4. (10 pts) Let C : y = 3x -x2 be a parabola. (a) (2 pts) Find an equation of the tangent line to C at the point (a, 3a -a2) on C which is in the first quadrant. (b) (8 pts) Among all triangles in the first quadrant formed by the x-axis, the y-axis, and the tangent line above, what triangle has the smallest area? (a, 3a - a- ) y = 3x - x2Strategy for Solving Related Rate Problems 1. Understand the Problem - draw a picture or two ( draw a snapshot ) picture - read the problem several times 2. Identify - known, given by the problem - unknown , what you want to find 3. Equation create an equation relating the known and unknown 4. Differentiate differentiale the equation from 3 with respect to the desired variable (often time ) * using implicit differentiation 5. Solve/Interpret solve for the unknown rate interpret the answer by writing a complete sentence with appropriate units Formulac Object Volume Surface Area Sphere Cone Sarah=V SA= ars | ar , where s =length of side Cylinder V =P-A SA= 2ar | 2arh Rectangular Prism V = lah SA =2(wh | (w | th) Cube SA = 6x2 3 @2020 Kelli Karcher, Joseph WellsImplicit Differentiation Suppose we have an implicit equation of a curve, in terms of variables a and y, and we wish to find dy 1 . Differentiate both sides of the equation with respect to * . 2. when differentiatingy with respect to x we must use chain rule. 3. solve for dy dax using algebra Note your answer may have y in itStrategy for Solving Optimization Problems I. 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. Exl : we first had to understand what it meant for us to find the distance between 2 points. we needed the graph of y= Text and maybe the triangle of ly1 2. Deline appropriate notation. Define relevant variables needed Give appropriate equations, Give constraint equation and equation to be optimized Ex 1 Defined d, ( x ,y ) , constraint equation y= 12x+4 optimization equation : d= 7x2 + yz 3. Identify the function to be optimized. This includes identifying an appropriate domain. without a domain we cannot appropriately solve these problems Ex] 1 = 7x2+ 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 6x 1 d= 1 x2+ 2 x+4 5. Find the absolute extrema of our function. It is not enough to find local extrema P(x ) 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, 72 )