Calculus Date: 2/6/2014 ID Check Objective: SWBAT solve related rates problems Do Now: Students Choice W Requests: SM #7-10 In class: Turn in Text Book pg 254 #5, 7, 9, 11, 13, 15 Text Book pg 254 #17, 19, 23, 29, 25 pg 232 #9, 17, 25 HW: prepare for quiz Announcements: Saturday Sessions Rm :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.

Calculus Date: 1/30/2014 ID Check Objective: SWBAT solve optimization problems Do Now: SM 116 #6 HW Requests: SM #1, 3, 5, 7-10 In class: Economic Optimization neDIRECTORY/maxmindirectory/MaxMin.html In Class: Go over problems from Test HW: Read Section 5.6 Handout Test Corrections due Friday Announcements: Saturday Sessions Rm 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.

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. area of ends lateral area We need another equation that relates r and h :

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

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. 

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: None In class: See overhead HW: 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