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Section 3.6: Critical Points and Extrema
Objectives: I can find the extrema (maximums and
minimums) of a function.
Definitions
Definition – Critical Point:
Definition – Absolute Max/Min:
Definition – Relative Max/Min:
• Where the function changes directions• (where a line tangent to the curve is either
horizontal or vertical)
• The largest/smallest value on the entire graph (over the entire domain)
• The largest/smallest value on a given interval (not necessarily over the entire domain)
Example 1:
Locate the extrema for the graph for g(x). Name and classify the extrema.
Absolute Maximum: Absolute Minimum: Relative Maximum (maxima): Relative Minimum (minima):
None (arrows!)
None (arrows!)
(-3, 13)
(2, -10)
Example 2 (You Try It!):
Locate the extrema for the graph for h(x). Name and classify the extrema.
Relative Maximum: (-8, 5) Relative Minimum: (7.5, -2.3 ish) Relative Maximum (maxima): (0, 3) Relative Minimum (minima): (-2.5ish, 2ish)
Example 3( add inc/dec):
Use a calculator to graph to determine and classify its extrema. Sketch a graph of the situation.
720105)( 23 xxxxf
Abs Max: none Abs Min: none Rel Max: (-2/3, 14.17) Rel Min: (2, -33)
Inc: {x < -2/3}Dec: {-2/3 < x < 2}Inc: {x > 2}
Example 4:
The function has critical points at x = 0 and x = 1. Classify each critical point and determine on which intervals it is increasing and decreasing. Sketch a graph of the situation.
34 43)( xxxh
Warmup Grab a “Foldable” packet (4 pages)
Cut off bottom (shaded) portion from each Staple together on top left and right corners Start warmup below
WarmupLocate and classify the extrema of f(x) = 3x4 – 6x + 7 and write the intervals in which the function is increasing/decreasing.
_Finding Maximums and Minimums
Finding a(n)… It means… Example…
Absolute Maximum
Absolute Minimum
Relative Maximum (Maxima)
Relative Minimum (Minima)
Highest point on entire domain
Lowest point on entire domain
Highest point inLocal area
Lowest point inLocal area
Section 3.5: Continuity and End Behavior
Objectives: Determine whether a function is continuous or
discontinuous. Identify the end behavior of functions. Determine whether a function is increasing or
decreasing on an interval.
Example 1(skip for now):
Determine whether the function f(x) = 3x2 + 7 is continuous at x = 1.
Does the function exist at the point?
f(1) = 3(1)2 + 7 = 10 Does the function have any domain
restrictions that might cause issues? Does the function approach ‘10’ from both
sides? Yuppers.
yup
nope
yuppers
CONTINUOUS
Example 2 and 3 (slip for now): Determine whether the function
is continuous at x = 1.
Determine whether the function is continuous at x = -2.
1
33)(
2
x
xxxf
2
4)(
2
x
xxf
Nope, domain restriction
Darn it….this one too…..(even though your calc might trick you)
Example 4: Find the intervals for which f(x) = 4x2 + 9 is
increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate. Dec: x < 0
Inc: x > 0
Chillin’ when x = 0
End behavior:
)(lim xfx
)(lim xfx
Example 5: Find the intervals for which
is increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate.
13)( 23 xxxxf
Dec: -.46 < x < .24Inc: x < -.46
End behavior:
)(lim xfx
)(lim xfx
Inc: x > .24
Example 6: Find the intervals for which
is increasing and/or decreasing, also determine its end behavior. Sketch a graph to illustrate.
365)( 3 xxxf
Dec: on the entire graph{x: all real numbers}
End behavior:
)(lim xfx
)(lim xfx
Section 3.7: Graphs of Rational Functions (Day 1)
Objectives: Graph rational functions. Determine vertical, horizontal, and oblique
asymptotes.
Example 1(from 3.6):
Determine whether the function f(x) = 3x2 + 7 is continuous at x = 1.
