SCAN0065

# SCAN0065 - 7 Use the Fundamental Theorem of Calculus to...

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Unformatted text preview: 7. Use the Fundamental Theorem of Calculus to show that 71/2 I cos(x) dx = 1 0 (without using your calculator). 8. Explain how you can tell, just by looking at the graph ofy = cos(x), that I: cos(x)dx = 0. 9. Verify that j"cos(x)dx = 0 0 by using the FTC (without using your calculator). 10. Based on your work in the past few problems, explain you know that there is some number, x1, between 7r/2 and it, such that j cos(x)dx = 0.686. 0 (Hint: You don't need to do any calculations. Just reflect on your results from the previous few problems.) 11. Are there are two different numbers, x1 and x2, each between 7r/2 and 7r such that I“ cos(x)dx = 0.686 and I“ cos(x)dx = 0.686? 0 0 Explain your answer. 12. Draw the graph of a function,ﬁ such that jjﬂxmxzjfﬁxwx. Indefinite Integrals The term “indeﬁnite integral” is often used to denote the set of all antiderivatives of a function,f, on some interval (a,b). The reason for the use of this term is that integration is closely related to antidifferentiation (by the FTC). We use the notation [(me ﬂx) dx to refer to the indefinite integral of the functionfon the interval (a,b) — meaning the set of all functions that are antiderivatives offon the interval (a,b). The Mean Value Theorem can be used to prove that if a function, F, is an antiderivative of the function f on a certain interval ((1,1)), then any antiderivative, G, of f on (a,b) must be of the form G(x) = F(x) + C where C can be any constant. Thus, for example, since we know that the function F(x) = %x2 is an antiderivative of the function ﬂx) : x on the interval (—oo,oo), then we know that any antiderivative, G, of f must be of the form G(x) = '72? + C (where C can be any constant). We describe the indefinite integral off as I xdx = -1—x2 + C. (wow) 2 In most cases, it is not necessary to include the interval ((1,1)) in the notation for the 14 ...
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