applications of differentiation curve sketching. why do we need this? the analysis of graphs...
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Applications of DifferentiationCurve Sketching
Why do we need this?The analysis of graphs involves looking at
“interesting” points and intervals and at horizontal and vertical asymptotes. We use calculus techniques to help us find all of the important aspects of the graph of a function so that we don’t have to plot a large number of points.
Curve Sketching
Graphing Skills AKA skills you will have when this chapter has ended
Domain and Range Symmetry x and y intercepts Asymptotes Extrema Inflection Points
Curve Sketching
4
Extrema on an IntervalExtrema of a Function ● Relative Extrema and Critical Numbers ● Finding Extrema on a Closed Interval
Curve Sketching
Global Extrema Example
Curve Sketching
Lets look at the following function:
How many horizontal tangent lines does this curve have?
What is the slope of a horizontal tangent?
Curve Sketching
Finding Horizontal Tangents
Differentiate:
Set the derivative equal to zero (slope =0):
Solve for x:
This is the x coordinate where you have a horizontal tangent on your graph.
3f x x x
23 1f x x
23 1 0f x x
2 13 1 0
3x x
Curve Sketching
Finding Horizontal Tangents
Differentiate:
Set the derivative equal to zero (slope =0):
Solve for x:
This is the x coordinate where you have a horizontal tangent on your graph.
4 33 4f x x x
3 212 12f x x x
3 212 12 0f x x x
3 2 212 12 0 12 1 0 0,1x x x x x
Curve Sketching
Finding Horizontal Tangents
Differentiate:
Set the derivative equal to zero (slope =0):
Solve for x:
This is the x coordinate where you have a horizontal tangent on your graph.
2
3
9 3xf x
x
3 2 2 2
6 4
18 9 3 3 9 81x x x x xf x
x x
2
4
9 810
xf x
x
22
4
9 810 9 81 0 3
xx x
x
Curve Sketching
Vertical Tangents Keep in mind where the derivative is
defined:
When x = 0 we have slope that is undefined
This is where we have a vertical tangent line.
This counts as a critical number as well and should be considered as such.
2
4
9 81xf x
x
Curve Sketching
Definition of ExtremaLet f be defined on an interval I containing c.
1. f(c) is the maximum of f on I if f(c) > f(x)
for all x on I.2. f(c) is the minimum of f on I if f(c) < f(x)
for all x on I.The maximum and minimum of a function on an
interval are the extreme values, or extrema, of the function on the interval.
The maximum and minimum of a function on an interval are also called the absolute maximum and absolute minimum on the interval, respectfully.
Curve Sketching
Definition of a Critical Number
Critical numbers are numbers you check to locate any extrema.Let f be defined at c.
If f ‘(c) = 0 or if f is undefined at c, then c is a critical number
Curve Sketching
Finding Extrema on a Closed Interval Step 1: find the critical numbers of f in
(a,b) Step 2: Evaluate f at each critical
number in (a,b) Step 3: Evaluate f at each endpoint of
[a,b] Step4: The least of the values is the
minimum, the greatest is the maximum.
Curve Sketching
Theorem: Relative Extrema Occur only at Critical Numbers
If f has a relative minimum or relative maximum at
x = c, then c is a critical number of f.
These are also known as locations of horizontal tangents
Curve Sketching
Critical NumbersLet f be defined at c. if f’(c)=0
or if f is not differentiable at c, then c is a critical number of f.
Relative extrema (max or min) occur only at a critical number.
