a summary of curve sketching · a summary of curve sketching in class example =−2 4+3 2...
TRANSCRIPT
A Summary of Curve
Sketching
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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)