MATH 1591 - Finl Review November 20, 2005 1 Evlution of Limits 1. the ε δ definition of limit. 2. properties of limits. 3. how to use the diret substitution to find limit. 4. how to use the dividing out tehnique to find limit. 5. how to use the rtionlizing tehnique to find limit. 6. how to use the Squeeze Theem to find limit. 7. how to determine infinite limits. 8. how to find the vertil symptotes. 9. how to find limits t infinity. 10. how to find the hizontl symptotes. 11. how to use the one-sided limits to prove tht limit does not exists. 12. the definition of ontinuity. 13. properties of ontinuity. 14. how to find disontinuous points of funtion. 2 Clultion of Derivtives 1. the definition of the derivtive. 2. how to use differentition rules nd fmuls to find the derivtive of funtion. 3. how to use the hin rule to find the derivtive of omposite funtion. 4. how to use the impliit differentition tehnique to find the derivtive of n impliit funtion. 5. how to use the logrithmi differentition tehnique to find the derivtive of funtion. 6. how to find higher-der derivtives 1
3 Clultion of Integrls 1. the definition of indefinite integrls. 2. the definition of definite integrls. 3. how to use integrtion rules nd fmuls to find the integrl of funtion. 4. how to rewrite integrnds to find the integrl of funtion. 5. how to reognize the pttern f[g(x)]g (x) to find the integrl of funtion(pge 322). 6. how to use the method of Chnge of Vribles to find the integrl of funtion (pge 325). 7. how to use the properties of even nd odd funtions to find the integrl of funtion (pge 330). 8. how to use the log integrtion rule: 1dx = ln x + C to find the integrl of funtion x (pge 342). 9. how to use trigonometri identities to find the integrl of funtion (pge 346). 10. how to use the long division method to find the integrl of funtion (pge 344). 11. how to use the inverse trigonometri integrtion rules to find the integrl of funtion (pge 350). 12. how to omplete squre to find the integrl of funtion (pge 351). 4 Fundmentl Theems in Clulus 1. the Intermedite Vlue Theem f ontinuous funtions (pge 93). 2. Rolle s Theem (on pge 206). 3. the differentil Men Vlue Theem (on pge 208). 4. the integrl Men Vlue Theem (pge 312). 5. the onnetion fmul of integrtion nd differentition: (pge 309). 6. the reltion between integrtion nd differentition: ( d x ) f(t)dt = f(x) dx f(x)dx = F (b) F () 2
5 Applitions of Differentition 1. the definition of tngent line of the grph of funtion. 2. how to find the slope of tngent line nd the tngent line eqution. 3. the definition of instntneous veloity. 4. how to find the veloity from position funtion. 5. how to find the bsolute mximum nd minimum? Follow the following three steps. find ritil numbers of funtion evlute the funtion t ritil numbers nd the endpoints ompre these vlues 6. how to find reltive mximum nd minimum? Use the First derivtive Test (on pge 214); Use the Seond derivtive Test (on pge 226) 7. how to find intervls of inrese, derese nd onvity? 8. how to find infletion points? 9. how to sketh the grph of funtion? find the vertil nd hizontl symptotes; find the intervls of inrese nd derese; find the lol minimum nd mximum; find the intervls of onvity nd infletion points; mke tble plot ritil points like mx, min nd infletion points nd then onnet them. 10. how to solve n optimiztion problem? Drw digrm; identify qunities; write the primry eqution f qunty Q tht is to be mximized minimized; write the seond eqution from given ondition; solve the seond eqution nd then redue the primry eqution to the one hving single vrible; find the mximum minimum of Q. 11. how to find relted rte from nother given rte? Drw digrm; Introdue nottion; write down the given infmtion; wht to find; find equtions; differentite the equtions with respet to time t; find the unknown rte. 3