Curve Sketching
Relative Extrema
Curve Sketching
Additional Extrema on a Closed Interval-examples:
Curve Sketching
Locate the Absolute Extrema
undefined. is or where 0any Find
5] [3, interval on the 2
Given
cfcft
txf
undefined. is or where 0 wherelocated are numbers critical
2
22
ff
ttf
.any for 0 not does xf
.2at undefined is xf
extrema.an oflocation thebecannot it so 5] [3, interval in thenot is 2
.3
55 and 33 interval. theof endpoints Check the ff
5] [3, interval on the minimum a is 5/3) (5, and maximum a is 3) (3,
19
Finding Extrema on a Closed Interval
Left Endpoint
Critical Number
Critical Number
Right Endpoint
f(-1)=7 f(0)=0 f(1)=-1Minimum
f(2)=16Maximum
Find the extrema:
Differentiate the function:
Find all values where the derivative is zero or UND:
2]. 1,[ interval on the 43 34 xxxf
23 1212 xxxf
01212 23 xxxf
0112 xx
1,0 when 0 xxfGiven endpoints, consider them too!!!
Curve Sketching
Homework for 3.1 Page 169-171#1-46 #61-64 are physics problems #65-70 are good thinking problems
Curve Sketching
First Derivative Test (Ch.3 S.3) Finding the critical numbers:
C is called a critical number for f if f’(c)=0 (horizontal tangent line) or f’(c) does not exist
Curve Sketching
Increasing or Decreasing functions A function f is called increasing on a
given interval (open or closed) if for x1 any and x2 in the interval, such that x2> x1, f(x2)>f(x1)
A function f is called decreasing on a given interval (open or closed) if for x1 any and x2 in the interval, such that x2> x1, f(x2)<f(x1)
Curve Sketching
Maximums, Minimums, Increasing, Decreasing Let the graph represent some function f.
1
2
3
1
1
2
2
2
3
a b d e fc
inimumxfef
xfdf
inimumxfcf
caxfbf
mfd, interval on the
maximumec, interval on the
mdb, interval on the
maximum, interval on the
decreasing is 0 .3
maxor min a has 0 .2
increasing is 0 .1
fxf
fxf
fxf
Curve Sketching
Copyright © Houghton Mifflin Company. All rights
reserved.
3-24
Definitions of Increasing and Decreasing Functions and Figure 3.15
Copyright © Houghton Mifflin Company. All rights
reserved.
3-25
Theorem 3.6 The First Derivative Test
Finding Intervals Where the Function is Increasing/Decreasing
A. 4 3( ) 2 1f x x x B. 2
8 16( )
xf x
x
C. 2 3( ) 2 3f x x x
Curve Sketching
Finding Intervals Where the Function is Increasing/Decreasing
A. 4 3( ) 2 1f x x x B. 2
8 16( )
xf x
x
C. 2 3( ) 2 3f x x x
Curve Sketching
Finding Intervals Where the Function is Increasing/Decreasing
A. 4 3( ) 2 1f x x x B. 2
8 16( )
xf x
x
C. 2 3( ) 2 3f x x x
Curve Sketching
Increasing and Decreasing Example
1. In each case, sketch a graph of a continuous function with the given properties. -- + -- A. ( 1) 0f and (3) 0f ( )f x | | -1 3
+ -- -- B. (1) 0g and (4)g is undefined ( )g x | | 1 4
+ -- + -- C. ( 2) 0h and (2) 0h ( )h x | | | (0)h is undefined -2 0 2
Curve Sketching
LOCAL (RELATIVE) EXTREME VALUES
If x = c is not at an endpoint, then f (c) is a local (relative) maximum value for f(x) if f(c) > or = f(x) for all x-values in a small open interval around x = c.
If x = c is not at an endpoint, then f (c) is a local (relative) minimum value for f(x) if f(c) < or = f(x) for all x-values in a small open interval around x = c.
Curve Sketching
Finding local extrema
A. 4 3( ) 2 1f x x x B. 2
8 16( )
xf x
x
C. 2 3( ) 2 3f x x x
Curve Sketching
Local Extrema Example See text for additional examples
Curve Sketching
Homework for 3.3
pg. 186 #9-38, 43-50, 65-84 (good for AP)
Curve Sketching
3.4 Concavity and the Second Derivative Test
Second Derivative Test
Copyright © Houghton Mifflin Company. All rights
reserved.