Does the function exist at the point?
f(1) = 3(1)2 + 7 = 10 Does the function have any domain
restrictions that might cause issues? Does the function approach ‘10’ from both
sides? Yuppers.
yup
nope
yuppers
CONTINUOUS
Example 2 and Example 3(from 3.6): Determine whether the function
is continuous at x = 1.
Determine whether the function is continuous at x = -2.
1
33)(
2
x
xxxf
2
4)(
2
x
xxf
Nope, domain restriction
Darn it….this one too…..(even though your calc might trick you)
Definition – Vertical Asymptote:
Essential (Infinite) Discontinuity
An asymptote in the vertical direction
A vertical asymptote ;)- Found from the denominators domain restrictions
Example 1:
Using answer the following: What is the vertical asymptote?
What is the limit of the function near the asymptote?
4
1
x
V.A. : x = 4
)(lim4
xfx
}4:{ xx
)(lim4
xfx
“from the left” “from the right”
Example 2:
Using answer the following: What is the vertical asymptote?
What is the limit of the function near the asymptote?
5
3
x
V.A. : x = 0
)(lim0
xfx
}0:{ xx
)(lim0
xfx
“from the left” “from the right”
Example 3:
Discuss the discontinuities and end behavior for the following graphs:
Vertical asymptote x = 0Horiz. Asymptote y = 0
End behav: as x goes to –infinity? + infinity?
Hole (removable) at (4, 6)
End behav: as x goes to –infinity? + infinity?
No discontinuities for this one.
End behav: as x goes to –infinity? + infinity?
Vertical asymptotes x = 2 and -2Horiz. Asymptote y = 0
End behav: as x goes to –infinity? + infinity?
Definitions
Definition – Horizontal Asymptote:
Definition – Removable Discontinuity:
Definition – Oblique Asymptot:
Comes from the end behavior (Limit!!!!)
Just a hole in the graph (factor to find)
When the asymptote is a diagonal line…stay tuned for this…
Revisit Example 3(from 3.6):
Determine whether the function is continuous at x = -2.
2
4)(
2
x
xxf
Horizontal AsymptotesOption 1: Same over Same
Option 2: Bigger over Smaller
Option 3: Smaller over Bigger
62
53
x
x
000,000,11000
2
xx
439
3534
24
xx
xx
2
42
x
x
3
2
x
x54
212
xx
x
#
0
Warm-up: Match up the Function, its graph, and the type
of discontinuity
Foldable
End Behavior (Horz Asym)
Exponents How to find Limits…
Same Power on top and bottom(Horizontal Asymptote)
Lower power on top(Horizontal Asymptote)
Higher power on top(Oblique Asymptote)
x
x3
265
4322
2
xx
xx #32
2
22
3
xx
x
x
# 0
x
x
2
3 2
#
x
35
2
0
Type Equation Graph
Removable (Hole/Point)
Essential/Infinite(Vertical Asymptote)
Jump(Piecewise!)
Types of Discontinuities
)3(
)3)(2(
x
xx
)3(
4
x
lafjkld
gowo
blahblah
lklkasdfsadf
xf
;
#
;;'
)(
outcancelssomething
outcancelsnothing
Example 4: Determine the asymptotes and limits for
2
13
x
x
End behavior:
3)(lim
xf
x
3)(lim
xfx
)(lim2
xfx
)(lim2
xfx
Vertical asymptote x = 2
Horiz. Asymptote y = 3
Example 5:
Determine the asymptotes for 56
52
xx
x
Horiz. Asymptote y = 0
(x+5)(x+1)
End behavior:
0)(lim
xfx
0)(lim
xfx
)(lim5
xfx
)(lim5
xfx
)(lim1
xfx
)(lim1
xfx
Vertical asymptotes x = -5 and x = -1
Example 6:
Determine the asymptotes for 63
12142
x
xx
Horiz. Asymptote none
End behavior:
)(lim xfx
0)(lim
xfx
)(lim2
xfx
)(lim2
xfx
Vertical asymptotes x = -2
Watch-me!!!!65
22
xx
x 65
22
xx
x
What now…
1. FINISH QUIZ CORRECTIVES
2. PICK UP Horizontal Asym Worksheet.
3. Do 3.7 *Day 1 HW
Warm-Up Grab the matching sheet and fill out.
