MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls, nd the Fundmentl Theorem of Clculus. In Clculus 2, we lern some pplictions of the definite integrl, some techniques for clculting or pproximting definite integrls, infinite sequences, infinite series, nd power series representtions of functions. A little review of the definite integrl Let s review wht we know bout the integrl b f(x) dx of function of one vrible. There re three interprettions of b f(x) dx tht I would like you to keep in mind. You cn review these interprettions of the definite integrl in Chpter 5 of the textbook. 1
MA 124 Jnury 18, 2018 The definition of the definite integrl For b f(x) dx, we prtition the intervl [, b] into subintervls (usully of equl width). In ech subintervl [x k 1, x k ], we pick test number x k, nd we consider the Riemnn sum n f(x k ) x. k=1 If the function f(x) is resonble (for exmple, continuous) on the intervl [, b], then the Riemnn sums converge to unique number s n. Tht is, lim n ( n f(x k ) x k=1 The definite integrl hs number of properties such s b f(x) dx + c b ) = f(x) dx = b c f(x) dx. f(x) dx. You should review these properties in Section 5.2 of your textbook. There is one property tht the definite integrl does NOT hve. Tht is, the integrl of the product does not necessrily equl the product of the integrls. 2
MA 124 Jnury 18, 2018 Wht is the difference between b f(x) dx nd f(x) dx? Exmple. 3π/2 0 sin x dx = nd sin x dx = The Fundmentl Theorem of Clculus As its nme implies, the Fundmentl Theorem of Clculus is the single most importnt theorem in clculus. It reltes two pprently unrelted concepts the derivtive ( rte of chnge) nd the definite integrl ( generlized summtion procedure). At the hert of the theorem is the concept of n re function N (signed re or net re) ssocited with continuous function f on the intervl [, b]. It is defined s N(x) = x f(t) dt. Note tht the independent vrible for N is the upper endpoint of integrtion. Theorem. The function N is differentible on (, b) nd N (x) = f(x). In other words, the function N is n ntiderivtive of f. This theorem is often clled Prt I of the Fundmentl Theorem. Prt II follows from Prt I, nd it is Prt II tht gives us method for clculting integrls. Theorem. Suppose f is continuous on [, b]. If F is ny ntiderivtive of f, then b f(x) dx = F (b) F (). Exmple. We know tht cos x is n ntiderivtive of sin x. Therefore, 3π/2 0 sin x dx = [ cos x ] 3π/2 0 3
MA 124 Jnury 18, 2018 The Method of Substitution We know how to compute the derivtives of typicl combintions of well-known functions such s power functions, rtionl functions, nd trigonometric functions. Unlike differentition, however, ntidifferentition is significntly less systemtic. The Fundmentl Theorem of Clculus gives us n bstrct wy to construct n ntiderivtive for ny continuous function, but we re not lwys ble to express the result in fmilir terms. Therefore, we pproch the problem of finding n ntiderivtive in stges. First, we identify smll list of functions tht re derivtives. These functions re our bsic building blocks. Then we use theorems to tret sums nd constnt multiples. Finlly, we try to use the computtionl theorems for computing derivtives in reverse. Now we discuss technique for ntidifferentition tht is the Chin Rule in reverse. Exmple. Consider the function sin(x 2 ). Then We obtin the indefinite integrl d dx sin(x2 ) = The most difficult spect to lerning the method of substitution is recognizing when it cn be used. We must be ble to determine if given function is the result of n ppliction of the Chin Rule. In the exmple bove, we must relize tht the function 2x cos(x 2 ) is the result of the Chin Rule pplied to the function sin(x 2 ). Here s how this kind of clcultion is done in more bstrct nottion. Using the Chin Rule, we hve d dx F (g(x)) = As result, we obtin the indefinite integrl In the exmple, F (x) = sin(x) nd g(x) = x 2. 4
MA 124 Jnury 18, 2018 Imgine strting out trying to clculte 2x cos(x 2 ) dx without knowing the nswer in dvnce. Then we would hve to identify tht g(x) = x 2 nd F (x) = cos x. Exmple. sec 2 x 2 tn x dx We cn see tht this is correct by differentiting. There is wy to simplify the procedure of identifying tht n integrnd hs the form F (g(x)) g (x). We refer to g(x) s chnge of vribles, nd we write u = g(x). So strting with F (g(x)) g (x) dx = F (g(x)) + C 5
MA 124 Jnury 18, 2018 Differentil nottion: We define the differentil du = F (u) du = F (u) + C. ( ) du dx. Then we get dx This method of producing ntiderivtives is clled the Method of Substitution or u-substitution becuse it is esiest to think of the inner function g(x) s new vrible. This new vrible is often denoted by the letter u. Exmple. x 9 x 2 dx You cn lwys check your nswer! 6