3-35
Definition of Concavity and Figure 3.24
The graph of f is called
concave up
on a given interval if for any two
points on the interval, the graph
of f lies below the chord that
connects these points.
Concavity
Concavity
The graph of f is called
concave down
on a given interval if for any two
points on the interval, the graph
of f lies above the chord that
connects these points.
Copyright © Houghton Mifflin Company. All rights
reserved.
3-38
Theorem 3.7 Test for Concavity
Copyright © Houghton Mifflin Company. All rights
reserved.
3-39
Definition of Point of Inflection and Figure 3.28
Copyright © Houghton Mifflin Company. All rights
reserved.
3-40
Theorem 3.8 Points of Inflection
Copyright © Houghton Mifflin Company. All rights
reserved.
3-41
Theorem 3.9 Second Derivative Test and Figure 3.31
Second Derivative Test
Finding the points of inflection:
C is called a point of interest for f if
f’’(c)=0 or f’(c) does not exist.
NOTE: The fact that f’’(x) = 0 alone does not
mean that the graph of f has an
inflection point at x.
We must find the points where the concavity
changes.
Points of Inflection
A point of inflection is a point
where the concavity of the graph
changes from concave up to
down or vice-versa.
Concavity and the Second Derivative Test
A. 4 3( ) 2 1f x x x B. 2
8 16( )
xf x
x
C. 2 3( ) 2 3f x x x
Curve Sketching
Homework for 3.4 P. 195 #1-52, 61-68 (good for AP prep)
Curve Sketching
Limits at Infinity (3.5)
Curve Sketching
What happens at x = 1?
What happens near x = 1?
As x approaches 1, g increases without bound, or g approaches infinity.
As x increases without bound, g approaches 0.
As x approaches infinity g approaches 0.
2
1
1g x
x
asymptotes
2
1
1g x
x
x x-1 1/(x-1)^2
0.9 -0.1 100.00
0.91 -0.09 123.46
0.92 -0.08 156.25
0.93 -0.07 204.08
0.94 -0.06 277.78
0.95 -0.05 400.00
0.96 -0.04 625.00
0.97 -0.03 1,111.11
0.98 -0.02 2,500.00
0.99 -0.01 10,000.00
1 0 Undefined
1.01 0.01 10,000.00
1.02 0.02 2,500.00
1.03 0.03 1,111.11
1.04 0.04 625.00
1.05 0.05 400.00
1.06 0.06 277.78
1.07 0.07 204.08
1.08 0.08 156.25
1.09 0.09 123.46
1.10 0.1 100.00
vertical asymptotes
2
1
1g x
x
x x-1 1/(x-1)^2
1 0 Undefined
2 1 1
5 4 0.25
10 9 0.01234567901234570
50 49 0.00041649312786339
100 99 0.00010203040506071
500 499 0.00000401604812832
1,000 999 0.00000100200300401
10,000 9999 0.00000001000200030
100,000 99999 0.00000000010000200
1,000,000 999999 0.00000000000100000
10,000,000 9999999 0.00000000000001000
100,000,000 99999999 0.00000000000000010
horizontal asymptotes
AKA Limits at infinity We know that
And we know that
So we can expand on this…
horizontal asymptotes
10lim
x x
2
10lim
x x
Theorem: If r is a positive rational number and c is
any real number, then
Furthermore, if is defined when x<0, then
horizontal asymptotes
0lim rx
c
x
0lim rx
c
x
rx
Theorem: The line y=L is a horizontal
asymptote of the graph of f if
or
Limits at Infinity
limx
f x L
limx
f x L
Examples: Finding a limit at infinity
Limits at Infinity
2
2lim 5x x
2
2lim 5 limx x x
5 0 5
Finding a limit at infinity
Direct substitution yields indeterminate form! Now we need to do algebra to evaluate the limit:
Divide each term by x:
Simplify:
Evaluate(keep in mind each 1/x0)
Limits at Infinity
2 1lim
1x
x
x
2 1
lim1x
x
x xx
x x
12
lim1
1x
x
x
2 0 2lim 2
1 0 1x
A comparison (AKA the shortcut!)
