1. Let y = f(x) = x2. Then consider the signed area function F(x) = ]of(t)dt, which you know represents the signed area below the curve from 0 to x. For example, F(2) represents the area below the curve y = f from 0 to 2, as show below: 4 3 2 -1 2A. Using your graphing calculator to numerically approximate the definite integrals, complete the data chart below: 15 F(x) = [3mm 43th Scoring (teacher will complete): Category Points Possible Points Earned Correct data entries 4 TOTAL 4 B. Now, you'll use your calculator to create an area function F(x). By \"area function,\" we mean a function where you enter an 1 value, and the output Is the area under the curvex) = 12. The function will be In the form rub and will represent the area function F(x). You can do this by doing a \"regression\" on your calculator; enter your data points (from the table you lled in), and your calculator will try to nd a function that matches those data points. 0n the TI-83, this is done by pressing STATIEDIT and then entering your data into the statistical register; a: values go in L1, Fm values go In L2. Then go STATICALCIA:Pereg, which will nd a function (In the form y = tab) that fits your data points. (See pages 12-27 in the TI-83 book If needed.) Remember that the calculator is approximating, so if it tells you that a = 12239999999 and b = 42000012001, it's legitimate to just take a = 123 and b = 12. Your function approximating x) ls: Scoring (teacher will complete): Category Pblnts Possible Points Earned Your equation for F(x) of the form orb 4 TOTAL 4 C. What is the relationship between your function approximating 17(1) and the antiderivative of x) = 12} how are they similar and how are they different? Soorlng (teacher will oomplete): Category Pblnts Possible Points Earned Correct relationship between ((1) and the antiderivative ofx) 3 TOTAL 3 D. Using the relationship you just described above and takingjx} = 4:3 + 19.1:2 +2x, take a guess at an equation for the function F(x) = I; 1min, the signed area function for f. In other words, take a uess at an area function that will give the area under the curve x) = 4x3 + 11x +2x. Your guess. For) = '5 3104': = Scoring (teacher will oomplete): Category mints Possible Points Earned Your guess for FOE) 2 TOTAL 2 E. Now, using your formula for F(x) (your guess in part D) and your calculator to numerically approximate the definite integrals, complete the table below, and discuss whether your formula for F(x) seems to work. x [of(t)di (calculator approximation) Your F(x) W N Scoring (teacher will complete) : Category Points Possible Points Earned Data - chart 3 Discussion UI N TOTAL F. Based on your work, can you draw any general conclusions concerning functions f(x) and signed area functions (definite integrals) [of()di and how they relate? Even if you can't, say something about what your work signifies. Scoring (teacher will complete) : Category Points Possible Points Earned Any thoughtful comment 2 TOTAL 2