Calculus: Optimization Problems PDF

© 2017 by McGraw-Hill Education. Permission required for reproduction or display.  CHAPTER Applications of Differentiation 3 3.1 LINEAR APPROXIMATIONS AND NEWTON’S METHOD 3.2 INDETERMINATE FORMS AND l’HÔPITAL’S RU...

© 2017 by McGraw-Hill Education. Permission required for reproduction or display.  CHAPTER Applications of Differentiation 3 3.1 LINEAR APPROXIMATIONS AND NEWTON’S METHOD 3.2 INDETERMINATE FORMS AND l’HÔPITAL’S RULE 3.3 MAXIMUM AND MINIMUM VALUES 3.4 INCREASING AND DECREASING FUNCTIONS 3.5 CONCAVITY AND THE SECOND DERIVATIVE TEST 3.6 OVERVIEW OF CURVE SKETCHING 3.7 OPTIMIZATION 3.8 RELATED RATES Slide 2  3.7 OPTIMIZATION Preliminaries In this section, we bring the power of the calculus to bear on a number of applied problems involving finding a maximum or a minimum. We start by giving a few general guidelines. If there’s a picture to draw, draw it! Don’t try to visualize how things look in your head. Put a picture down on paper and label it. Determine what the variables are and how they are related. Decide what quantity needs to be maximized or minimized. Slide 3  3.7 OPTIMIZATION Preliminaries Write an expression for the quantity to be maximized or minimized in terms of only one variable. To do this, you may need to solve for any other variables in terms of this one variable. Determine the minimum and maximum allowable values (if any) of the variable you’re using. Solve the problem and be sure to answer the question that is asked. Slide 4  3.7 OPTIMIZATION EXAMPLE 7.1 Constructing a Rectangular Garden of Maximum Area You have 40 (linear) meters of fencing with which to enclose a rectangular space for a garden. Find the largest area that can be enclosed with this much fencing and the dimensions of the corresponding garden. Slide 5  3.7 OPTIMIZATION EXAMPLE 7.1 Constructing a Rectangular Garden of Maximum Area Solution We see the maximum value of the function Slide 6  3.7 OPTIMIZATION EXAMPLE 7.1 Constructing a Rectangular Garden of Maximum Area Solution Since x is a distance, we must have 0 ≤ x. Further, since the perimeter is 40, we must have x ≤ 20. The only critical number is x = 10 and this is in the interval under consideration. Slide 7  3.7 OPTIMIZATION EXAMPLE 7.1 Constructing a Rectangular Garden of Maximum Area Solution The maximum and minimum values of a continuous function on a closed and bounded interval must occur at either the endpoints or a critical number. maximum The rectangle of perimeter 40 m with maximum area is a square 10 m on a side. Slide 8  3.7 OPTIMIZATION EXAMPLE 7.3 Finding the Closest Point on a Parabola Find the point on the parabola y = 9 − x2 closest to the point (3, 9). Slide 9  3.7 OPTIMIZATION EXAMPLE 7.3 Finding the Closest Point on a Parabola Solution Slide 10  3.7 OPTIMIZATION EXAMPLE 7.3 Finding the Closest Point on a Parabola Solution Instead of minimizing d(x) directly, we minimize the square of d(x): Slide 11  3.7 OPTIMIZATION EXAMPLE 7.3 Finding the Closest Point on a Parabola Solution Recognize that x = 1 is a zero of f (x), which makes (x − 1) a factor. So, x = 1 is a critical number. In fact, it’s the only critical number, since (2x2 + 2x + 3) has no zeros. Now compare the value of f at the endpoints and the critical number. minimum. Slide 12  3.7 OPTIMIZATION EXAMPLE 7.3 Finding the Closest Point on a Parabola Solution Thus, the minimum value of f (x) is 5. This says that the minimum distance from the point (3, 9) to the parabola is and the closest point on the parabola is (1, 8). Slide 13  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction The state is building a new highway to link an existing bridge with a turnpike interchange, located 8 km to the east and 8 km to the south of the bridge. There is a 5-km-wide stretch of marshland adjacent to the bridge that must be crossed. The highway costs AED 10 million per km to build over the marsh and AED 7 million per km to build over dry land. Slide 14  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction How far to the east of the bridge should the highway be when it crosses out of the marsh? Slide 15  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction Solution Slide 16  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction Solution Slide 17  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction Solution Note that the only critical numbers are where C’(x) = 0. Slide 18  3.7 OPTIMIZATION EXAMPLE 7.6 Minimizing the Cost of Highway Construction Solution Compare the value of C(x) at the endpoints and at this one critical number: The highway should be about 3.56 km to the east of the bridge when it crosses over the marsh. Slide 19