Direct substitution yields infinity/infinity! Do algebra:
Notice a pattern?????
Limits at Infinity
2
2 5lim
3 1x
x
x
2
2
2 5lim
3 1x
x
x
3
2
2 5lim
3 1x
x
x
2 2
2
2 2
2 5
lim3 1x
xx xxx x
2
2
2 5
lim1
3x
x x
x
00
3
2
2 2
2
2 2
2 5
lim3 1x
x
x xx
x x
3
3 3
2
3 3
2 5
lim3 1x
x
x xx
x x
2
2
52
lim1
3x
x
x
2
3
2!!
0undefined
3
3
52
lim3 1x
x
x x
A comparison (AKA the shortcut!)
Direct substitution yields infinity/infinity! Do algebra:
Bottom Heavy Balanced Top Heavy
Limits at Infinity
2
2 5lim
3 1x
x
x
2
2
2 5lim
3 1x
x
x
3
2
2 5lim
3 1x
x
x
02
3 !!undefined
Limits with 2 horizontal asymptotes
Note: for
Note: for
Homework Page 205 #2-38E
Curve Sketching
Curve Sketching 3.6 The following slides combine all of our
new graphing and analysis skills with our precalculus skills
Example:
Curve Sketching
3 2( ) 6 9 1f x x x x
Asymptotes:
Polynomial functions do not have asymptotes.
a) Vertical: No vertical asymptotes because f(x) is continuous for all x.
b) Horizontal: No horizontal asymptotes because f(x) is unbounded as x goes to positive or negative infinity .
3 2( ) 6 9 1f x x x x
Curve Sketching
Intercepts:
a) y-intercepts: f(0)=1 y-intercept: (0,1)
b) x-intercepts: difficult to find – use TI-89
Curve Sketching
3 2( ) 6 9 1f x x x x
Critical numbers:
a) Take the first derivative: b) Set it equal to zero:
c) Solve for x:
x=1, x=3
Curve Sketching
3 2( ) 6 9 1f x x x x
Critical Points:
a)Critical numbersx=1 and x=3
b) Find corresponding values of y:
c) Critical points: (1,5) and (3,1)
Curve Sketching
3 2( ) 6 9 1f x x x x
Increasing/Decreasing:
a) Take the first derivative: b) Set it equal to zero:
c) Solve for x:
x=1, x=3
Curve Sketching
3 2( ) 6 9 1f x x x x
d) Where is f(x)undefined? Nowheree) Sign analysis: Plot the numbers found above on a number line.
Choose test values for each interval created and evaluate the first derivative:
Increasing/Decreasing:
2'(0) 3(0) 12(0) 9 9f Positive ( )f x is increasing on ( ,1) .
2'(2) 3(2) 12(2) 9 3f Negative ( )f x is decreasing on (1,3) .
2'(4) 3(4) 12(4) 9 9f Positive ( )f x is increasing on (3, ) .
f '(x)1 3
+ _ +
3 2( ) 6 9 1f x x x x
Extrema
f’(x) changes from positive to negative at x=1 and from negative to positive at x=3 so and are local extrema of f(x).
Note: Values of corresponding to local extrema of must: Be critical values of the first derivative – values at which
equals zero or is undefined, Lie in the domain of the function, and Be values at which the sign of the first derivative
changes.
Curve Sketching
3 2( ) 6 9 1f x x x x
f(x) is increasing before x=1 and decreasing after x=1.
So (1,5) is a maximum.
f(x) is decreasing before x=3 and increasing after x=3.
So (3,1) is a minimum.
