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# Lesson 22: Optimization II (Section 041 handout)

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Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few more good examples.

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### Lesson 22: Optimization II (Section 041 handout)

1. 1. Section 4.5 Optimization II V63.0121.041, Calculus I New York University November 24, 2010 Announcements No recitation this week Quiz 4 on §§4.1–4.4 next week in recitation Happy Thanksgiving! Announcements No recitation this week Quiz 4 on §§4.1–4.4 next week in recitation Happy Thanksgiving! V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 2 / 25 Objectives Given a problem requiring optimization, identify the objective functions, variables, and constraints. Solve optimization problems with calculus. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 3 / 25 Notes Notes Notes 1 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
2. 2. Outline Recall More examples Addition Distance Triangles Economics The Statue of Liberty V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 4 / 25 Checklist for optimization problems 1. Understand the Problem What is known? What is unknown? What are the conditions? 2. Draw a diagram 3. Introduce Notation 4. Express the “objective function” Q in terms of the other symbols 5. If Q is a function of more than one “decision variable”, use the given information to eliminate all but one of them. 6. Find the absolute maximum (or minimum, depending on the problem) of the function on its domain. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 5 / 25 Recall: The Closed Interval Method See Section 4.1 The Closed Interval Method To ﬁnd the extreme values of a function f on [a, b], we need to: Evaluate f at the endpoints a and b Evaluate f at the critical points x where either f (x) = 0 or f is not diﬀerentiable at x. The points with the largest function value are the global maximum points The points with the smallest/most negative function value are the global minimum points. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 6 / 25 Notes Notes Notes 2 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
3. 3. Recall: The First Derivative Test See Section 4.3 Theorem (The First Derivative Test) Let f be continuous on (a, b) and c a critical point of f in (a, b). If f changes from negative to positive at c, then c is a local minimum. If f changes from positive to negative at c, then c is a local maximum. If f does not change sign at c, then c is not a local extremum. Corollary If f < 0 for all x < c and f (x) > 0 for all x > c, then c is the global minimum of f on (a, b). If f < 0 for all x > c and f (x) > 0 for all x < c, then c is the global maximum of f on (a, b). V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 7 / 25 Recall: The Second Derivative Test See Section 4.3 Theorem (The Second Derivative Test) Let f , f , and f be continuous on [a, b]. Let c be in (a, b) with f (c) = 0. If f (c) < 0, then f (c) is a local maximum. If f (c) > 0, then f (c) is a local minimum. Warning If f (c) = 0, the second derivative test is inconclusive (this does not mean c is neither; we just don’t know yet). Corollary If f (c) = 0 and f (x) > 0 for all x, then c is the global minimum of f If f (c) = 0 and f (x) < 0 for all x, then c is the global maximum of fV63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 8 / 25 Which to use when? CIM 1DT 2DT Pro – no need for inequalities – gets global extrema automatically – works on non-closed, non-bounded intervals – only one derivative – works on non-closed, non-bounded intervals – no need for inequalities Con – only for closed bounded intervals – Uses inequalities – More work at boundary than CIM – More derivatives – less conclusive than 1DT – more work at boundary than CIM Use CIM if it applies: the domain is a closed, bounded interval If domain is not closed or not bounded, use 2DT if you like to take derivatives, or 1DT if you like to compare signs. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 9 / 25 Notes Notes Notes 3 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
4. 4. Outline Recall More examples Addition Distance Triangles Economics The Statue of Liberty V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 10 / 25 Addition with a constraint Example Find two positive numbers x and y with xy = 16 and x + y as small as possible. Solution V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 11 / 25 Distance Example Find the point P on the parabola y = x2 closest to the point (3, 0). V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 12 / 25 Notes Notes Notes 4 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
5. 5. Distance problem minimization step V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 13 / 25 Remark We’ve used each of the methods (CIM, 1DT, 2DT) so far. Notice how we argued that the critical points were absolute extremes even though 1DT and 2DT only tell you relative/local extremes. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 14 / 25 A problem with a triangle Example Find the rectangle of maximal area inscribed in a 3-4-5 right triangle with two sides on legs of the triangle and one vertex on the hypotenuse. Solution 3 4 5 y x V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 15 / 25 Notes Notes Notes 5 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
6. 6. Triangle Problem maximization step V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 16 / 25 An Economics problem Example Let r be the monthly rent per unit in an apartment building with 100 units. A survey reveals that all units can be rented when r = 900 and that one unit becomes vacant with each 10 increase in rent. Suppose the average monthly maintenance costs per occupied unit is \$100/month. What rent should be charged to maximize proﬁt? Solution V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 17 / 25 Economics Problem Finishing the model and maximizing V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 18 / 25 Notes Notes Notes 6 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
7. 7. The Statue of Liberty Example The Statue of Liberty stands on top of a pedestal which is on top of on old fort. The top of the pedestal is 47 m above ground level. The statue itself measures 46 m from the top of the pedestal to the tip of the torch. What distance should one stand away from the statue in order to maximize the view of the statue? That is, what distance will maximize the portion of the viewer’s vision taken up by the statue? V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 19 / 25 The Statue of Liberty Seting up the model Solution V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 20 / 25 The Statue of Liberty Finding the derivative V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 21 / 25 Notes Notes Notes 7 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
8. 8. The Statue of Liberty Finding the critical points V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 22 / 25 The Statue of Liberty Final answer V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 23 / 25 The Statue of Liberty Discussion The length b(a + b) is the geometric mean of the two distances measured from the ground—to the top of the pedestal (a) and the top of the statue (a + b). The geometric mean is of two numbers is always between them and greater than or equal to their average. V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 24 / 25 Notes Notes Notes 8 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010
9. 9. Summary Remember the checklist Ask yourself: what is the objective? Remember your geometry: similar triangles right triangles trigonometric functions V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 25 / 25 Notes Notes Notes 9 Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010