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# 5.3 - x(b Calculate g x using FTC1(c Calculate g x by...

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5.3: The Fundamental Theorem of Calculus (Dated: September 23, 2011) Let f be integrable on [ a , b ]. Then it makes sense to de- ﬁne the function g ( x ) = Z x a f ( t ) dt , a x b . If f is positive, then g can be conceived of as the "area so far" function. If f changes sign, then g ( x ) is the net area under the graph of f from a to x . Example 1 (Exercise 5.3.3 in the text) Refer to the text. Answer: (a) g (0) = 0 , g (1) = 2 , g (2) = 5 , g (3) = 7 , and g (6) = 3 . (b) g is increasing on [0,3] . (c) g has a maximum at x = 3 . Theorem 1 (Fundamental Theorem of Calculus, Part 1 (FTC1)) Let f be continuous on [ a , b ] . Then g ( x ) = Z x a f ( t ) dt , a x b , is continuous on [ a , b ] and differentiable on ( a , b ) . Further- more, g 0 ( x ) = f ( x ). Note that the last equation can be written as d dx Z x a f ( t ) dt = f ( x ). Theorem 2 (Fundamental Theorem of Calcalus, Part 2 (FTC2)) If f is continuous on [ a , b ] , then Z b a f ( x ) dx = F ( b ) - F ( a ), where F is any antiderivative of f , i.e., a function such that F 0 = f . Note that the last equation can be written as Z b a F 0 ( x ) dx = F ( b ) - F ( a ). Notation: F ( b ) - F ( a ) = F ( x )] b a = F ( x ) | b a = [ F ( x )] b a . Example 2 (Exercise 5.3.5 in the text) Let g ( x ) = R x 1 t 2 dt. (a) Sketch the area represented by g
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Unformatted text preview: ( x ) . (b) Calculate g ( x ) using FTC1. (c) Calculate g ( x ) by evaluating the integral using FTC2 and then differentiating. Example 3 (Exercises 5.3.13, 5.3.15, 5.3.17, and 5.3.55 in the text) Calculate the derivatives using FTC1. 13: h ( x ) = R 1/ x 2 arctan t dt 15: y ( x ) = R tan x p t + p t dt 17: y ( x ) = R 1 1-3 x u 3 1 + u 2 du 55: y ( x ) = R x 3 p x p t sin t dt Example 4 (Exercises 5.3.19-5.3.42 in the text) Evaluate the integrals. 19: R 2-1 ( x 3-2 x ) dx 25: R 2 1 3 t 4 dt 26: R 2 π π cos θ d θ 30: R 2 ( y-1)(2 y + 1) dy 32: R π /4 sec θ tan θ d θ 33: R 2 1 (1 + 2 y ) 2 dy 39: R 1-1 e u + 1 du 40: R 2 1 4 + u 2 u 3 du 41: R π f ( x ) dx, where f ( x ) = ( sin x if ≤ x < π /2, cos x if π /2 ≤ x ≤ π ....
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