# Activities

• • • ##### Subject Area

• Math: Calculus: Derivatives

• ##### Author 9-12

45 Minutes

Derive™ 6

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Exploring the Fundamental Theorem of Calculus

#### Activity Overview

In this Derive™ activity, students investigate the Fundamental Theorem of Calculus and explore examples of Riemann Sums for approximating the Definite Integral: the Midpoint Sum, the Left Hand Endpoint Sum, the Right Hand Endpoint Sum, The Trapezoidal Sum, and Simpson's Approximating Sum.

#### Before the Activity

• See the attached DFW file for detailed instructions for this activity
• Print pages from the attached DFW file for your class
• #### During the Activity

Distribute the pages to the class.

• Set up a 2D plot window
• Construct the expressions for the Midpoint Sum and other Riemann Sums, in stages

• Plot a polynomial function
• Observe an expression whose graph is a sequence of rectangles over an interval having the value of the function at the midpoint of each interval as the height of the rectangle above that interval
• Graph the result
• Evaluate the area lying between the resulting graph and the x-axis
• Realize that if a function is positive over an interval, the Midpoint Sum is the area of a sequence of rectangles (constructed as illustrated in the graph)
• Notice that the area of the rectangles is almost the same as the area between the graph and the x-axis
• Understand that the definite integral of a function over an interval is the difference of the area of the region lying above the x-axis and below the graph of the function, and the area of the region lying below the x-axis and above the graph of the function

• Use the Derive software to compute the value of an integral over an interval, and compare the result with the value calculated with the help of the Midpoint Sum formula

• Graph a polynomial function
• Set the lower value of the interval as a constant and let the upper value be a variable
• Use the formula for the Midpoint Sum to approximate the integral
• Record the action of the Midpoint Sum when the value of the function is zero, negative, positive, minimum, and maximum

• Understand that as the limit of the maximum width of the subintervals making up the partition goes to zero, the variable sum will have the function as its derivative
• #### After the Activity

Students answer questions listed on the activity sheet.

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary