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13 Nov 2019
Math 1510 Preview A.14: Analysis Objective There are many situations where functions arise as area functions for some continuous integrand, that is F(x)- 0) d In this activity you will use the tools of calculus to analyze such a function. Background If the integrand(r) has an elementary antiderivative then the second part of the Fundamentall Theorenm of Calculus gives a formula for the area function F(x). However, many functions do not have an elementary antiderivative. In such a situation we can use the first part of the Fundamental Theorem to get information about F through its derivative. You should review section 5.3 and make sure you understand both parts of the Fundamental Theorem. You will also want to use a CAS (for example Wolfram Alpha) for some of your computations. CAUTION: Some systems (including Alpha) will report an antiderivative for the integral below but they are not useful in this problem because they incorporate assumptions that are not valid here. For calculating numeric estimates of area using the midpoint rule you may use the Desmos example The Midpoint Rule Steps: You will analyze the function sin ()+1 1. Domains. Describe the domains of the area function F(x) and the integrand s). 2. Values. Use the midpoint rule to estimate the following values: 6 F(x) 3. Symmetry. Does F(x) have any properties such as symmetry (even or odd) or periodicity? Page 1 of 2
Math 1510 Preview A.14: Analysis Objective There are many situations where functions arise as area functions for some continuous integrand, that is F(x)- 0) d In this activity you will use the tools of calculus to analyze such a function. Background If the integrand(r) has an elementary antiderivative then the second part of the Fundamentall Theorenm of Calculus gives a formula for the area function F(x). However, many functions do not have an elementary antiderivative. In such a situation we can use the first part of the Fundamental Theorem to get information about F through its derivative. You should review section 5.3 and make sure you understand both parts of the Fundamental Theorem. You will also want to use a CAS (for example Wolfram Alpha) for some of your computations. CAUTION: Some systems (including Alpha) will report an antiderivative for the integral below but they are not useful in this problem because they incorporate assumptions that are not valid here. For calculating numeric estimates of area using the midpoint rule you may use the Desmos example The Midpoint Rule Steps: You will analyze the function sin ()+1 1. Domains. Describe the domains of the area function F(x) and the integrand s). 2. Values. Use the midpoint rule to estimate the following values: 6 F(x) 3. Symmetry. Does F(x) have any properties such as symmetry (even or odd) or periodicity? Page 1 of 2