2. (12 pts) A line goes through the origin and a point on the curve y = xzeBx, for x 2 0. Find the maximum slope of such a line. At what x-value does it occur? Strategy for Solving Optimization Problems 1. Understand the problem. Read the problem multiple times. understand what is given and what we need to find. This may include drawing a picture when appropriate. Ex1 : we first had to understand what it meant for us to find the distance between 2 points. we needed the Day] graph of y= Text and maybe the triangle ly1 2. Define appropriate notation. (8,07 Define relevant variables needed Give appropriate equations, Give constraint equation and equation to be optimized Ex 1 Defined d , ( x ,y ) , constraint equation y = 72*+4 optimization equation : d= 7x 2+ 92 3. Identify the function to be optimized. This includes identifying an appropriate domain. without a domain we cannot appropriately solve these problems Ex 1 1 = 7 x 2 + 92 420, x 2 - 2 domain 4. Represent the function to be optimized as a single variable equation. we only know how to find absolute extrema of single variable equations, Therefore, we use our constraint equation to write our optimization equation in terms of I vanable Gx 1 d= 7/ x2+2 x+4 5. Find the absolute extrema of our function. It is not enough to find local extrema P(c) Notice Phas a local max profit at x= 2 weeks +> x butan absolute max profit at x= 12 weeks. 12 soifwe want to find when we have max profit, we get *= 12 weeks . 6. Make sure you answer the question given. The question may ask for cost, area, dimensions. write a sentence answering the specific question asked Ex ] we had to provide the coordinate , (x,y) = (-1,12)Newton's Method Step 1 Guess an initial value, an, so that f(a ) is very close to 0. we want to pick x, so that it is relatively close to the zero we want to approximate Step 2 Use the tangent line approximation at a = an to determine the zero of the tangent line to the graph at = = 31, = =12. Note the explicit formula for doing so is f(CI) 12 = CI f(II) Note: We require f'(:1) 0. O we cannot divicle by o & the slope of tangent line is O, so it will not Step 3 Repeat Step 2 using 22 to find 23, then a3 to find CA, ..., then ,, to find a. . convergo Keep repeating until you achieve the desired level of precision. The formula for the (n + 1) approximation is given by we can see when n= 1 f(:,) we get f'(En) (f'(:.) # 0). f ( x1 ) X2 = XI f'(x1 )