lecture29hout

lecture29hout - MATH 006 Calculus and Linear Algebra...

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Unformatted text preview: MATH 006 Calculus and Linear Algebra (Lecture 29) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 14 Outline 1 Optimization Albert Ku (HKUST) MATH 006 2 / 14 Optimization Optimization Problems Strategy for solving optimization problems: Step 1: Introduce variables, look for relationships among the variables, and construct a mathematical model of the form Maximize (or minimize) f ( x ) on the interval I . Step 2: Find the critical value(s) of f ( x ). Step 3: Find the absolute maximum (or minimize) value of f ( x ) on the interval I and the value(s) of x where this occurs. Step 4: Use the solution to the mathematical model to answer all the questions asked in the problem. Albert Ku (HKUST) MATH 006 3 / 14 Optimization Example 1 Example A homeowner has $320 to spend on building a fence around a rectangular garden. Three sides of the fence will be constructed with wire fencing at a cost of $2 per linear foot. The fourth side is to be constructed with wood fencing at a cost of $6 per linear foot. Find the dimensions and the area of the largest garden that can be enclosed with $320 worth of fencing. Solution Let x and y be the lengths of the fence as shown in the following figure: Albert Ku (HKUST) MATH 006 4 / 14 Optimization Solution (cont’d) Since the total cost of the fence C is $320, we have C = 2 y + 2 x + 2 y + 6 x = 8 x + 4 y = 320 ⇒ y = 80- 2 x Now, we need to maximize the area of the garden, denoted by...
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lecture29hout - MATH 006 Calculus and Linear Algebra...

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