chapter 3: applications of the derivative and curve...

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How have you sketched graphs in the past?

Polynomials: Zeros (x­ints) & y­int; Relative/Absolute Extrema (min/max)

Rationals: Asymptotes, intercepts, holes.

Chapter 3: Applications of the Derivative and Curve Sketching

3.1 Increasing and Decreasing Functions

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You will still be responsible for the following (We will discuss min/max by hand)

Zeros: set y=0; Factor & solve or Calculator

y­int: set x=0 (evaluate)

Vert Asym: Denominator = 0 and Numerator ≠ 0

Horiz. Asym.

Holes: Numerator and Den equal Zero (typically).

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The first thing we will add to this list is describing how the curve is behaving over certain intervals (a, b); We will describe whether the curve is increasing, decreasing, or neither.

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Increasing Interval:

A function is said to be increasing on the interval (a, b) if the y value of any point is higher than that of the point to the left of it. In calc...

A function is increasing iff...

f(x1) < f(x2) for all x1 < x2

on the interval.

From this we can get a similarly worded definition of decreasing intervals.

How will we determine if a graph is increasing or decreasing? Will we plot a plethora of points and examine y­values? Not accurate enough!

What describes how a function moves?

Derivatives!!!

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Derivative: Describes slope Slope is positive → Increasing Slope is negative → Decreasing Slope is zero → Constant (Horiz)

If we have the derivative of a function, we can describe how the function moves/changes.

How do I find where we change from inc to dec?

How do we find the various intervals?

,

Our intervals are formed by places where the derivative either equals zero, or is undefined. These places are called critical values.

Once we find critical values, we test the derivative in each interval (positive or negative) to determine if the original function is increasing or decreasing.

This is the First­Derivative­Test

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IF...

1. x=c is a critical value from the first derivative

2. f(c) is defined (it exists. no asymptote, no hole.)

3. First­Der­Test yields sign change over x=c

THEN...

x=c represents a minimum or a maximum of f(x)

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Describe the inc/dec behavior of the following functions.

Describe / classify critical values.

Use chart.

chart:

When is a critical value a minimum or a maximum?

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chart:

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There are times when an undefined derivative does NOT necessarily imply an ASYM.

Ex]

Domain:

Zeros:

y­int:

We find the locations of asymptotes from the original function!!

If a function is undefined at c, the derivative will be undefined at c.

If a function is defined at c, the derivative can be defined or undefined.

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Determine if y = (x+4)3 is increasing everywhere.

Domain:

Zeros:

y­int:

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Pg 184 Ex 3

Find & classify critical points. Describe Increasing/Decreasing intervals.

Domain:

Zeros:

y­int:

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Ex]

Domain:

Zeros:

y­int:

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Pg 175: 5­33 EOO

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Finding Absolute Extrema on Closed Intervals [a,b]

1. Find critical values on [a, b]

2. Evaluate critical values, a, and b, in function.

3. Find highest / lowest values.

3.2 (Part of it) Extrema and the First­Derivative Test

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Ex] Find absolute extrema on [3,5] for

y = t ­ 2t

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Pg 175: 5­33 EOO

Pg 184: 19, 21, 23