CALCULUS CHAPTER 4 NOTES SECTION 4-1 FINDING EXTREMA...

CALCULUS CHAPTER 4 NOTES

SECTION 4-1 FINDING EXTREMA (Day 1)

Absolute Maximums/Minimums: Also called _________________ extrema.

Absolute Max’s/Mins are always determined by the ____ coordinate. The _____________________ or _______________________ function value.

Find the absolute maximum and minimum values of the function:

𝒚 = 𝒙𝟐 + 𝟏 𝒇𝒐𝒓 [𝟎, 𝟐] max: ____ min ____ 𝒚 = 𝒙𝟐 + 𝟏 𝒇𝒐𝒓 (𝟎, 𝟐] max:____ min:____

𝒚 = 𝒙𝟐 + 𝟏 𝒇𝒐𝒓 [𝟎, 𝟐) max: ____ min ____ 𝒚 = 𝒙𝟐 + 𝟏 𝒇𝒐𝒓 (𝟎, 𝟐) max:____ min:____

Local Max/Mins: Also called ____________________ extrema.

Note: All extrema occur at the following places:

1. At ______________________. 2. Where the derivative equals ____. 3. Where the derivative is _____________________ (not differentiable).

Identify (label) all of the absolute and local extrema of the function below.

Critical Point:

Max/Mins can occur at CP’s, but a CP may not be a max or a min.

𝒚 = 𝟒 − 𝒙𝟐 𝒚 = |𝒙 − 𝟐| 𝒚 = 𝒙𝟐

𝟑⁄ 𝒚 = 𝒙𝟑

ASSIGNMENT: Page 184 #1 - 5, 7 – 10, 19, 21, 45 - 48


SECTION 4-2 The Mean Value Theorem (Day 1)

Mean Value Theorem:

2 Conditions that must be first met:

Then at least one point c exists where:

Other ways of saying the MVT:

Graphically:

Example: For the equation f(x) = ln(x – 1) on the interval [2, 4].

a. Show graphically how it is both continuous and differentiable on [2, 4].

b. Find c that satisfies the MVT.

Example: Find the equation of the secant line of the curve: 𝒚 = 𝒙 + 𝟏

𝒙 for . 𝟓 ≤ 𝒙 ≤ 𝟐.

Find the equation of the tangent line that is parallel to the secant line above.

ASSIGNMENT: Page 192 #15, 16, 20, 21, 39


SECTION 4-2 (Day 2) 1st and 2nd Derivative Test

1st Derivative Test:

When a graph is increasing…

When a graph is decreasing…

Find the extrema and the intervals of Increasing and decreasing for 𝒚 = 𝟐𝒙𝟑 − 𝟑𝒙𝟐 + 𝟑

Critical Points:

Intervals of Increasing/Decreasing:

2nd Derivative Test:

Find the extrema and the intervals of Increasing/Decreasing for 𝒚 = 𝒙

𝒙𝟐− 𝟒

ASSIGNMENT: Page 192 #1, 2, 8, 10, 11


SECTION 4-2 (Day 3) Antiderivatives

Finding an Antiderivative:

∫ 𝒙𝒏 =

Different terms for Antiderivative:

Take the derivative of the following: 𝒚 = 𝟑𝒙𝟐 − 𝟕 𝒚 = 𝟑𝒙𝟐 + 𝟓 𝒚 = 𝟑𝒙𝟐

What did you notice?

Constant of Integration:

Find f given the following derivatives.

𝒇′(𝒙) = 𝟒𝒙𝟑 𝒇′(𝒙) = 𝒆𝒙 𝒇′(𝒙) = 𝟑 𝒇′(𝒙) = 𝟏

𝒙

𝒇′(𝒙) = 𝒄𝒐𝒔 𝒙 𝒇′(𝒙) = 𝒔𝒆𝒄 𝒙 𝒕𝒂𝒏 𝒙 𝒇′(𝒙) = 𝟏

√𝟏− 𝒙𝟐

Find f if : 𝒇′(𝒙) = 𝟏

𝟒𝒙𝟑𝟒

and P(1, -2) lies on f.

Recall the relationship between position, velocity and acceleration:

Given the velocity of a particle described by 𝒗(𝒕) = 𝟑𝒕 + 𝟐. Find the position function if s(0) = 4.

ASSIGNMENT: Page 192 #25 – 31, 34, 35, 36, 37


SECTION 4-3 (Day 1) Inflection Points and Concavity

What Concavity means:

Inflection Points:

Determining Concavity: If 𝒇"(𝒙) > 0:

If 𝒇"(𝒙) < 0:

Example: Determine the inflection point(s) and the intervals for concavity for:

𝒚 = 𝟒𝒙𝟑 + 𝟐𝟏𝒙𝟐 + 𝟑𝟔𝒙 − 𝟐𝟎

Critical Points:

Inflection Point(s):

Intervals of Concavity:

Refer to the chart explaining the relationships between position, velocity and acceleration.

ASSIGNMENT: Page 203 – 204 #1 and 2 (b), 7 – 9 (c, d, f)


SECTION 4-3 (Day 2) Connecting f, f’, f”

Sketch the graph of f given the following:

*P

f”

f ’

Relationships:

f(x) f’(x) f”(x)

Critical Points

Increasing

Decreasing

Inflection Points

Concave Up

Concave Down

Using the information below, sketch the graph of f, an even function that is continuous [-3,3]:

x 0 1 2 (0,1) (1,2) (2,3) f 2 0 -1 + - -

f’ DNE 0 DNE - - + f” DNE 0 DNE + - -

ASSIGNMENT: Page 204 – 205 #31, 32, 33, 35, 41,


SECTION 4-4 (Day 1) Modeling and Optimization

What does it mean to optimize something?

