A Summary of Curve

Sketching

Copyright © Cengage Learning. All rights reserved.

Homework Questions!?

Determining Concavity Page 192 #1-9 odd

Finding the points of inflection Page 192 #19-29 odd

Using the second derivative test (to find extrema) page 192 #31-41

odd

#51, 53, 57 (understanding the graphs)

#65 Application

Learning Target today yesterday

I can find the second derivative a function and apply it to determine concavity and find points of inflection

Learning Target Today

I can analyze and sketch the graph of

a function using algebra, first

derivatives and second derivatives!

Analyzing the Graph of a Function

Analyzing the Graph of a

FunctionWhen you are sketching the graph of a function, either by hand

or with a graphing utility, remember that normally you cannot

show the entire graph.

The decision as to which part of the graph you choose to show is

often crucial.

Don’t worry

about

notes on

this slide (in

my opinion)

…can be

found on

page 206

Analyzing the Graph of a

FunctionFor instance, which of the viewing windows in Figure

3.44 better represents the graph of f(x) = x3 – 25x2 + 74x

– 20?

Figure 3.44

Don’t worry

about

notes on

this slide (in

my opinion)

…can be

found on

page 206

Analyzing the Graph of a

FunctionBy seeing both views, it is clear that the second viewing window

gives a more complete representation of the graph.

But would a third viewing window reveal other interesting

portions of the graph?

To answer this, you need to use calculus to interpret the first and

second derivatives.

Don’t worry

about

notes on

this slide (in

my opinion)

…can be

found on

page 206

Here are some guidelines for determining a good viewing

window for the graph of a function.

Analyzing the Graph of a

Function

Don’t worry

about

notes on

this slide (in

my opinion)

…can be

found on

page 206

Information you will want/need to find (it is

extremely unlikely you would need to do all of

these steps on an AP test…though you may

need to do a few…and which you will not

know prior

Don’t worry

about

notes on

this slide (in

my opinion)

…can be

found on

page 206

Take Notes on the next slide!!

A summary of Curve Sketching

What you’ll need to do

X-intercepts

Y-intercepts

Domain (typically all real numbers…unless a rational function)

Vertical asymptotes if rational

Horizontal asymptotes if rational (or end behavior if not rational)

Symmetry (is f(x)=f(-x) or is f(x)= -f(x) or neither)

First derivative…AND critical points

Second derivative…AND points of inflection

Set up and test intervals to determine extrema and concavity

Take Notes on this slide!!

Example 1 – Sketching the Graph of a Rational

Function

Analyze and sketch the graph of

Solution:

Don’t worry

about notes on

this slide or next 6

(in my opinion)

…example 1 in

book…page 207

We will do an

example

together

I slightly

rearranged order

from the

book…cause I

thought it made

more sense.

Example 1 – Solution cont’dcont’d

Example 1 – Solution cont’dThe table shows how the test intervals are used to determine several

characteristics of the graph.cont’d

Example 1 – Solution

The graph of f is shown in Figure 3.45.

Figure 3.45

cont’d

More examples can be found in

section 3.6 of your text

Rational Function example (example 2…page 208)

Radical function example (example 3 and 4…page 209)

Trig Function example (example 6…page 211)

We will do the polynomial function example (example 5, page 210) next

Note…the example function in example 2 uses a slant asymptote

In Figure 3.48, note that the graph of f approaches the slant

asymptote y = x as x approaches ∞ 𝑜𝑟 −∞

Figure 3.48

Analyzing the Graph of a

FunctionThe graph of a rational function (having no common factors and

whose denominator is of degree 1 or greater) has a slant

asymptote if the degree of the numerator exceeds the degree of

the denominator by exactly 1.

To find the slant asymptote, use long division to rewrite the

rational function as the sum of a first-degree polynomial and

another rational function.

A summary of Curve Sketching

in class example 𝑦 = −2𝑥4 + 3𝑥2

X-intercepts

Y-intercepts

Domain (typically all real numbers…unless a rational function)

Vertical asymptotes if rational

Horizontal asymptotes if rational (or end behavior if not rational)

Symmetry (is f(x)=f(-x) or is f(x)= -f(x) or neither)

First derivative…AND critical points

Second derivative…AND points of inflection

Set up and test intervals to determine extrema and concavity

Take Notes on this slide!!

Assignment for Friday

Page 212

#1-4 all

#9-21 odd

Calc practice questions #25-29 odd

#55-59 odd (very quick questions…connects your analysis skills to

graphs)

Rest of period!!

AP calc practice.

By my estimation (after going over a few old AP tests)…we have

learned information enough to answer about 20 of the 54 points in an

extended response pack AND about 16 or so of the 45 multiple choice

questions.

Let’s use the 2013 extended response to look at the questions we can

do AND the 2008 multiple choice section.

I very much recommend going through these tests on your

own time…try the following method

1. Print out the test

2. Highlight the parts we potentially are able to do (I will give you those

numbers in a second)

3. Try them.

4. Look at the answers (on AP website…link on www.scubamoose.weebly.com )

5. Study the answers…compare to your answers.

6. In a day or two…come back to the test…try it again on fresh sheet of paper.

7. Repeat steps 4-6 as often as is necessary.

8. I can give you a breakdown of other test years as well…and will continue to

do so as we learn more through the year (integration being a huge topic we

will begin covering next!)

2013 Test…extended response…questions which assess the stuff we’ve learnedwww.scubamoose.weebly.com

1a chp 2 Questions 1 and 2 are calculator OK

1c chp 2

2a chp 2

2c chp 3

2d chp 3

3a chp 3 Question 3-6 are calculator not OK

3b chp 3

4a chp 3

4c chp 3

4d chp 2

6a (algebra…though the notation is scary…wasn’t sure to list this one or not)

2008 test (multiple choice)…question over

material we have studiedGoogle AP calculus AB 2008 multiple choice www.google.com

Questions

3,6,8,11,18,20,21,24,25,26,28 (no calc)

78,80,82,84 (calculator OK)