3.7: Optimization Homework: p.220 17, 19, 21, 29, 33, 47 Standards EK2.3C3 – The derivative can be used to solve optimization problems, that is, finding a maximum or minimum value of a function over a given interval Learning Objectives: Solve applied minimum and maximum problems

Does 𝑽= 𝒔 𝟑 or does 𝑽= 𝒔 𝟐 ⋅𝒉 or does 𝑽=𝒍⋅𝒘⋅𝒉? Concept 1: Formulas Does 𝑽= 𝒔 𝟑 or does 𝑽= 𝒔 𝟐 ⋅𝒉 or does 𝑽=𝒍⋅𝒘⋅𝒉?

The Equation/Formula that is to be used in the optimization Concept 1: Vocabulary Primary Equation The Equation/Formula that is to be used in the optimization

Teacher Example 1: Finding Maximum Volume An open top box with a square base needs to have a surface area of 108 cubic inches. How large would the sides have to be to maximize the volume?

Teacher Example 1: Finding Maximum Volume Step 1: Volume Formula Step 2: Surface Area Step 3: Single Variable Conversion Step 4: Feasible Region Step 5: Maximize Step 6: Solve for the variable Step 7: Evaluate

Teacher Example 1: Finding Maximum Volume

Concept 1 (Continued): Guidelines for Solving Optimization Problems On page 216, bottom of the page, enter the appropriate information into your notes. 2 minutes + 10 minutes for SLE

Student Led Example 1: Finding Maximum Volume Handout: Problem 1

Student Led Example 1: Handout – Problem 1 This image is here to ASSIST and has nothing to do with the actual problem…they are, however, ridiculously similar.

Student Led Example 1: Finding Maximum Volume

Student Led Example 1: Finding Maximum Volume 𝑽= 𝟑𝟎−𝟐𝒙 𝟏𝟒−𝟐𝒙 ⋅𝒙 𝑽 ′ =𝟏𝟐 𝒙 𝟐 −𝟏𝟕𝟔𝒙+𝟒𝟐𝟎 =𝟒(𝟑𝒙−𝟑𝟓)(𝒙−𝟑) 10 Minutes

Student Led Example 1: Finding Maximum Volume 𝟎≤𝒙≤𝟕 Feasible Region: Critical Points: 𝒙=𝟑

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? What quantity needs to be minimized? Or, in other words, what is the PRIMARY EQUATION?

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? 𝒅= 𝒙−𝟎 𝟐 + 𝒚−𝟐 𝟐 𝒚=𝟒− 𝒙 𝟐 𝒅= 𝒙 𝟐 + 𝟒− 𝒙 𝟐 −𝟐 𝟐 𝒅= 𝒙 𝟒 −𝟑 𝒙 𝟐 +𝟒

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? 𝒅 is smallest when the radicand is smallest. 𝒙 𝟒 −𝟑 𝒙 𝟐 +𝟒 ′ =𝟒 𝒙 𝟑 −𝟔𝒙 =𝟐𝒙 𝒙 𝟐 −𝟑 Critical values occur at 𝒙= 𝟎,± 𝟑 𝟐

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ?

Teacher Example 2: Minimum Distance Which points on the graph 𝒚=𝟒− 𝒙 𝟐 are closest to the point 𝟎,𝟐 ? 𝒇 𝟑 𝟐 = 𝟓 𝟐 𝒇 − 𝟑 𝟐 = 𝟓 𝟐

Student Led Example 2: Finding Minimum Distance Handout: Problem 2

Student Led Example 2: Finding Minimum Distance Function to be minimized: 𝒅= 𝒙−𝟕 𝟐 + 𝒙 𝟐 Critical Points: 𝒙= 𝟏𝟑 𝟐 Feasible Region: −∞<𝒙<∞

Teacher Example 3: Finding Minimum Length Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?

Teacher Example 3: Finding Minimum Length Pictures! 

Teacher Example 3: Finding Minimum Length Function to be Minimized: 𝑾=𝒚+𝒛

Teacher Example 3: Finding Minimum Length Function to be Minimized: 𝑾=𝒚+𝒛 𝒚= 𝒙 𝟐 +𝟏𝟒𝟒 𝒛= 𝟑𝟎−𝒙 𝟐 + 𝟐 𝟖 𝟐 = 𝒙 𝟐 −𝟔𝟎𝒙+𝟏𝟔𝟖𝟒

Teacher Example 3: Finding Minimum Length Function to be Minimized: 𝑾=𝒚+𝒛 𝑊 ′ = 𝑦 ′ +𝑧′ 𝑧 ′ = 𝑥−30 𝑥 2 −60𝑥+1684 𝑦 ′ = 𝑥 𝑥 2 +144 𝑑𝑊 𝑑𝑥 = 𝑥 𝑥 2 +144 + 𝑥−30 𝑥 2 −60𝑥+1684

Teacher Example 3: Finding Minimum Length Function to be Minimized: 𝑾=𝒚+𝒛 𝑥 𝑥 2 +144 + 𝑥−30 𝑥 2 −60𝑥+1684 =0 After some ridiculous algebra we get… 𝑥={9,−22.5}

Teacher Example 3: Finding Minimum Length Function to be Minimized: 𝑾=𝒚+𝒛 𝑥= 9,−22.5 𝑊 𝑥 =𝑦 𝑥 +𝑧 𝑥 𝑦 9 = 9 2 +144 = 81+144 = 225 =15 𝑧 𝑥 = 9 2 −60 9 +1684 = 81−540+1684 = 1225 =35 𝑊 9 =50

Student Led Example 3: Finding Minimum Length Handout: Problem 3

Exit Task – Complete if not done Enter into your notes: “Guidelines for Solving Applied Minimum and Maximum problems” on page 216