5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship


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1 5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is continuous on the intervl [, b], n f(t) = F (t) (f is the erivtive of F ), then b f(t)t = b F (t)t = F (b) F () In other wors, the integrl of the erivtive gives the chnge in vlues of the originl function! We hve lrey seen this ie when we foun the re uner the curve of velocity. Properties of the Integrl: For the following, ssume ll functions re continuous (so tht the efinite integrl exists). (1) b (f(x) + g(x)) x = b + b (2) b c = c b for ny constnt c. (3) b = b (4) = 0 (5) b + c b = c (6) If f(x) is n o function, then = 0 (7) If f(x) is n even function, then = 2 (8) If f(x) 0, then b 0 (similr for f(x) 0) (9) If m f(x) M, tht is f(x) is boune bove n below, then m(b ) b M(b ) (Like Riemnn sums with one rectngle). (10) If f(x) g(x), then b b (11) The verge vlue of function is efine by Averge Vlue = 1 b 0 b This is the height of rectngle whose re is equl to the re uner the curve of f(x).
2 (12) The re between two curves is the ifference in their res (ssume f(x) g(x)): Are between f(x) n g(x) = b (f(x) g(x)) x (13) Horizontl Shift property: b = b+h f(x h) x +h (14) Horizontl Stretch property: b = 1 2 2b f(x/2) x (similr for other stretches/compressions). 2 NOT Properties The following re NOT properties of integrls (except in very specific cses), so o NOT use them!!! (1) b f(x) b b (2) b f(x) b b (3) b = 0 oes NOT men f(x) = 0 (the opposite is true, however: b 0 x = 0) (4) b = b oes NOT men f(x) = g(x) (the opposite is gin true). Definition. If F (x) = f(x), then we sy tht F (x) is n ntierivtive of f(x). Exmple. An ntierivtive of f(x) = 2x is F (x) = x 2. Exmple. Another ntierivtive of f(x) = 2x is F (x) = x Exercise. Cn you think of n ntierivtive of e x? Remrk. Since the erivtive of ny constnt is 0, then we cn ny constnt to n ntierivtive to mke nother ntierivtive. So if F (x) is n ntierivtive for f(x), then so is F (x) + C for ny number C (see exmples bove). Hence the ntierivtives of function ctully form fmily of functions. Exercise. The grph of f (x) is given below. Sketch grphs of the ntierivtive f(x) such tht (i) f(0) = 0 n (ii) f(0) = 1.
3 Exercise. Sketch n ntierivtive of the function g(x) = e x2 (bell curve function). Using the Funmentl Theorem of Clculus, we cn compute the exct vlues of the ntierivtive. This is becuse the chnge in the vlues of the ntierivtive is given by the re uner the curve of the erivtive. Recll the formul, f(b) f() = b f (x)x Using the grph of f (x) from the exercise bove, try to clculte the exct vlues of the function f(x) in this wy (tht is, fin f(1), f(2),, f(5)). Assume f(0) = 0 s in prt (i) of the exercise. Constructing Antierivtives n the Inefinite Integrl Fct 1: If F (x) = 0, then F (x) = C, for some number C. Fct 2: If F (x) is n ntierivtive for F (x), then ny ntierivtive is given by F (x)+c for some vlue of C. Definition. Suppose tht F (x) is n ntierivtive for f(x). Then the Inefinite Integrl is given by: f(x)x = F (x) + C Note: you must lwys inclue the +C. Also, while the efinite integrl outputs number, the inefinite integrl outputs fmily of functions. (1) Constnt functions: (note tht k = 0 is vli). kx = xk + C
4 (2) Other simple functions (compre with the corresponing erivtive rules): () Linerity of the erivtive/ linerity of the integrl x (c f 1f(x) + c 2 g(x)) = c 1 x + c g 2 x c 1 f(x) + c 2 g(x)x = c 1 f(x)x + c 2 g(x)x + C (b) Power rule for erivtives/ power rule for integrls x (xn ) = nx n 1 x n x = xn+1 n C (c) Chin rule for erivtives/ usubstitution f g (f g) x g x f u u = f u + C () Prouct rule for erivtives/ integrtion by prts f (f(x)g(x)) = g(x) x x uv = uv + f(x) g x vu + C (3) A specil cse of the power rule. When n = 1, 1 x = ln x + C x Note. For the lst integrl, n bsolute vlue is inclue to mke it efine for ll rel numbers One finl use of ntierivtives is to clculte the vlue of efinite integrl. If F (x) is n ntierivtive for f(x), then by the Funmentl Theorem of Clculus. b f(x)x = F (b) F (), Exercise. Write the generl ntierivtive: f(t) = 2t 3 + 3t 2 + t
5 g(t) = 5 t + et h(t) = 3 t Chllenge: f(x) = n i=1 x n
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