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1 For the following problems. consider the function f(x} = 5x2 -- 2X l- 4 over the interval [0. 5]. The graph is given below. Our goal is to find the area under this curve over the interval [0.6]. We will do this in a Few stages. In Problem 1. we will make an estimate using a small number of rectangles. in Problem 2. we will get a better estimate by using a large number of rectangles. In Problem 3. we will find a formula for an approximation using a variable number of rectangles. And in Problem 4. we use a limit to find the exact area under the curve. Remember to show your lwrork on each part of this homework. Even if the calculation is simple. your work must show how you got your answer. including any formulas you used. 12 H 7 Problem 1: Approximating Area with Three Rectangles {15 points] In this problem. we want to approximate the area under the graph of y = {(x}. above the x-axis. over the interval [0. 6] by using in = 3 rectangles and right endpoints. a} (2 points} What is the width. x. of each rectangle used in this approximation? Ax .. b} (2 points} List the grid points. x0. x1. x2. x3. used in this approximation. Grid points: c} (3 points} List the sample points. x-f. x.X'. used in this approximation. (Remember that we are using right endpoints} Sample points: d) (3 points} Illustrate this area approximation by sketching the corresponding rectangles in the graph above. e} (5 points} Calculate this area approximation. Area Approximation: Problem 3: Finding a Formula for an Approximation with a Variable Number of Rectangles {13 points) If you want to evaluate many different approximations of the same function. you can leave the number of rectangles as a variable n. The process of finding the approximation is the same as abOve. but the final answer will have the n variable left. This problem will walk you through this process. a} (2 points} Find a formula for the width. Ax. of each rectangle used in this approximation. x: b) (3 points} Find a formula for the grid point xk. (The only variables appearing in your expression should be k and n.) X 2- | c} (2 points} Find a formula for the sample point Kg. (The only variables appearing in your expression should be it and n. Remember that we are using right endpoints.) r> Dc: of the Riemann sum found in Problem 3(f} above. (Your final answer should have no variables left in it. Don't forget to check the form and show your work. You may use I'Hopital's Rule.) Limit Value: c} (2 points} Explain what the value of this limit from part (b) represents in the context of this problem. Be as specific as you can (i.e. do not say "the function". instead give the name of the function.)