Cumulative Review of Calculus

Transcription

1 Cumulative Review of Calculu. Uing the limit definition of the lope of a tangent, determine the lope of the tangent to each curve at the given point. a. f 5,, 5 f,, f, f 5,,,. The poition, in metre, of an object i given by t t t, where t i the time in econ a. Determine the average velocity from t to t. Determine the intantaneou velocity at t. h. If lim repreent the lope of the tangent to y f at, hs0 h what i the equation of f?. An object i dropped from the obervation deck of the Skylon Tower in Niagara Fall, Ontario. The ditance, in metre, from the deck at t econd i given by d t.9t. a. Determine the average rate of change in ditance with repect to time from t to t. Determine the intantaneou rate of change in ditance with repect to time at. The height of the obervation deck i.9 m. How fat i the object moving when it hit the ground? 5. The model P t t t etimate the population of fih in a reervoir, where P repreent the population, in thouand, and t i the number of year ince 000. a. Determine the average rate of population change between 000 and 008. Etimate the rate at which the population wa changing at the tart of y y = f() 8. a. Given the graph of f at the left, determine the following: i. f iii. lim f S ii. lim f S iv. lim f S Doe lim f eit? Jutify your anwer. S 7. Conider the following function:, if f, if 5, if 7 Determine where f i dicontinuou, and jutify your anwer. CHAPTERS 5 7

2 8. Ue algebraic method to evaluate each limit (if it eit). a. lim lim S0 5 S lim S lim V S 8 V V lim lim S S0 9. Determine the derivative of each function from firt principle. a. f f 0. Determine the derivative of each function. a. y 5 y 5 5 y y 5 y y 5. Determine the equation of the tangent to y 8 at the point,.. Determine the lope of the tangent to y 9 9 at the point where the curve interect the line y.. In 980, the population of Littletown, Ontario, wa 00. After a time t, in year, the population wa given by p t t t 00. a. Determine p t, the function that decribe the rate of change of the population at time t. Determine the rate of change of the population at the tart of 990. At the beginning of what year wa the rate of change of the population 0 people per year?. Determine f and f for each function. a. f 5 5 f V f f 5. Determine the etreme value of each function on the given interval. a. f, f e e, 0, f 9 f in, 0, p, 8 CUMULATIVE REVIEW OF CALCULUS

3 . The poition, at time t, in econd, of an object moving along a line i given by t t 0.5t t for 0 t 8. a. Determine the velocity and the acceleration at any time t. When i the object tationary? When i it advancing? When i it retreating? At what time, t, i the velocity not changing? At what time, t, i the velocity decreaing? At what time, t, i the velocity increaing? 7. A farmer ha 750 m of fencing. The farmer want to encloe a rectangular area on all four ide, and then divide it into four pen of equal ize with the fencing parallel to one ide of the rectangl What i the larget poible area of each of the four pen? 8. A cylindrical metal can i made to hold 500 ml of oup. Determine the dimenion of the can that will minimize the amount of metal require (Aume that the top and ide of the can are made from metal of the ame thickne.) 9. A cylindrical container, with a volume of 000 cm, i being contructed to hold candie. The cot of the bae and lid i $0.005cm, and the cot of the ide wall i $0.005cm. Determine the dimenion of the cheapet poible container. 0. An open rectangular bo ha a quare bae, with each ide meauring centimetre. a. If the length, width, and depth have a um of 0 cm, find the depth in term of. Determine the maimum poible volume you could have when contructing a bo with thee pecification. Then determine the dimenion that produce thi maimum volum. The price of MP player i p 50, where N. If the total revenue, R, i given by R p, determine the value of that correpond to the maimum poible total revenu. An epre railroad train between two citie carrie paenger per year for a one-way fare of $50. If the fare goe up, riderhip will decreae becaue more people will driv It i etimated that each $0 increae in the fare will reult in 000 fewer paenger per year. What fare will maimize revenue?. A travel agent currently ha 80 people igned up for a tour. The price of a ticket i $5000 per peron. The agency ha chartered a plane eating 50 people at a cot of $ Additional cot to the agency are incidental fee of $00 per peron. For each $0 that the price i lowered, one new peron will ign up. How much hould the price per peron be lowered to maimize the profit for the agency? CHAPTERS 5 9

