Calculus Date: 2/6/2014 ID Check Objective: SWBAT solve related rates problemsDo Now: Students ChoiceW Requests: SM 117-119 #7-10In class: Turn in Text Book pg 254 #5, 7, 9, 11, 13, 15Text Book pg 254 #17, 19, 23, 29, 25 pg 232 #9, 17, 25

HW: prepare for quiz

Announcements: Saturday SessionsRm 241 10-12:50

Life Is Just A Minute

Life is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!

By Dr. Benjamin Elijah Mays, Past President of Morehouse College

Grading Take Home Test

• 2 points Primary Equation• 2 points Secondary Equation• 2 points Derivative• 2 points Solve for zero and • 1 point correct answer and units• 1 point check max or min

• Write problem corrections on a separate sheet of paper.

•PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) •and both the x- and y-axes. Find the dimensions of the triangle with the shortest •hypotenuse. Click HERE to see a detailed solution to problem 6.

•PROBLEM 7 : Find the point (x, y) on the graph of y= nearest the point (4, 0). Click HERE to see a detailed solution to problem 7.

•PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom• costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of •the most economical can.

•Click HERE to see a detailed solution to problem 8.

Calculus Optimization•PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

•PROBLEM 7 : Find the point (x, y) on the graph of y= nearest the point (4, 0).

•PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

Calculus Optimization•PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the •triangle with the shortest hypotenuse.

•PROBLEM 7 : Find the point (x, y) on the graph of y= nearest the point (4, 0).

•PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

Calculus Optimization•PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the •triangle with the shortest hypotenuse.

•PROBLEM 7 : Find the point (x, y) on the graph of y= nearest the point (4, 0).

•PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

•PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the •triangle with the shortest hypotenuse.

•PROBLEM 7 : Find the point (x, y) on the graph of y= nearest the point (4, 0).

•PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

Calculus Date: 1/30/2014 ID Check Objective: SWBAT solve optimization problemsDo Now: SM 116 #6HW Requests: SM 117-119 #1, 3, 5, 7-10

In class: Economic Optimizationhttps://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/maxmindirectory/MaxMin.html In Class: Go over problems from TestHW: Read Section 5.6 HandoutTest Corrections due Friday

Announcements: Saturday SessionsRm 241 11-1

PROBLEM 6 : Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

PROBLEM 7 : Find the point (x, y) on the graph of y=√ nearest the point (4, 0). 𝑥PROBLEM 8 : A cylindrical can is to hold 20 π m.3 The material for the top and bottom costs $10/m.2 and material for the side costs $8/m.2 Find the radius r and height h of the most economical can.

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?

We can minimize the material by minimizing the area.

22 2A r rh area of

endslateralarea

We need another equation that relates r and h:

2V r h

31 L 1000 cm21000 r h

2

1000h

r

22

10 02

02A r r

r

2 20002A r

r

2

20004A r

r

Example 5: What dimensions for a one liter cylindrical can will use the least amount of material?

22 2A r rh area of

endslateralarea

2V r h

31 L 1000 cm21000 r h

2

1000h

r

22

10 02

02A r r

r

2 20002A r

r

2

20004A r

r

2

20000 4 r

r

2

20004 r

r

32000 4 r

3500r

3500

r

5.42 cmr

2

1000

5.42h

10.83 cmh

To find the maximum (or minimum) value of a function:

1 Write it in terms of one variable.

2 Find the first derivative and set it equal to zero.

3 Check the end points if necessary.

If the end points could be the maximum or minimum, you have to check.

Notes:

If the function that you want to optimize has more than one variable, use substitution to rewrite the function.

If you are not sure that the extreme you’ve found is a maximum or a minimum, you have to check.

p

Consider all triangles formed by lines passing through the point (8/9, 3) and both the x- and y-axes. Find the dimensions of the triangle with the shortest hypotenuse.

MAXIMUM/MINIMUM PROBLEMS

Maximum/minimum optimization problems illustrate one of the most important applications of the first derivative.

Many students find these problems intimidating because they are "word" problems, and because there does not appear to be a pattern to these problems. If you are patient you can minimize your anxiety and maximize your success with these problems by following these guidelines :

GUIDELINES FOR SOLVING MAX./MIN. PROBLEMS 1. Read each problem slowly and carefully.

Read the problem at least three times before trying to solve it. Sometimes words can be ambiguous. It is imperative to know exactly what the problem is asking. If you misread the problem or hurry through it, you have NO chance of solving it correctly.

MAXIMUM/MINIMUM PROBLEMS

3. Define variables to be used and carefully label your picture or diagram with these variables.

This step is very important because it leads directly or indirectly to the creation of mathematical equations.

4. Write down all equations which are related to your problem or diagram. Clearly denote that equation which you are asked to maximize or minimize.

Experience will show you that MOST optimization problems will begin with two equations. One equation is a "constraint"

equation and the other is the "optimization" equation. The "constraint" equation is used to solve for one of the variables. This is then substituted into the "optimization" equation before differentiation occurs. Some problems may have NO constraint equation. Some problems may have two or more constraint equations. Our notes call the constraint equation the secondary equation and the optimization equation the primary equation.

MAXIMUM/MINIMUM PROBLEMS

5. Before differentiating, make sure that the optimization equation is a function of only one variable.

6. Differentiate equation and set equal to zero. Make sure your answer is in the domain of your function.

7. Verify that your result is a maximum or minimum value using the first or second derivative test for extrema or you can use graphing if allowed. If you have a closed interval remember for the max and min you must test the endpoints.

8. Be sure to write a sentence explaining clearly your results.

Refer to page 108 in your student manual.

Calculus Date: 1/16/2014 ID Check Objective: SWBAT solve optimization problems

HW Requests: NoneIn class: See overheadHW: Selected Problems SM

Announcements: Life Is Just A Minute

Life is just a minute—only sixty seconds in it.Forced upon you—can't refuse it.Didn't seek it—didn't choose it.But it's up to you to use it.You must suffer if you lose it.Give an account if you abuse it.Just a tiny, little minute,But eternity is in it!

By Dr. Benjamin Elijah Mays, Past President of Morehouse College