Review Exercises. lim 5 x. lim. x x 9 x. lim. 4 x. sin 2. ln cos. x sin x

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1 MATHEMATICS 0-0-RE Dierential Calculus Martin Huard Winter 08 Review Eercises. Find the ollowing its. (Do not use l Hôpital s Rul. a) b) g) j) m) sin h) k) n) cos 0 sin. Find the ollowing its. (l Hôpital s Rule is permitte e ln cos a) b) 0 0 sin cscsin 0 0 e c) 6 ) i) l) 0 csc cot o) c) ) sin. For the unction whose graph is given, ind the value o the ollowing its (or unction), or eplain why it does not eist. a) b) c) g) j) 0 h) k) m) the intervals o continuity ) 0 i) l)

2 Math 0. Discuss the continuity o discontinuity. Support your answers. a) c) deined below. I sin i i ln i 0. Using the deinition o the derivative, ind Winter 08 Martin Huard is discontinuous, give the type o 6 i b) a) b) i 6 i 0 c). 6. Find a) b) c) tan ) g) sin cos h) i) sec tan j) k) ln arctan l) log tan m) sin n) csce o) arcsin p) tan q) ln ln r) s) cot u) sin sec sec t) arccos arccot ln arcsin sin arccsc w) v) ln ln ln arcsec ) y) arcsec e e sin cos z) arctan e cot csc 6

3 Math 0. Find dy d. e a) yarccsc c) y y sin cos y ) y b) ln y y y y y e y 8. Find the equation o the tangent line to the curve at the given point. at a) c) ln at b) 9. Find the point(s) on the graph o the unction parallel to the line y. at 0, y y at where the tangent line is 0. Find the equation o the tangent line(s) to the graph o ln the line y.. Find d y d i b) y ln sin a) y sec c) y arcsec y that is parallel to. The number o viewers o a television series introduced several years ago is approimated by the unction 60 N 6 where N denotes the number o weekly viewers o the series in the th week. Find the rate o increase o the weekly audience at the end o week. 0. The demand unction or a certain commodity is given by p. Find the instantaneous e rate o change o the number sold with respect to the price when.. When the price o a certain commodity is p dollars per unit, customers demand hundred units o the commodity, where p p 9 How ast is the demand changing with respect to the time when the price is $ per unit and is decreasing at the rate o 0 cents per month? Winter 08 Martin Huard

4 Math 0. At noon, ship A is 0 km west o ship B. Ship A is sailing east at km/h and ship B is sailing north at km/h. How ast is the distance between the ships changing at :00 P.M.? 6. A spotlight on the ground shines on a wall m away. I a man m tall walks rom the spotlight toward the building at a speed o.6 m/s, how hast is the length o his shadow on the building decreasing when he is m rom the building?. Find the absolute maimum and minimum values o (i they eist) on the given interval. a) on [-,6] b) 0, c) on ln arctan on, 6, on 8. A hotel has 0 luury units that it will rent out during the peak season at $00 per day. From eperience management knows that one unit will become vacant or each $0 increase in charge per day. What rent should be charged to maimize revenue? 9. An apple orchard produces annual revenue o $0 per tree when it is planted with 000 trees. Due to overcrowding, the annual revenue per tree is reduced by cents or each additional tree planted. I the cost o maintaining each tree is $0 per year, how many trees should be planted in order to maimize total proit orm the orchard? 0. As a result o a time-and-motion study, a manuacturer determines that between starting time (8 A.M.) and lunchtime ( noon), the average worker has produced p units o the product ater working t hours, where p t 9t t. At what time does the point o diminishing returns occur or the average worker?. A closed rectangular container with a square base is to have a volume o 000 cm. It costs twice as much per square centimeter or the top and bottom as it does or the sides. Find the dimension o the container o least cost.. A garage owner inds that she can sell tires per week at p dollars per tire, where p and the cost C in dollars o obtaining tires per week to sell is epressed by 00 C. a) Find the marginal cost, revenue and proit. b) How many tires must she sell to minimize the average cost? c) Find the number o tires she must sell per week and the price she should charge per tire to maimize proit.. A certain airline requires that cylindrical packages carried on an airplane by passengers be such that the sum o its length and girth (distance around the cylinder) is at most 0 cm. Find the dimensions o the cylindrical package o greatest volume that meets this requirement. Winter 08 Martin Huard