4 . For each function, determine the derivative, all the critical number, and the interval of increae and decrea a. y 5 0 y y 0 y 5. For each of the following, determine the equation of any horizontal, vertical, or oblique aymptote and all local etrema: a. y 8 y 9. Ue the algorithm for curve ketching to ketch the graph of each function. a. f y 7. Determine the derivative of each function. a. f e 5 y 8 f e y e in 8. Determine the equation of the tangent to the curve y e at. m 9. In a reearch laboratory, a dih of bacteria i infected with a particular diea The equation N d 5de d 5 model the number of bacteria, N, that will be infected after d day. a. How many day will pa before the maimum number of bacteria will be infected? Determine the maimum number of bacteria that will be infecte 0. Determine the derivative of each function. a. y in co 5 y in co y in y tan tan y in y in co. A tool hed, 50 cm high and 00 cm deep, i built againt a wall. Calculate the hortet ladder that can reach from the ground, over the hed, to the wall behin. A corridor that i m wide make a right-angle turn, a hown on the left. Find the longet rod that can be carried horizontally around thi corner. Round your anwer to the nearet tenth of a metr m 70 CUMULATIVE REVIEW OF CALCULUS

5 Review Eercie, pp. 5. a. e e e e t. a. 0 ln 0 ln 5 5 ln 5 ln ln. a.. a. The function ha a horizontal tangent at (, e). So thi point could be poible local ma or min. 5. a. 0 The lope of the tangent to f at the point with -coordinate i 0.. a. e 0e y e e d e e e e e e 5 e e t 5 a ln 5 b co 8 in ec in co ec tan e in co co in Now, e e e e e e e y e e e e e e e e e d y ln 0 y 0 0. about 0.98 m per unit of time. a. t 0 After 0 day, about 0.5 mice are infected per day. Eentially, almot 0 mice are infected per day when t 0.. a. c c. a. 9e e e e e e 5e 5 e 5. a. 5 ln ln ln 5 5. a.. co a p b 5 ln e 0 ln 0 ln co co in co in co in in co in y p 0 7. v d dt ; Thu, v 8co 0pt 0p 80p co 0pt The acceleration at any time t i a dv dt d dt. Hence, a 80pin 0pt 0p 800p in 0pt. Now, d dt 00p 800p in 0pt 00p 8 in 0pt diplacement: 5, velocity: 0, acceleration: 0 p 9. each angle rad, or m..5 m. 5.9 ft. a. f 8 in 8 co f co ec tan in ec Chapter 5 Tet, p.. a. e e ln e e co 5 in 5 in co ec. y, The tangent line i the given lin.. y a. at v t 0ke kt k0e kt kvt Thu, the acceleration i a contant multiple of the velocity. A the velocity of the particle decreae, the acceleration increae by a factor of k. 0 cm> ln k ; 5k 5. a. f in co f cc cot cc in. abolute ma:, abolute min: minimum: Q,, no maimum e R 9. a. p, 5p, p increaing: 5p p ; 5p decreaing: p and p p local maimum at p local ; minimum at 5p Cumulative Review of Calculu, pp a. 0 ln. a. m> 5 m>.. f a. 9. m> 9. m> 5.55 m> p p 0 p p p p p p y 58 A n w e r

6 5. a fih> year 000 fih> year. a. i. ii. iii. iv. No, lim f doe not eit. In order for the limit to eit, lim f S and lim f mut eit and they S mut be the am In thi cae, lim f q, but S lim f q, o lim f S S doe not eit. 7. f i dicontinuou at. lim f 5, but lim f. S S 8. a a. S 0. a y a. p t t people per year 00. a. f 5 5 ; f 0 0 f f ; f f ; 5 f 5; f 0 5. a. maimum: 8, minimum: maimum: 9 minimum:, e maimum:, minimum: e maimum: 5, minimum:. a. vt 9t 8t, at 8t 8 tationary when t or t, advancing when vt 7 0, and retreating when vt 0 t.5 0 t.5.5t m r. cm, h 8. cm 9. r.8 cm, h 7.5 cm 0. a cm ;.7 cm by.7 cm by. cm.. $70 or $80. $0. a. 0 0, d i critical number, Increae:, Decreae: 7, d i critical number, Increae: 7, Decreae: d, ; are critical number, Increae:, 7, Decreae: The function ha d. no critical number. The function i decreaing everywhere it i defined, that i,. 5. a. y 0 i a horizontal aymptot ; are the vertical aymptote. There i no oblique aymptot Q0, 8 i a local maimum. 9 R There are no horizontal aymptote. ; are the vertical aymptote. y i an oblique aymptot, i a local maimum,, i a local minimum.. a. y y = + 7. a. 0e 5 e ln 8 co e in y e e a. 5 day 7 0. a. co 5 in 5 8 co in co in co co. about.8 m. about 8.5 m Chapter ec tan ec in co co Review of Prerequiite Skill, p. 7. a. y =.. a. AB 9.7, B.5, C 5.5 A 97.9, B 9.7, C 5.. Z 50, XZ 7., YZ R, S 0, T. 5.8 km km 8.. km cm 0 y A n w e r 59