MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives by known rules. 5. Find the eqution of the tngent line t point. 6. Use implicit differentition. 7. Clculte severl definite nd indefinite integrls by vrious integrtion-techniques. 8. Find the re between two curves. 9. Anlyze function. () Find intervls of increse nd decrese. (b) Find intervls of concvity. (c) Find locl extreme vlues. (d) Find inflection points. 10. Applictions () Solve problems involving exponentil growth or continuously compounded interest. (b) Solve relted-rtes problem. (c) Solve n optimiztion-problem. (d) Optimize revenue, cost, nd profit. (e) Find the net-chnge of quntity over n intervl. (f) Find the Gini Index, nd interpret the result. 1
2 Formuls 1. point-slope form for the eqution of line: y y 0 = m(x x 0 ) 2. The limit-definition of derivtive: f (x) = lim h 0 f(x + h) f(x) h 3. Bsic differentition rules: constnts, pulling out constnt, dding or subtrcting 4. Product, Quotient, nd Chin Rules: (f g) = f g + fg, ( ) f = f g fg g g 2, d [ ] f(g(x)) = f (g(x)) g (x) 5. Derivtives nd Indefinite Integrls of Common Functions: (x n ) = nx n 1 x n 1 = n+1 xn+1 + C if n 1 (e x ) = e x e x = e x + C (ln(x)) = 1 1 = ln x + C x x 6. Fundmentl Theorem of Clculus: f(x) = F (b) F () if F (x) = f(x). 7. Integrtion-by-Prts: u dv = uv v du; Choose u by LAE. 8. Averge vlue of function: f vg = 1 b 9. Are between two curves: 10. Continuous Compound Interest: A = P e rt f(x) [ f(x) g(x) ] if f(x) g(x) on [, b] 11. Revenue: R = p x, where p is the price per item nd x is the number of items 12. Profit: P = R C, where R is revenue nd C is cost [ ] [ f(x) f ] (x) 13. L Hospitl s Rule: lim = lim x g(x) x g if the limit hs indeterminte form 0 (x) 0 or 14. Gini Index: G = 1 0 [x f(x)] 2
3 Summry 3.1 Functions: Ch. 1 1. Understnd wht function is nd how to determine whether n eqution describes function (Verticl Line Test). 2. Be fmilir with the bsic shpes of the grphs of common functions; for instnce, you should be ble to distinguish between f(x) = x 2 nd g(x) = x 3. 3. Know wht vrious chnges to n eqution will do to trnsform the grph (verticl/horizontl trnsltions, reflections, stretches, or compressions). These re summrized on pge 63 of the textbook. 3.2 Limits: 2.1-2.3 1. Properties of Limits: Limit of constnt or identity function Limit of sum, difference, product, or quotient of functions Pulling constnt outside limit 2. Limits t Point: If function cn be evluted t point without dividing by 0, then the limit of the function t the point equls the function-vlue. This is the key ide behind continuity. Infinite Limits: If, for rtionl function, the limit of the denomintor s x c is 0, nd the limit of the numertor is not 0, then the function will be going to infinity. However, it could be going to positive infinity from one side nd negtive infinity from the other, so the limit my not exist. Recll tht limit only exists if its right- nd left-sided limits both exist nd re equl. 3. Limits t Infinity: Theorems 2-4 on pges 146-148 in the textbook In generl, if the numertor is becoming lrger nd lrger, then the whole function is going to infinity (positive or negtive); lso, if the denomintor is becoming lrger nd lrger, then the function is going to 0. The limit s x of polynomil is the sme s the limit s x of the first term of the polynomil. 4. Asymptotes: Verticl Asymptotes: Verticl symptotes correspond to infinite limits. They occur when the denomintor is equl to 0. When finding verticl symptotes, find the x-vlues for which the denomintor is 0, then test the numertor for ech of these vlues. If the numertor is lso 0 t zero of the denomintor, then tht point is hole. If not, it is verticl symptote. 3
Horizontl Asymptotes: Horizontl symptotes correspond to limits t infinity, so follow these sme rules if you re sked to find the limit t infinity of rtionl function. When finding the horizontl symptote of rtionl function if there is one, check the degrees of the polynomils in the numertor nd denomintor: () If the degree of the numertor is greter, then there is no horizontl symptote. (b) If the degree of the denomintor is greter, then the line y = 0 is the horizontl symptote. (c) If their degrees re equl, then the rtio of the leding coefficients is the y-vlue t which the horizontl symptote occurs. 3.3 Derivtives: 2.4-2.5, 3.1-3.5 1. Know wht derivtive ctully mens; try expressing wht derivtive mens in your own words. 2. Be ble to use the limit-definition in order to find the derivtive for simple polynomil function (the Four-Step Process). 3. Polynomils: Know the power rule nd how to use it. Know how to del with constnts when differentiting. 4. Exponentil nd Logrithmic Functions: rules. 5. Product Rule 6. Quotient Rule d (ex ) = e x nd d [ ] 1 ln(x) =. Memorize these x 7. Chin Rule: Know how to recognize the inner function nd the outer function. Use the chin rule to differentite composite functions. 8. Implicit Differentition: finding dy given n implicit eqution involving x nd y. () Differentite both sides of the eqution with respect to x. (b) Split up the eqution nd differentite ech term seprtely. (c) For the terms involving only x or constnts, use norml rules of differentition. (d) For terms involving y, differentite s usul, then multiply the result with dy by the chin rule. (e) Solve the resulting eqution for dy. 4