5 Step Schema for Max/Min Problems:

1.

2.

3.

4.

5.

Example: A rectangular dog pen is constructed using a barn wall as one side and 60m of fencing for the other three sides. Find the dimensions of the pen that give the greatest area. What is the maximum area?

Barn wall

Open Box Problem: An open box is to be constructed from a square piece of cardboard 12 inches on each side by cutting a square corner and folding up the sides. Find the dimensions that maximize the volume. Find the maximum volume.

ASSIGNMENT: Page 214 – 215 #2, 3, 7, 9



Poster Design: You are designing a poster to contain 250 in2 of printing with a 5 inch margin on top and bottom and a 2 inch margin on each side. What overall dimensions will minimize the amount of paper used?

Example: A rectangle has its bas on the x-axis and its upper two vertices on the parabola y = 8 – x2. What is the largest area the rectangle can have and what are its dimensions?

y = 8 – x2.

ASSIGNMENT: Page 214 – 215 #6, 13, 14


SECTION 4-5 (Day 1) Linear Approximations and Newton’s Method

Linear Approximation: Also called _______________________________________

Used for:

y = f(x)

slope = f’(a)

(a, f(a))

a

Point-Slope Formula for the equation of a line:

Using the notation from the graph above:

Using L(x) instead of y:

So…a Linearization is just:

Example: Find the linearization of 𝒚 = √𝟏 + 𝒙 for all x near 0. Then find an approximation for f(.2).

Newton’s Method:

Newton’s Method always starts with a guess (called x0)

y = f(x)

.

actual root x1 x0

Example: Use Newton’s Method to approximate the root of 𝒇(𝒙) = 𝒙𝟑 + 𝟑𝒙 + 𝟏.

Use 𝒙𝟎 = 𝟏

𝟐 as your first guess.

ASSIGNMENT: Page 229 #1 – 4 (Explain), #15 (use x0 = ½), #16 (Use x0 = 1 and x0 = -2), #18 (Use x0 = -1 and x0 = 1)


SECTION 4-5 (Day 2) Differentials

Writing equations in differential form (separating variables):

Find dy:

𝒚 = 𝒙𝟑 + 𝟓𝒙 𝒚 = 𝒙𝟐 𝒔𝒊𝒏 𝟐𝒙 𝒚 = 𝒕𝒂𝒏−𝟏(𝒙𝟑)

The Mongolian Barbeque: The radius of the steel plate at the Mongolian Barbeque increases from 5 to 5. 04 feet. Estimate using differentials, the change in the resulting area.

ASSIGNMENT: Page 229 – 230 #19, 21, 22, 25, 38, 39


SECTION 4-6 (Day 1) Related Rates

What is a Rate? A derivative w/r to ________________.

What are Related Rates? How one rate ______________________ in relation with how another rate ________________________.

The Gulf Oil Spill: Assume an oil spill from a ruptured oil rig spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius is 1000 ft? Include units in your answer.

Formula:

r = dr/dt =

Falling Ladder Problem: A 10 foot ladder, leaning against a wall slips so that its base moves away from the wall at a rate of 2 ft/sec. How fast will the top of the ladder be moving down the wall when the base is 8 ft from the wall? Include units in your answer.

Formula:

dx/dt:

Tank Rates: How rapidly (at what rate) will the fluid inside a vertical cylindrical tank with a radius of 10 m drop if we pump the fluid out at a rate of 300 m3/min? Include units in your answer.

Volume of a cylinder: 𝑽 = 𝝅𝒓𝟐𝒉

dV/dt =

ASSIGNMENT: Page 237 #1 – 3, 8, 9a, 13, 14, 20, 21a

CHAPTER FOUR ASSIGNMENTS

SECTION 4-1 FINDING EXTREMA (Day 1)

ASSIGNMENT: Page 184 #1 - 5, 7 – 10, 19, 21, 45 - 48

SECTION 4-2 The Mean Value Theorem (Day 1)

ASSIGNMENT: Page 192 #15, 16, 20, 21, 39

SECTION 4-2 (Day 2) 1st and 2nd Derivative Test

ASSIGNMENT: Page 192 #1, 2, 8, 10, 11

SECTION 4-2 (Day 3) Antiderivatives

ASSIGNMENT: Page 192 #25 – 31, 34, 35, 36, 37

SECTION 4-3 (Day 1) Inflection Points and Concavity

ASSIGNMENT: Page 203 – 204 #1 and 2 (b), 7 – 9 (c, d, f)

SECTION 4-3 (Day 2) Connecting f, f’, f”

ASSIGNMENT: Page 204 – 205 #31, 32, 33, 35, 41,

CHAPTER FOUR REVIEW SHEET

CHAPTER FOUR QUIZ


ASSIGNMENT: Page 214 – 215 #2, 3, 7, 9


ASSIGNMENT: Page 214 – 215 #6, 13, 14

SECTION 4-5 (Day 1) Linear Approximations and Newton’s Method

ASSIGNMENT: Page 229 #1 – 4 (Explain), #15 (use x0 = ½), #16 (Use x0 = 1 and x0 = -2), #18 (Use x0 = -1 and x0 = 1)

SECTION 4-5 (Day 2) Differentials

ASSIGNMENT: Page 229 – 230 #19, 21, 22, 25, 38, 39

SECTION 4-6 (Day 1) Related Rates

ASSIGNMENT: Page 237 #1 – 3, 8, 9a, 13, 14, 20, 21a

CHAPTER FOUR REVIEW WORKSHEET #2

CHAPTER FOUR QUIZ #2

Date post:	30-Jan-2021
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

CALCULUS CHAPTER 4 NOTES SECTION 4-1 FINDING EXTREMA...

Documents