CALCULUS I (in a nutshell)

Transcription

1 We ve spent this semester lerning two min concepts: 1. DERIVATION 2. INTEGRATION CALCULUS I (in nutshell) DERIVATION: The derivtive of function f(x) with respect to the vrible x is FUNCTION denoted f (x) or df dx. The vlue of the derivtive t given point x = tells us how the function f(x) is growing t the point. More precisely, f () is the slope of the tngent line to the grph of f t the point (, f()). For this reson, the sign of the derivtive is connected to the fct tht the function f is incresing or decresing. In prticulr: f is incresing when f is positive; f is decresing when f is negtive. The points where the derivtive is 0 re specil, nd re clled criticl points. Here the tngent line to the grph of the function is horizontl. A criticl point P cn be: MAXIMUM, if f is positive before P nd negtive fter P ; MINIMUM, if f is negtive before P nd positive fter P ; n INFLECTION POINT with horizontl inflectionl tngent if f mntins the sme sign before nd fter P. Creful: Points of mximum nd minimum cn lso occur elsewhere: besides where f = 0 you should lwys check lso the following points, to see if they re mxim or minim: points where the derivtive is not defined!! endpoints of the domin of the function. (if included!) 1

2 The derivtive of the derivtive of the function f is clled the SECOND derivtive of f nd denoted by f. The second derivtive gives us informtion bout how the grph of the function bends : f is CONCAVE UP if f is positive; f is CONCAVE DOWN if f is negtive. Studying concvity cn lso help us decide whether criticl point is mximum or minimum. Let P be criticl point: if f (P ) is positive, then P is MINIMUM; if f (P ) is negtive, then P is MAXIMUM; if f (P ) = 0, then I do not know!! Also, concvity is relted to whether the grph of your function is bove or below the tngent line t given point: If f is concve up t point P, then f is bove the tngent line to the grph t the point (P, f(p )). If f is concve down t point P, then f is below the tngent line to the grph t the point (P, f(p )). Points where f goes from positive to negtive re clled INFLECTION POINTS. In prticulr, if P is n inflection point, then f (P ) = 0. (But f (P ) = 0 does NOT llow us to conclude tht P is n inflection point!!) Remember, the second derivtive is the first derivtive of the first derivtive, so, the second derivtive going from positive to negtive corresponds to the first derivtive hving mximum, nd vice-vers, if f goes from negtive to positive, then f hs minimum. Therefore, INFLECTION POINTS for f correspond exctly to MAXIMA nd MINIMA for f. Computing derivtives is not too bd. There re few functions for which we hd to compute derivtives by hnd, using the definition of derivtive s the limit of rte of chnge. Then we developed bunch of rules tht help us tking the derivtives of more complex functions, strting from the esy ones we computed by hnd. In prticulr, let me remind you: 2

3 the derivtive of sum of functions IS the sum of the derivtives of ech summnd; the derivtive of constnt times function IS tht sme constnt times the derivtive of the function; the derivtive of product of two functions IS NOT the product of the derivtives of the fctors; there is rule (PRODUCT RULE) tht llows us to tke the derivtive of product of functions; the derivtive of quotient of two functions IS NOT the quotient of the derivtives of the fctors; there is rule (QUOTIENT RULE) tht llows us to tke the derivtive of quotient of functions; there is rule tht llows us to del with composition of functions. This rule is clled the CHAIN RULE. The CHAIN RULE is undoubtedly the coolest nd most importnt of ll of the bove. It is EXTREMELY IMPOR- TANT to know how to use it well!!!!! There re two equivlent wys of stting the chin rule: 1. (f(g(x))) = f (g(x))g (x) 2... df dx = df dg dg dx The second form of the chin rule is the one tht is most suited to problems of relted rtes nd implicit differentition, so I reccommend tht you re very wrmly cquinted with it. INTEGRATION: We hve tlked bout two different types of integrls: 1. DEFINITE INTEGRALS. 2. INDEFINITE INTEGRALS (.k. ANTIDERIVATIVES). A DEFINITE INTEGRAL is NUMBER nd it represents n re. Well, more thn n re, to be honest. It is n re with sign. By f(x)dx we men the re between the grph of the function f(x) nd the x-xis. This re cquires sign ccording to: whether f(x) is bove (+) or under (-) the x-xis. wheter is before b (+) or b is before (-). 3

4 The fct tht the integrl is n intelligent re with sign, llows it to hve these beutiful properties: 1. you cn brek up n intervl into pieces. The integrl over the whole intervl is the sum of the integrl over ech piece. Yes, even if chopping up your intervl consists in subtrcting n intervl from nother!! The formul is: f(x)dx + c f(x)dx = c b f(x)dx. 2. you cn brek up the integrl of sum of functions into the sum of the integrls. In formul: (f(x) + g(x))dx = f(x)dx + g(x)dx. 3. if you multiply function by constnt number (5, 17, 23 or even the mysticl c or N!!), such number cn hrmlessly come out of the integrl sign: Nf(x)dx = N f(x)dx Huge Cution: the beutiful rules pretty much end here. Don t be cretive in mking up new ones. Sttements such s the integrl of product is the product of the integrls, or function cn come out of the integrl sign under certin circumstnces re dredfully wrong!!! Symmetry is cool. We like symmetry. The following observtions often come in hndy! Even functions re symmetric bout the y-xis. Therefore the integrl of n even function on n intervl of the form [, ] is twice the integrl of the sme function on the intervl [0, ]. Odd functions re symmetric with respect to the origin. The integrl of n odd function on n intervl of the form [, ] is lwys 0. An INDEFINITE INTEGRAL, on the other hnd, is NOT number, but FUNCTION...ctully not quite even tht...it is fmily of functions. The indefinite integrl of the function f(x) consists of ll possible functions hving derivtive equl to f(x). Such functions re clled ANTIDERIVATIVES of f(x). If we tke ny ntiderivtive of f(x) nd shift it verticlly, you obtin nother ntiderivtive of f(x) (the slope of the tngent line t ech point does not chnge, since lso the tngent lines re simply shifted verticlly!!!). As mtter of fct, by doing so you find ALL ntiderivtives of f(x). Hence, if in ny possible wy you cn get your hnds on one ntiderivtive F (x) of f(x), then you cn sfely conclude: f(x)dx = F (x) + C, where C is just ny ol constnt. 4

5 Finding ntiderivtives is quite tricky business...bsiclly so fr the only tool we hve up our sleeve is...guessing. However, if you do hve guess for n ntiderivtive of f(x), then checking whether the guess is right is no problem. Just tke the derivtive of the ntiderivtive nd see if you get f(x) bck. For exmple: is YES, becuse x n dx = xn+1 n C? ( ) d x n+1 dx n C = x n!! Finlly, wht brings the two types of integrtion together is the If F(x) is n ntiderivtive of f(x), then!!! FUNDAMENTAL THEOREM OF CALCULUS!!! f(x)dx = F (b) F (). Now, remember, definite integrl is number, not function, therefore you must lwys remember to plug in the pproprite vlues into the ntiderivtive. The right thing to do is, first plug in F the vlue tht ppers in the upper side of the integrl sign, then subtrct F evluted t the bottom extreme of integrtion. If it s esy, (nd lot of times it is!!) check whether the sign of the integrl you obtin mkes sense! A useful wy to interpret the FUNDAMENTAL THEOREM OF CALCULUS is tht it tells you tht, between point nd point b, the ntiderivtive F (x) grows exctly by the re between the grph of f(x) nd the x-xis between nd b. All this sid, GOOD LUCK GUYS! Do your best, I hope you enjoyed this semester. I sure enjoyed you! Renzo 5