Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity
Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums)...
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Transcript of Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums)...
![Page 1: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/1.jpg)
Section 3.6: Critical Points and Extrema
Objectives: I can find the extrema (maximums and
minimums) of a function.
![Page 2: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/2.jpg)
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)
![Page 3: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/3.jpg)
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)
![Page 4: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/4.jpg)
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)
![Page 5: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/5.jpg)
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}
![Page 6: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/6.jpg)
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
![Page 7: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/7.jpg)
![Page 8: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/8.jpg)
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.
![Page 9: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/9.jpg)
_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
![Page 10: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/10.jpg)
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.
![Page 11: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/11.jpg)
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
![Page 12: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/12.jpg)
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)
![Page 13: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/13.jpg)
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
![Page 14: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/14.jpg)
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
![Page 15: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/15.jpg)
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
![Page 16: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/16.jpg)
![Page 17: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/17.jpg)
Section 3.7: Graphs of Rational Functions (Day 1)
Objectives: Graph rational functions. Determine vertical, horizontal, and oblique
asymptotes.
![Page 18: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/18.jpg)
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
![Page 19: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/19.jpg)
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)
![Page 20: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/20.jpg)
Definition – Vertical Asymptote:
Essential (Infinite) Discontinuity
An asymptote in the vertical direction
A vertical asymptote ;)- Found from the denominators domain restrictions
![Page 21: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/21.jpg)
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”
![Page 22: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/22.jpg)
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”
![Page 23: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/23.jpg)
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?
![Page 24: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/24.jpg)
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…
![Page 25: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/25.jpg)
Revisit Example 3(from 3.6):
Determine whether the function is continuous at x = -2.
2
4)(
2
x
xxf
![Page 26: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/26.jpg)
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
![Page 27: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/27.jpg)
Warm-up: Match up the Function, its graph, and the type
of discontinuity
![Page 28: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/28.jpg)
Foldable
![Page 29: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/29.jpg)
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
![Page 30: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/30.jpg)
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
![Page 31: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/31.jpg)
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
![Page 32: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/32.jpg)
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
![Page 33: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/33.jpg)
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
![Page 34: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/34.jpg)
Watch-me!!!!65
22
xx
x 65
22
xx
x
![Page 35: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/35.jpg)
What now…
1. FINISH QUIZ CORRECTIVES
2. PICK UP Horizontal Asym Worksheet.
3. Do 3.7 *Day 1 HW
![Page 36: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/36.jpg)
Warm-Up Grab the matching sheet and fill out.
![Page 37: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/37.jpg)
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
![Page 38: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/38.jpg)
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 ;)
![Page 39: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/39.jpg)
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
![Page 40: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/40.jpg)
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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.
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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…
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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
![Page 44: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/44.jpg)
Section 3.8: Direct, Inverse, and Joint Variation
Objectives: Solve problems involving direct, inverse, and
joint variation.
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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…)
![Page 46: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/46.jpg)
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
![Page 47: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/47.jpg)
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
![Page 48: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/48.jpg)
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
![Page 49: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/49.jpg)
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!
![Page 50: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/50.jpg)
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
![Page 51: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/51.jpg)
Definition – Joint Variation:
When more than two variables are related to one another through Multiplication….There is still a constant of variation
cxzy
![Page 52: Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums) of a function. I can find the extrema (maximums and.](https://reader031.fdocuments.in/reader031/viewer/2022013004/56649dba5503460f94aab17e/html5/thumbnails/52.jpg)
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