1. Optimization Problems

2. A farmer has 2400 ft of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? 1.Identify the quantity that is to be maximized or minimized, Q. Maximize area: A 2.Write down a formula or expression for the quantity you need to optimize, Q. A = l ∙ w 3.If necessary, rewrite your expression in terms of just one variable. 4.Differentiate your expression for Q with respect to the other variable.5.Analyze the problem carefully to identify the absolute maximum or minimum of Q. For a rectangle: A = length x width l w Using all the fencing: l + 2w =2400 So: l =2400 − 2w A = (2400 − 2w) ∙ w Rewrite: A = 2400w − 2w 2 First option: a sign analysis: A’ 600 +− w = 600 ft yields an absolute maximum of the area because A’ > 0 when 0 600 Second option: comparing critical numbers and endpoints: Common sense: 0 < w < 1200 w06001200 A07200000 Again, w = 600 ft yields an absolute maximum of the area. Also: l = 1200 ft. 0 (

3. Third option: second derivative test: Notice that A’= 2400 − 4w = 0 has only one solution: w = 6 Since A” = −4 < 0, the graph of A is always concave down and there is only one point with a horizontal tangent line, at w = 6. That point MUST be an absolute maximum.

4. Find the point on the parabola y 2 = 2x that is closest to the point (1, 4). 1.Identify the quantity that is to be maximized or minimized, Q. Minimize distance: p 2.Write down a formula or expression for the quantity you need to optimize, Q. 3.If necessary, rewrite your expression in terms of just one variable. 4.Differentiate your expression for Q with respect to the other variable. Distance formula: Using those points: The point (x, y) is on the graph of the parabola: y 2 = 2x. So x = y 2 /2 Discuss extrema for radicals… (x, y)

5. First option: sign analysis: 2 −+ y = 2 yields an absolute minimum of the distance because i’ 0 when y > 2. This means that the distance always decreases for y 2 5.Analyze the problem carefully to identify the absolute maximum or minimum of Q. (x, y) Find the point on the parabola y 2 = 2x that is closest to the point (1, 4). The point closest point on the graph of y 2 = 2x to (1, 4) is (2, 2). Second option: second derivative test: