Section 3.6: Critical Points and Extrema Objectives: I can find the extrema (maximums and minimums)...

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