5 Math 0. A solid is ormed by adjoining two hemispheres to the ends o a right circular cylinder. The total volume o the solid is cubic centimeters. Find the radius o the cylinder that produces the minimum surace area.. The demand unction or a certain brand o compact disc is p where p is the wholesale unit price in dollars and is the quantity demanded each week, measured in units o a thousand. Compute the elasticity o demand and determine whether the demand is inelastic, unitary, or elastic when The demand unction or a product is p demand is elastic, inelastic, and o unit elasticity., 0. Find the intervals on which the. Apple has determined that the number o songs sold (in millions) on itunes in a day is related to the price o a song (in $) by the equation p. a) Calculate the price elasticity o demand when the price is $.8 per song. Is the demand elastic or inelastic? b) Find the price and demand when the demand is o unit elasticity. c) Find the revenue unction. Find the maimum revenue, and the price and number o songs sold at which this occurs. I Apple wants to increase its revenue, should they lower or raise the price o a song, which is now at $.9? 8. Sketch the graph o the ollowing unction by inding the domain, symmetry, intercepts, asymptotes, intervals o increase decrease, local maimum and minimum, concavity and points o inlection. a) 6 b) c) e 9. Use dierentials (or, equivalently, a linear approimation) to estimate the number. a) b) O c) tan. 0. The cost (in dollars) or producing units o a commodity is given by C ln 0 while the demand is given by p 0 e 0 producing and selling the st unit using dierentials.. Approimate the cost, revenue and proit or Winter 08 Martin Huard

6 Math 0 Answers. a) g) m). a) 0 6 b) 6 c) h) 0 i) n) b) o) c) 9 e Winter 08 Martin Huard 6 j) 6 86 ) k) undeined l). a) b) c) - - ) g) - h) i) j) k) l) 0 m),,,,,0, 0,,,. a) is continuous on,0, 0,,,. with an ininite discontinuity at ) 0 removable discontinuity at,,,,, with an ininite discontinuity at b) is continuous on. 8 6 e and a a jump discontinuity at,,,,,0, 0, with ininite discontinuities at c) is continuous on and 0. and a jump discontinuity at,,,,, with a jump discontinuity at is continuous on removable discontinuity at.. Note: All answers in this question must be ound using a) b) c) 6 6. a) b) ln c) tan sec ) g) 9 and a h) sin cos sin cos i) 0sec tan 6sec tan k) 9 arctan n) e csce cot e j) l) sec csc ln o) q) ln sec tan r) sec m) sin cos tan p) sin sec ln(sin ) s) csc cot 0 ln u) cot v) w) ln sin y) cos 9 cot 8tan csc sec. a) e cot csc y y y y z) ln ln ln b) c) cos cos y sin sin y y ) arccot arccos t) ) y y y ye y ye arcsec e e 9 8. a) y 9 b) y c) y y 9. 0, and, y ln y ln0 9 and

7 Math 0. a) sec 8 sec tan sec tan sec sin cos b) c) sin. viewers per week. e. Increasing by Decreasing at a rate o. a) Abs ma o units per month. 6 m/s Abs min o 0 e y units/$ 0 0 km/h b) Abs ma o ln Abs min o ln c) Abs min o No abs ma Abs ma o No abs min 9 8. $00 per unit trees 0. At 9:00 A.M.. 0 cm by 0 cm by 0 cm C 0 R 9 6 P. a) b) 0 tires c) tires at $. Radius o 0 6 cm and height o 0 cm.. 6. Elastic on 0,, Unit elasticity at 9 and Inelastic on, 0. a) 0, Elastic b) million songs at $ million songs at $ per song lower the price to $ 8. a) D: Sym: Even -int: 6, 0, 6 y-int: y 0 No Asymptotes is on,0, and on, 0, is on, and Rel. Ma:0,0 Rel. Min: 6, 9 6, 9 I.P.,,,, b) D: / 0, Sym: None on,, -int: y-int: None V.A 0, H.A. y is on,0 0, and on,, is on,0, and on 0, Rel. Ma:,0 No Rel. Min. No I.P. cm. 0 Inelastic. R per song c) per song Winter 08 Martin Huard

8 Math 0 c) D: -int: V.A / 0 Sym: None y-int: y 0 is on,0, and on 0,, is on and on Rel. Ma:,,8 Rel. Min: No I.P. D: Sym: Odd -int: 0 y-int: H.A. y 0 y 0, 0,0 is on and on,, is on 6 6,0, and on, 6 6 0,, Rel. Min:, Rel. Ma:, e e I.P.,, 0,0,, e e D: Sym: None -int: 0 y-int: y 0 No asymptotes. is on and on, is on, 0, and on,0 Rel. Min:, No rel. Ma: I.P.,, 0,0 9. a) 8 b) c) tan $ $ and $ 0 80 Winter 08 Martin Huard 8