3.4 Applictions of Derivtives: 3.6, 4.1-4.3, 4.5-4.6 1. Relted Rtes () Strt with n eqution linking two or more functions of time, x(t) nd y(t) for exmple. (b) Differentite the eqution implicitly with respect to t. (c) Given ll pieces of informtion but one, find the remining rte; for exmple, given x, y, nd dy, find. You my hve to use the originl eqution in order to find y, though. dt dt 2. Mxim nd Minim () First Derivtive Test: Extrem occur t criticl points, which re the points where the first derivtive is 0 or not defined while the function is still defined. i. Find f (x), the first derivtive. ii. Find the x-vlues for which f (x) = 0 or f (x) does not exist. iii. The resulting points re the criticl points. iv. Crete sign-chrt for f (x), splitting up the number line t the criticl points. v. Mke conclusion bout ech criticl point bsed on the sign chrt: If f (x) > 0 on the left nd f (x) < 0 on the right, then tht point gives mximum. If f (x) < 0 on the left nd f (x) > 0 on the right, then tht point gives minimum. If f (x) does not chnge sign, then tht point is neither mximum nor minimum. (b) Second Derivtive Test: Extrem occur t criticl points where the second derivtive is not 0; the second derivtive describes concvity. i. Find the criticl points. ii. Find f (x), the second derivtive. iii. Evlute f (x) t ech of the criticl points. A. If f (c) > 0, then the function is concve up there, so tht point gives minimum. B. If f (c) < 0, then the function is concve down there, so tht point gives mximum. C. If f (c) = 0, then tht point is neither mximum nor minimum. (c) Locl vs. Absolute: The first nd second derivtive tests give two wys to find locl extrem. In order to find the bsolute extrem: i. On closed intervl: Find ll criticl points, then evlute the function t the criticl points nd the endpoints of the intervl to find the bsolute mximum nd minimum over tht intervl. ii. Over ll rel numbers: A function hs n bsolute extremum over ll rel numbers if there is one nd only one criticl point. 5
3.5 Integrls: 5.1-5.2, 5.4-5.5, 6.3 1. Indefinite Integrls () Be ble to find nti-derivtives of common functions, including power functions, exponentil functions, nd logrithmic functions. (b) Remember to tke nti-derivtives: do not tke derivtives when you re sked to find n indefinite integrl. (c) Remember to include the constnt of integrtion when you find n indefinite integrl. (d) Know how to perform u-substitution, nd prctice picking u nd finding du. 2. Definite Integrls () Evluting definite integrls: Find the indefinite integrl; tht is, find the prticulr nti-derivtive of the integrnd. Use the Fundmentl Theorem to plug in the limits of integrtion. (b) Remember tht n integrl with the sme vlue for the lower nd upper limits of integrtion is lwys 0, tht is, f(x) = 0. 3. Integrtion-Techniques () Know how to use the bsic rules of integrtion. (b) Know the nti-derivtives of common functions. (c) Know how to use the Fundmentl Theorem of Clculus. (d) Use u-substitution if the integrnd is not common function. (e) Use integrtion-by-prts if the integrnd is not common function nd u-substitution does not work. 3.6 Applictions of Integrls: 5.5, 6.1 1. Net-Chnge Theorem: If Q (t) represents the rte of chnge of some quntity, then Q (t) dt = Q(b) Q() represents the net-chnge or the totl chnge from time t = to time t = b. 2. Averge vlue: 1 b f(x) finds the verge vlue of f(x) on the intervl [, b]. f(x) gives the re of rectngle with the sme re s the region under the curve. The height of this rectngle is the verge vlue. Dividing the re of the rectngle by its width gives the height; the width of the rectngle is b. 6
3. Are under curve Here, we tret res bove the x-xis s positive nd res below the x-xis s negtive. Just tking the integrl of the function gives the re under the curve; tht is, Are = f(x). 4. Are between two curves Here, we tret ll res s positive. Steps: 5. Gini Index () Find where the two curves intersect if t ll. Do this by setting the functions equl to ech other nd solving for x. (b) If you re given n intervl nd one or more of the intersection points lies on tht intervl, then split the intervl t those points, mking severl smller subintervls. (c) Over ech subintervl, find which function is bove nd which is below by picking test vlue nd evluting both functions. (d) Set up n integrl over ech subintervl with the pproprite order bsed on the upper nd lower functions. Add these integrls together. (e) Solve the integrtion problem to find the re between the two curves. If 0 x is the percentge of popultion nd 0 f(x) 1 is the percentge of income cross given country, then y = f(x) is Lorenz curve. If y = f(x) is given Lorenz curve, then its Gini index is G = If G 0, then income is closer to being distributed eqully. If G 1, then income is closer to being distributed uneqully. 1 0 [x f(x)]. For recommended exercises, refer to previous reviews, previous exms, or written s- Remrk: signments. Disclimer: In ddition to this review-sheet, nything presented in clss even if it ws not explicitly written on the bord, ssigned s online homework, or found in the book is fir gme for the exm. I will not try to trick you, but there my be importnt informtion which we hve covered which I hve not included on this hnd-out. This is my disclimer.... ;) 7