6 Applitions of Integrtion 1. how to use definite integrls to lulte the re of region bounded by urves. 2. how to use the Disk Method: V = π V = π ( [R(x)] 2 [r(x)] 2) dx ( [R(y)] 2 [r(y)] 2) dy to find the volume of solid of revolution bout line. 3. how to use the r length fmul: s = s = 1 + [f (x)] 2 dx 1 + [g (y)] 2 dy to find the length of the r given by y = f(x) by x = g(y). 4. how to use the re fmul: S = 2π s = 2π r(x) 1 + [f (x)] 2 dx r(y) 1 + [g (y)] 2 dy to find the re of surfe of revolution of the grph bout n xis given by y = f(x) by x = g(y). 5. how to use the fmul M x = ρ M y = ρ [ f(x) + g(x) 2 x[f(x) g(x)]dx ] [f(x) g(x)]dx, to find the moments nd enter of mss of plnr lmin bounded by the grphs of y = f(x), y = g(x), nd x b. 4
7 Smple Problems 7.1 Evlution of Limits 1. Use the diret substitution to find the following limits. () lim(3x 3 4x 2 + 3). x 1 x + 1 (b) lim x 3 x 4. ( πx ) () lim sin. x 1 2 (d) lim x 1 ln x e x. 2. Use the dividing out tehnique to find the following limits. () lim x 5 x 5 x 2 25. (b) lim x 3 x 2 x 6 x 2 5x + 6. e 2x 1 () lim x 0 e x 1. x + 1 2 (d) lim ln. x 3 x 3 3. Use the rtionlizing tehnique to find the following limits. () lim x 4 (b) lim x 0 x + 5 3. x 4 x + 5 5. x 4. Use the Squeeze Theem to find the following limits. () lim x 0 x 2 sin x. (b) lim x 0 x 2 sin 1 x. 5. Determine the infinite limits. () (b) x(1 + x) lim x 1 + 1 x. x 3 lim x 2 + x 2. 6. Find the vertil symptotes. 5
() f(x) = 1 x 2. (b) f(x) = 2+x x 2 (1 x). 7. Find limits t infinity. () lim x 2x + 4 3x 2 + 1. (b) lim x 2x 2 + 1 3x 2 + 5. 8. Find the hizontl symptotes. () f(x) = 1 1+e x. (b) f(x) = 3x2 x 2 +2. 9. Use the one-sided limits to prove tht the limit lim x 0 h(x) does not exists, where 10. F the funtion f(x) = x it is ontinuous. x 2 x x 2 + 1 if x < 0, h(x) = x 2 1 if x 0., find ll disontinuous points nd ll intervls on whih 7.2 Clultion of Derivtives 1. Use differentition rules nd fmuls to find the derivtives of following funtions. () y = 5. (b) y = x 9. () y = 6 x. (d) y = 3x 3 + 4x 5 + 7x 2. (e) y = 5e x + 6 sin x + 7 os x. (f) y = x 3 os x. (g) y = (x 3 + 2)(x 4 2x 3 + 1). (h) y = sin x. x 2 (i) y = x. x 2 +1 2. Use the hin rule to find the derivtives of the following funtions. () f(x) = (9x + 7) 2/3 (b) f(x) = 8 (x+3) 3 6
( () f(x) = (d) f(x) = 1 x+2 ) 3 3x 2 1 2x+5 (e) f(x) = se(x 3 ) (f) f(x) = sin(os x) (g) f(x) = e 3/x2 (h) f(x) = sin 2 (2x) (i) f(x) = (ln x) 4 (j) f(x) = 5 x (k) f(x) = rtn(x 2 1 (l) f(x) = log 5 ( 1 x 2 (m) f(x) = x6 ln x (n) y = 5 x/2 sin(2x) + 10 log 4 x x 3. Use the fundmentl fmul ( d x ) f(t)dt = f(x) dx to find F (x): (1) F (x) = x 1 4 t dt, x (2) F (x) = sin t 2 dt. 4. Use the impliit differentition tehnique to find the derivtives of the following impliit funtions. () ye x + xe y = xy. (b) x 2 + y 2 = 1. 5. Use the logrithmi differentition tehnique to find the derivtives of the following funtions. () y = (x 1)(x 2)(x 3). (b) y = x x. 6. Find the seond derivtive of the funtion. () y = 4x 3/2. (b) y = x. () y = sin(2x). 0 7
7.3 Clultion of Integrls 1. Rewrite integrnds nd then integrte term by term to find the integrls. () 1 0 (4x3 + 6x 2 1)dx (b) 3 x(x 4) () x 3 +x+1 x 2. Reognize the pttern f[g(x)]g (x) to find the integrls (pge 322). () x 2 (x 3 + 5) 4 (b) sin(3x) () π 0 x sin x 2 3. Use the method of Chnge of Vribles to find the following integrls (pge 325). () (x + 1) 2 x (b) 5 1 x 2x 1 4. Use the properties of even nd odd funtions to find the integrls (pge 330). () π/2 π/2 sin2 x os x (b) π/2 sin x os x π/2 5. Use the log integrtion rule: 1dx = ln x + C to find the integrls (pge 342). x () (ln x) 2 x (b) os x 1+sin x 6. Use trigonometri identities to find the integrls (pge 346). () tn x (b) se x 7. Use the long division method to find the integrls (pge 344). () x 3 6x 20 x+5 8. Use the inverse trigonometri integrtion rules to find the integrls (pge 350). () 3 1 4x 2 9. Complete squre to find the integrls (pge 351). () 2 2 (b) dx. x 2 +4x+13 2dx. x 2 +4x 8