First Derivative Test
Curve Sketching
3 2( ) 6 9 1f x x x x
SECOND DERIVATIVE TEST
Alternate method to find relative max/min:a) Take the second derivative:
b) Substitute x-coordinates of extrema: (negative local max)(positive local min)
c) Label your point(s): local max: (1,5) local min: (3,1)
Curve Sketching
3 2( ) 6 9 1f x x x x
Concave Up/Down
6. Concave up/Concave down: a) Set ( )f x equal to zero: 6 12 0x b) Solve for x : 2x c) Where is ( )f x undefined? Nowhere d) Sign analysis: Plot the numbers found above on a number line. Choose test values for each interval created and evaluate the second derivative.
''(1) 6(1) 12 6 0f ( )f x is concave down on ( ,2) . ''(3) 6(3) 12 6 0f ( )f x is concave up on (2, ) .
f ''(x)
+
2
_
Curve Sketching
3 2( ) 6 9 1f x x x x
Inflection Points
7. Find any inflection points: a) For which values of x (found in 6) is ( )f x defined? 2x b) Find corresponding value of y : 3 2(2) (2) 6(2) 9(2) 1 3f c) ( )f x changes from concave up to concave down at 2x , so (2,3) is an inflection point. Inflection point: (2,3)
Note: Values of x corresponding to inflection points of f must: • be critical values of the second derivative – values at which ( )f x equals zero or is undefined, • lie in the domain of f , and
• be values at which the sign of the second derivative changes
Curve Sketching
3 2( ) 6 9 1f x x x x
Create a table of the values obtained:
Curve Sketching
x 0 1 2 3
f(x)
f’(x)
f’’(x)
9. Plot all points on the coordinate plane and sketch in the rest of the graph. Be sure to include all maximum points, minimum points, and inflection points:
_ _ _ _ _ _
5 4 2 1 -1
| | | | | | | | | -3 -2 -1 1 2 3 4 5 6
f(x) = x - 6x + 9x + 13 2
Curve Sketching
2 2 4( )
2
x xf x
x
Graphing Functions
Domain and Range Domain: evaluate where the function is
defined and undefined, values of discontinuity (under radicals, holes, etc.), nondifferentiability (sharp turns, vertical tangent lines and points of discontinuity).
Range: evaluate the values of y after graphing. Look for minimum and maximum values and discontinuities vertically.
Domain and Range
Finding the Range of a FunctionWe will look at this AFTER we graph the
equation using calculus, you can graph it on the calculator now if you like.
Curve Sketching
SymmetryAbout the y-axis:
Replace x with –x and you get the same equation after you work the algebra
About the x-axis:Replace y with –y and you get the same equation after you work the algebra
About the origin:Replace x with –x and y with –y and get the same equation after you work the algebra
Curve Sketching
Graphing Functions
2
2
2
2 2
2 4( )
2
2 4( )
2
2 4
2
2 4 2 4? ?
2 2
x xf x
x
x xf x
x
x x
x
x x x x
x x
Do we have symmetry about the y-axis?
replace x with –x and we get:
Are these equivalent equations?
NO! So we do not have symmetry about the y-axis.
Curve Sketching
Graphing Functions2
2
2
2 2
2 4
2
2 4
2
2 4
2
2 4 2 4? ?
2 2
x xy
x
x xy
x
x xy
x
x x x x
x x
Do we have symmetry about the x-axis?
replace y with –y and we get:
Are these equivalent equations?
NO! So we do not have symmetry about the x-axis.
Curve Sketching
Graphing Functions
2
2
2
2
2 2
2 4
2
2 4
2
2 4
2
2 4
2
2 4 2 4? ?
2 2
x xy
x
x xy
x
x xy
x
x xy
x
x x x x
x x
Do we have symmetry about the origin?
replace x with –x and y with –y and we get:
Are these equivalent equations?