2,2 xx
2x2x )2)(2(
)2(
xx
x
0)(lim
xfxHorizontal asym = 0
)(lim)(lim22
xfxfxx
none)
2
1,0(Y-int: plug in x = 0
x-int: plug in y = 0You get an error…therefore…
1. Factor2. Domain Restr.3. Asym? Hole?4. Hor. Asym5. Intercepts6. Shifted/Graph7. Limits
4x4x
none 4
)2)(4(
x
xx
)(lim xfx
Horizontal asym. (there is none for this problem)
6)(lim6)(lim44
xfxfxx
)0,2()2,0(
Y-int: plug in x = 0
x-int: plug in y = 0
This is just a line ;)
10,0 xx
none
10,0 xx )10(
10
xx0)(lim
xf
x
Horizontal asym.
6)(lim6)(lim00
xfxfxx
errornone :
errornone :Y-int: plug in x = 0
x-int: plug in y = 0 6)(lim6)(lim1010
xfxfxx
Warm Up
Compare the graphs below. Include discussions of Critical points, extrema, increasing and decreasing intervals, holes, asymptotes, etc.
Also, write ALL the limits of the functions! ALL.
Warm UpCompare the graphs below. Include discussions
of Critical points, extrema, increasing and
decreasing intervals, holes, asymptotes, etc.Continuous Removable (Hole) 2 Essentials (V.A.’s)
This equation must have a domain restriction that cancels out…This equation must have a domain restriction that DOESN’T cancel out…
Warm Up
Compare the graphs below. Include discussions of Critical points, extrema, increasing and decreasing intervals, holes, asymptotes, etc.
Also, write ALL the limits of the functions! ALL.
)(lim xfX
)(lim xfX
)(lim xfX
)(lim xfX
5)(lim4
xfX
5)(lim4
xf
X
1)(lim
xfX
)(lim3
xfX
)(lim3
xfX
)(lim3
xfX
)(lim3
xfX
Section 3.8: Direct, Inverse, and Joint Variation
Objectives: Solve problems involving direct, inverse, and
joint variation.
Definitions
Definition – Direct Variation:
Definition – Constant of Variation:
When two variables are related to one another through the Multiplication of a constant (a number).
xyex 3: tyex4
1:
4:
qyex
The constant (number) from above.
(most of the time you will have to find it…)
Example 1:Suppose y varies directly as x and y = 45 when
x = 2.5 Find the constant of variation and write an
equation.
Use the equation to find the value of y when x = 4.
cxy )5.2(45 c 18c
xy 18
)4(18y 72
Example 2:When an object such as a car is accelerating, twice the distance (d) it travels varies
directly with the square of the time (t). One car accelerating for 4 minutes travels 1440 feet.
Write an equation of direct variation relating travel distance to time elapsed. Then sketch a graph of the equation.
Use the equation to find the distance traveled by the car in 8 minutes.
22 ctd 2)4()1440(2 c 180c 21802 td
290td
2)8(90d 5760d
Example 3:
If y varies directly as the square of x and y = 30, when x = 4, find x when y = 270.
2cxy 2)4(30 c
8
15c 2
8
15xy
2
8
15270 x 12x
Definitions
Definition – Inverse Proportion:When two variables are related to one another through division. There is still a constant of variation
x
cy Notice: the x is on the bottom!
Example 4:
If y varies inversely as x and y = 14, when x = 3, find x when y = 30.
x
cy
314
c 42c
xy
42
Definition – Joint Variation:
When more than two variables are related to one another through Multiplication….There is still a constant of variation
cxzy
Example 5: In physics, the work (W) done in charging a capacitor varies jointly as the charge
(q) and the voltage (V). Find the equation of joint variation if a capacitor with a charge of 0.004 coulomb and a voltage of 100 volts performs 0.20 joule of work.
cqVW )100)(004(.2. c qVW2
1
