increasing or decreasing there.

Critical Numbers

Definition:

The critical numbers for a function f arethose numbers c in the domain of f for which𝑓′ 𝑐 = 0 or 𝑓′ 𝑐 does not exist.

A critical point is a point whose 𝑥 -coordinate is the critical number c, and whose𝑦-coordinate is 𝑓(𝑐).

APPLYING THE TEST

Locate the critical numbers for f on a numberline, as well as any points where f is undefined.These points determine several open intervals.

Choose a value of 𝑥 in each of the intervalsdetermined in Step 1. Use these values to decidewhether 𝑓′(𝑥) > 0 or 𝑓′(𝑥) < 0 in that interval.

Use the test above to decide whether f isincreasing or decreasing on the interval.

Example 1:

Given 𝑓 𝑥 = 𝑥3 + 6𝑥2 + 9𝑥

a) Find the open interval where the given function is increasing or decreasing.

b) Locate all points where the tangent line is horizontal.

c) Graph the function.

Seatwork

Given 𝑓 𝑥 = 2𝑥3 − 15𝑥2 + 36𝑥 − 24

a) Find the open intervals where the given function is increasing or decreasing.

b) Locate all points where the tangent line is horizontal.

c) Graph the function.

Assignment

DIRECTION:

Answer 5.1 Exercises on your book (Calculuswith Applications, 9th Edition) nos. 18, 19, 20 and21 page 314.