7.4 Applitions of Differentition 1. Find the eqution of the tngent line to the grph of f(x) = x x+1 t the point (0,0). 2. Clvin nd Hobbes re t their dds offie nd spot Susie stnding on the sidewlk 256 ft below. They deide it would be relly funny to drop wter blloon on her hed. If the distne the blloon flls in t seonds is given by s(t) = 16t 2, wht is the blloons veloity when it hits Susie? Wht is its elertion? (Hint: In der to hit Susie, the blloon must trvel 256 ft). 3. Find the reltive nd bsolute extreme vlues of f(x) = x x on [0,4]. 4. F f(x) = x 4 + 4x 3, (i) find the intervls of inrese derese; (ii) find the lol mximum nd minimum vlues; (iii) find the intervls of onvity nd the infletion points; (iv) sketh the grph of f. 5. (2E0D (Two equtions nd zero derivtive) method) A retngulr pge is to ontin 36 squre inhes of print. The mrgins on eh side re to be 1.5 inhes. Find the dimensions of the pge suh tht the lest mount of pper is used. 6. A frmer plns to fene retngulr psture djent to river. The psture must ontin 180,000 squre meters in der to provide enough grss f the herd. Wht dimensions would require the lest mount of fening if no fening is needed long the river? 7. The rdius r of irle is inresing t rte of 3 entimeters per minute. Find the rte of hnge of the re when r = 6 entimeters. 7.5 Applitions of Integrtion 1. Find the re of the region bounded by the grphs of the equtions y = 1 + 3 x, x = 0, x = 8, y = 0. 2. Find the re of the region bounded by the grphs of two funtions f(x) = x 2 nd f(x) = x 3. 3. Use the Disk Method: V = π V = π ( [R(x)] 2 [r(x)] 2) dx ( [R(y)] 2 [r(y)] 2) dy to find the volumes of the following solids of revolution bout line. () Find the volume of the solid fmed by revolving the region bounded by y = 4 x 2, y = 0, x = 0 bout the x-xis. 9
(b) Find the volume of the solid fmed by revolving the region bounded by x = y 2 + 4y, y = 1, x = 0 bout the y-xis. () Find the volume of the solid fmed by revolving the region bounded by y = x, y = 0, x = 4 bout the y-xis. (d) Find the volume of the solid fmed by revolving the region bounded by y = x, y = 0, x = 4 bout the line x = 6. 4. Use the r length fmul: s = s = 1 + [f (x)] 2 dx 1 + [g (y)] 2 dy to find the r length of the grph of the funtion over the indited intervl. () y = 2 3 x3/2 + 1, [0, 1]. (b) y = ln(sin x), [ π 4, 3π 4 ]. 5. Use the re fmul: S = 2π s = 2π r(x) 1 + [f (x)] 2 dx r(y) 1 + [g (y)] 2 dy to find the re of surfe generted by revolving the urve bout the x-xis. () y = 1 3 x3, [0, 3]. (b) y = x3 + 1, [1, 2]. 6 2x 6. Use the fmul M x = ρ M y = ρ [ f(x) + g(x) 2 x[f(x) g(x)]dx ] [f(x) g(x)]dx, to find the enter f the lmins of unifm density ρ bounded by the grphs of the equtions. () y = x, y = 0, x = 4. (b) y = x 2, y = x 3. 10
7.6 Fundmentl Theems in Clulus The problems in this setion re optionl. 1. Use the Intermedite Vlue Theem to show tht there is root of the eqution in the intervl (0, 1). x 3 10x 2 + 1 = 0 2. Use Rolle s Theem (on pge 206) to prove tht if > 0 nd n is ny positive integer, then the polynomil funtion p(x) = x 2n+1 + x + b nnot hve two rel roots. 3. Let p(x) = Ax 2 + Bx + C. Prove tht f ny intervl [, b], the vlue gurnteed by the differentil Men Vlue Theem (on pge 208) is the midpoint of the intervl. 4. Let G(x) = x [ s 0 s f(t)dt] ds, where f is ontinuous f ll rel t. Find 0 () G(0). (b) G (0). () G (0). (d) G (0). 11