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5 4 3 2 1 -1 0 1 2 3 0 0 Calculus AB Assignment Exploring the Relationship Between the Derivative and the Antiderivative 1 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. NOTE : Area Functions and Area Function Notation The expression F ( x ) = ∫ x f ( t ) dt symbolizes a signed area function. This function evaluates the signed area under the curve between 0 and x . Note that on the left-hand side of the = , the variable is x . The t on the right-hand side is a "dummy variable" that represents all the values that x may take. You can call this dummy variable anything you like without changing f ( x ), AS LONG AS you don't call the dummy variable x again! For example, F ( x ) = ∫ x f ( t ) dt = ∫ x f ( u ) du = ∫ x f ( s ) ds . You should NOT write F ( x ) = ∫ x f ( x ) dx . 0 0 0 0 1 . Let y = f ( x ) = x 2 . Then consider the signed area function F ( x ) = ∫ x f ( 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: A. Using your graphing calculator to numerically approximate the definite integrals, complete the data chart below : x F ( x ) = ∫ x f ( t ) dt 0 1 1 2 4 3 9 4 16 Scoring (teacher will complete): Category Points Possible Points Earned Correct data entries 4 TOTAL 4 AP Calculus Assignment: Exploring the Relationship Between the Derivative and the Antiderivative TI-83 and TI-84 Version

Calculus AB Assignment Exploring the Relationship Between the Derivative and the Antiderivative 2 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. 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 x value, and the output is the area under the curve f ( x ) = x 2 . The function will be in the form ax b 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 filled in), and your calculator will try to find a function that matches those data points. On the TI-83, this is done by pressing STATlEDIT and then entering your data into the statistical register; x values go in L1, F ( x ) values go in L2. Then go STATlCALClA:PwrReg, which will find a function (in the form y = ax b ) 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 = 122.99999999 and b = — 12.00000001 , it's legitimate to just take a = 123 and b = — 12 . Your function approximating F ( x ) is : AP Calculus Assignment: Exploring the Relationship Between the Derivative and the Antiderivative TI-83 and TI-84 Version

0 0 Calculus AB Assignment Exploring the Relationship Between the Derivative and the Antiderivative 4 Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated. Correct relationship between F ( x ) and the antiderivative of f ( x ) 3 TOTAL 3 D. Using the relationship you just described above and taking f ( x ) = — 4 x 3 + 12 x 2 + 2 x , take a guess at an equation for the function F ( x ) = ∫ x f ( t ) dt , the signed area function for f . In other words, take a guess at an area function that will give the area under the curve f ( x ) = — 4 x 3 + 12 x 2 + 2 x . Your guess . F ( x ) = ∫ x f ( t ) dt = Scoring (teacher will complete): Category Points Possible Points Earned Your guess for F ( x ) 2 TOTAL 2