NO! So we do not have symmetry about the origin.
Curve Sketching
x and y-intercepts x-intercepts: Set top of the fraction
equal to 0 in the original equation and solve for x values.
y-intercepts: Substitute 0 in for x and solve for y values.
x-intercepts
2
2
2
2
2
2 4( )
2
2 40
2
0 2 4
4
2
2 2 4 1 4
2 1
2 4 16
2 1
???
x xf x
x
x x
x
x x
b b acx
a
x
x
Replace y with 0 and we get ???
so do we cross the x-axis?
y-intercepts
2
2
2 4( )
2
0 2 0 4(0)
0 2
42
2
x xf x
x
f
Replace x with 0 and we get y = -2
so we cross the y-axis at (0,-2)
Curve Sketching
Horizontal AsymptotesThe line y=L is called a horizontal asymptote for f if the limit as x approaches infinity from both the left and/or right is equal to L
In other words, find the original equation as x approaches positive or negative infinity.
Limits at Infinity
Curve Sketching
Horizontal Asymptotes2
2 2
2
2
2
2 4( )
2
( ) ?
12 4
12
2 41
1 12
1
0
x xf x
xx
f x
x x xx
x
x x
x x
As x goes to infinity what happens to our equation?
It goes to infinity as well. So we have no horizontal asymptotes
Curve Sketching
Vertical Asymptotes The line x=c is called a vertical asymptote
for f at x=c if the limit goes to positive or negative infinity as x approaches c from the positive and/or negative side.
This is the complicated way of saying set the denominator (if there is one) to zero and find the x values.
Infinite Limits
Curve Sketching
Vertical Asymptotes
2 2 4( )
22 0
2
x xf x
xx
x
We set the denominator to
equal zero
and find that we have an asymptote
at x = 2
Curve Sketching
Slant AsymptotesGiven a function that consists of a
composition of functions:
If there are no common factors in the numerator and denominator, perform long division. The NONfractional part gives you the equation of the slant asymptotes.
Curve Sketching
Graphing Functions2
2
2 4( )
24
22 2 4
x xf x
x
xx
x x x
Performing long division:
we get a non-fractional part of
the answer
of y = x for our slanted asymptote
Curve Sketching
First Derivative Test
212
1 12 2
1 22
2
2
2 2
2 2
22
2
2 4( ) 2 4 2
2
( ) 2 4 2 2 4 2
2 2 2 2 4 1 2
2 2 2 4
2 2
2 2 2 2 4 4
2 2
40 4 0
2
( 4)
x xf x x x x
xd d
f x x x x x x xdx dx
x x x x x
x x x
x x
x x x x x x
x x
x xx x
x
x x
0
0, 4x x
Take the derivative of the function,
set it equal to 0 and find the corresponding values of x.
These are our CRITICAL POINTS
(Potential extrema)
Graphing Functions
2
2
2
2 2
4
2
4
2 2
4
3
2 4( )
2
4( )
2
2 4 2 4 2 2( )
2
2 4 2 4 2
2
2 6 8 2 8
2
2 80 2 8 0 4
2
x xf x
x
x xf x
x
x x x x xf x
x
x x x x
x
x x x x
x
xx x
x
Take the second derivative
Set it equal to 0
Solve for x
This is our potential point of inflection
Graphing Functions2 2 4
( )2
x xf x
x
Here we compile all of the information to enable us to give a pretty accurate graph WITHOUT technology
Organize the Informationx f(x) f’(x) f’’(x) Conclusion
-∞<x<0 + - Increasing, concave down
X=0 -2 0 - Relative Max
0<x<2 - - Decreasing, concave down
X=2 Undef Undef Undef Vertical asymptote
2<x<4 - + Decreasing, concave up
X=4 6 0 + Relative min
4<x<∞ + + Increasing, concave up
Homework: Page 215 #1-45, (67-71 are good thinking problems)
Curve Sketching