University of Sioux Falls. MAT204/205 Calculus I/II
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1 University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques df d i. If f hs the onstnt vlue f ( x = x, then = ( = 0. dx dx d n n 1 ii. (Power rule If n is n integer (not zero, then ( x = nx. dx iii. If u is differentile funtion of x, nd is onstnt, then d du ( u =. dx dx iv. If u nd v re differentile funtions of x, then their sum u + v is differentile t every point where u nd v re oth differentile. At suh d du dv points, ( u + v = +. dx dx dx v. (Produt rule If u nd v re differentile funtions of x, then so is their d dv du produt uv, nd ( uv = u + v. dx dx dx vi. (Quotient rule If u nd v re differentile funtions of x nd if v ( x 0, then the quotient u / v is differentile t x, nd du dv v u d u ( = dx dx. 2 dx v v Setion 3.2. Integrtion tehniques i. Bsi formuls ii. Simplifying sustitution iii. Completing the squre iv. Expnding power nd using Trigonometri identity v. Eliminting squre root vi. Seprting frtion vii. Multiplying y form of 1 2. Mke numeril pproximtions of derivtives nd integrls Setion 8.1
2 f ( x f ( x0. Numeril pproximtion of the derivtive Clulte x1 x0 is the point in question nd x 1 is some point lose to x 0. 1 where x 0 Setion 2.1. Numeril pproximtion of the integrl Clulte the prtil sum f ( k x, insted of the infinite sum. Setion Anlyze the ehvior of funtion (e.g., find reltive (lol mxim nd minim, onvity. Definition of reltive (lol mxim nd minim i. A funtion f hs lol mximum vlue t n interior point of its domin if f ( x f ( for ll x in some open intervl ontining. ii. A funtion f hs lol minimum vlue t n interior point of its domin if f ( x f ( for ll x in some open intervl ontining.. The First Derivtive Theorem for Lol Extreme Vlues If f hs lol mximum or minimum vlue t n interior point of its domin, nd if f is defined t, then f ( = 0.. An interior point of the domin of funtion f where f is zero or undefined is ritil point of f. d. First Derivtive Test for Lol Extrem Suppose tht is ritil point of ontinuous funtion f, nd tht f is differentile t every point in some intervl ontining exept possily t itself. Moving ross from left to right, i. if f hnges from negtive to positive t then f hs lol minimum t ; ii. if f hnges from positive to negtive t then f hs lol mximum t ; iii. if f does not hnge sign t, then f hs no lol extremum t. Setion 4.3 e. The grph of differentile funtion y = f (x is i. onve up on n open intervl I if f is inresing on I ii. onve down on n open I if f is deresing on I. e twiedifferentile on n intervl I. i. If f > 0 on I, the grph of f over I is onve up. ii. If f < 0 on I, the grph of f over I is onve down. g. The Seond Derivtive Test for Lol Extrem Suppose f is ontinuous on n open intervl tht ontins x =. i. If f ( = 0 nd f ( < 0, then f hs lol mximum t x =. ii. If f ( = 0 nd f ( > 0, then f hs lol minimum t x =. f. The Seond Derivtive Test for Convity Let y = f (x n k = 1
3 iii. If f ( = 0 nd f ( = 0, then the test fils. It ould hve lol mximum, minimum or neither. 4. Solve prolems involving relted rtes, rtes of hnge, pproximtion of roots of funtion. Relted Rtes Strtegy i. Drw piture. Use t for time. Assume tht ll vriles re differentile funtions of t. ii. Write down the numeril informtion (in terms of the symols you hve hosen. iii. Write down wht you re sked to find (usully rte, expressed s derivtive. iv. Write n eqution tht reltes the vriles. You my hve to omine two or more equtions to get single eqution tht reltes the vrile whose rte you wnt to the vriles whose rtes you know. v. Differentite with respet to t. Then express the rte you wnt in terms of the rte nd vriles whose vlues you know. vi. Evlute. Use known vlues to find the unknown rte. Setion 3.7. Rtes of Chnge i. The instntneous rte of hnge of f with respet to x t x 0 is the f ( x0 + h f ( x0 derivtive f ( x0 = lim provided the limit exists. h 0 h ii. Veloity (instntneous veloity is the derivtive of position with respet to time. iii. Speed is the solute vlue of veloity. iv. Aelertion is the derivtive of veloity with respet to time. v. Jerk is the derivtive of elertion with respet to time. vi. In mnufturing opertion, the ost of prodution (x is funtion of x, the numer of units produed. The mrginl ost of prodution is the rte of hnge of ost with respet to level prodution, so it is d / dx. Setion 3.3. Approximtion of Roots - Newton s Method i. Guess first pproximtion to solution of the eqution f ( x = 0. A grph of y = f (x my help. ii. Use the first pproximtion to get seond, the seond to get third, nd f ( xn so on, using the formul xn + 1 = xn, if f ( xn 0. f ( xn Setion Solve pplied minim/mxim prolems (optimiztion prolems. Red the prolem. Red the prolem until you understnd it. Wht is given? Wht is the unknown quntity to e optimized?. Drw piture. Lel ny prt tht my e importnt to the prolem.. Introdue vriles. List every reltion in the piture nd in the prolem s n eqution or lgeri expression, nd identify the unknown vrile.
4 d. Write n eqution for the unknown quntity. If you n, express the unknown s funtion of single vrile or in two equtions in two unknowns. This my require onsiderle mnipultion. e. Test the ritil points nd endpoints in the domin of the unknown. Use wht you know out the shpe of the funtion s grph. Use the first nd seond derivtives to identify nd lssify the funtion s ritil points. Setion Understnd nd e le to use the Men Vlue Theorem nd the Fundmentl Theorem of Clulus. Men Vlue Theorem Suppose y = f (x is ontinuous on losed intervl [, ] nd differentile on the intervl s interior (,. Then there is t lest one f ( f ( point in (, t whih = f (. Setion 4.2. Fundmentl Theorem of Clulus i. If f is ontinuous on [, ] then F ( x = f ( t dt is ontinuous on [, ] nd differentile on (, nd its derivtive is f(x; x d F ( x = f ( t dt = f ( x. dx ii. If f is ontinuous t every point of [, ] nd F is ny ntiderivtive of f on [, ], then f ( x dx = F ( F (. 7. Demonstrte n intuitive understnding of the proess of integrtion 8. Evlute improper Integrls. Infinite Limits of Integrtion i. Upper limit - ( x dx = lim f f ( x dx ii. Lower limit - f ( x dx = lim f ( x dx iii. Both limits - ( x dx = lim f ( x dx + lim. Integrnd Beomes Infinite f f ( x dx i. Upper endpoint - ( x dx = lim f f ( x dx ii. Lower endpoint - f ( x dx = lim + f ( x dx iii. Interior point - d d ( x dx = f ( x dx + f f ( x dx x Setion 5.4 Setion 5.1
5 Setion 8.8
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