MCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate.

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1 MCVU Final Eam Review Answe (o Solution) Pactice Questions Conside the function f () defined b the following gaph Find a) f ( ) c) f ( ) f ( ) d) f ( ) Evaluate the following its a) ( ) c) sin d) π / π cos Conside the function if < f ( ) if if > Find the following its, if the eit a) f ( t) f ( t) c) f ( t) Evaluate a) c) e) 8 7 d) f) Evaluate a) c) e) 6 t d) t t f) t t t t 6 Evaluate 8 a) 8 c) / 8 Page of Designed b Teodou Gugoiu

2 7 Conside the function f () defined b the following gaph Fo each value of a, classif the function f () as continuous, having a jump, a emovable o an infinite discontinuit a) a a c) a d) a e) a 8 Fo what value of the constant c is the function c if < f ( ) c if continuous at eve numbe? 9 Fo each function, calculate fist the slope of the tangent line with the fomula f ( ) f ( a) m a a then find the equation of the tangent line at the given point a) f ( ), at P (,) f ( ), at P (, ) Fo each case find the slope of the tangent line at the f ( a h) f ( a) geneal point P( a, f ( a)) using m h h a) f ( ) f ( ) c) f ( ) d) f ( ) Fo each case, find the ARC ove the given inteval a) f ( ), [, ] f ( ), [,] Fo each case, find the IRC at the given numbe a) f ( ), at f ( ), at Fo each case, find the aveage velocit ove the given inteval a) s ( t) t t, [,] s( t) t t, [,] Fo each case, find the instantaneous velocit at the given moment of time a) s( t) t t, at t s( t) t t, at t Page of Designed b Teodou Gugoiu

3 Use the fist pinciples method to find the deivative of each function State the domain of each function and its deivative a) f ( ) f ( ) c) f ( ) d) f ( ) e) f ( ) Fo the function f () defined gaphicall below, find the values whee the function f is not diffeentiable and the eplain wh Use the powe ule to diffeentiate a) f ( ) f ( ) c) f ( ) d) f ( ) e) g) f ( ) h) f ( ) f) f ( ) i) f ( ) f ( ) Diffeentiate a) f ( ) sin f ( ) cos c) f ( ) e d) f ( ) ln Diffeentiate a) f ( ) f ( ) c) f ( ) sin cos d) f ( ) e ln 6 Find the equation of the tangent line to the cuve at the point T (, ) 7 Find the equation of the tangent line(s) with the slope m 6 to the cuve 8 At what points on the hpebola is the tangent line paallel to the line Page of Designed b Teodou Gugoiu

4 9 Diffeentiate a) f ( ) ( )( ) f ( ) ( ) c) sin d) e e) e ln Diffeentiate a) f ( ) ( ) f ( ) ( ) c) f ( ) ( ) d) f ( ) sin e) f ( ) (ln ) Diffeentiate, then simplif a) f ( ) f ( ) sin sin c) f ( ) d) f ( ) cos e) f ( ) f) ln e ln f ( ) sin d d du Fo each case, use to find the deivative of d du d f ( g( )) a) u, u u, u Use chain ule to diffeentiate a) [( ) ] c) ( ) ( ) Use the genealied diffeentiation ules to find the deivative of each function a) ( ) c) sin d) cos e) ln f) e Fo each case, find the fist and the second deivative a) c) d) e) f) g) sin h) e i) ln j) log 6 Fo each case, find the velocit and the acceleation functions a) s ( t) t t s ( t) t t t t c) s ( t) t 7 Fo each case, find the moments of time at which the object is at est a) s t t ( ) t s( t) t 6t Page of Designed b Teodou Gugoiu

5 Fo each case, use the fist deivative sign to find the intevals of incease o decease a) f ( ) f ( ) c) f ( ) d) f ( ) ( ) Find the intevals of incease o decease a) f ( ) ln f ( ) e c) f ( ) e d) f ( ) sin Fo each case, find the citical points a) f ( ) f ( ) c) f ( ) d) f ( ) e) f ( ) f) f ( ) Fo each case, find the citical points a) f ( ) sin f ( ) tan c) f ( ) e d) f ( ) ln e) f ( ) ln f) f ( ) e Fo each case, find an local etema using the fist deivative test a) f ( ) f ( ) ( ) c) f ( ) d) f ( ) e) f ( ) f) f ( ) 6 Fo each case, find the absolute etema (maimum o minimum) points a) f ( ), fo [,] f ( ), fo [,] c) f ( ), fo [, ] d) f ( ) cos, fo [ π /,π ] e) f ( ) log, fo [,] f) f ( ) e, fo [,] g) f ( ) sin, fo [,π ] 7 Fo each case, find the intevals of concavit a) f ( ) 6 f ( ) ( ) c) e) f ( ) d) f ( ) ( )( ) f ( ) e f) f ( ) ln g) f ( ) ln h) f ( ) cos Page of Designed b Teodou Gugoiu

6 8 Fo each case, find the points of inflection a) f ( ) f ( ) c) / f ( ) ( ) d) f ( ) ( ) ( ) e) f ( ) ln f) f ( ) sin 9 Find c given that the gaph of point of inflection at (, f ()) f ( ) c / has a Use the second deivative test to find the local maimum and minimum values of each function a) f ( ) 6 f ( ) 6 c) f ( ) d) f ( ) ( ) Find the local minimum and maimum values fo a) 6 Second Deivative compute f ''( ) find points whee f ''( ) o f ''( ) DNE find points of inflection find intevals of concavit upwad/downwad check the local etema using the second deivative test 7 Sketching use boken lines to daw the asmptotes plot - and - intecepts, etema, and inflection points daw the cuve nea the asmptotes sketch the cuve Sketch the gaph of the following polnomial functions a) f ( ) 6 f ( ) c) f ( ) ( ) d) f ( ) ( ) e) f ( ) ( )( ) Sketch the gaph of the following ational functions a) f ( ) f ( ) c) f ( ) d) f ( ) A ectangle has a peimete of m What length and width should it have so that its aea is a maimum What is the maimum value of its aea? If 7cm of mateial is available to make a bo with a squae base and open top, find the dimensions of the bo that give the lagest volume of the bo What is the maimum value of the volume? 6 A ectangula piece of pape with peimete cm is to be olled to fom a clindical tube Find the dimensions of the pape that will poduce a tube with maimum volume 7 A fame wants to fence an aea of,m in a ectangula field and divide it in half with a fence paallel to one of the sides of the ectangle How can be done so as to minimie the cost of the fence? 8 A metal clinde containe with an open top is to hold ft If thee is no waste in constuction, find the dimensions that equie the least amount of mateial Page 6 of Designed b Teodou Gugoiu

7 Conside the cube ABCDEFGH with the side length equal to cm Find the magnitude of the following vectos a) AB BD c) BH Pove o dispove each statement a) If a b then a b If a b then a b Two vectos ae defined b a N[ E] and b N[9 ] Find the sum vecto s a b the diffeence vecto d a b Two vectos ae defined b a km[ W ] and b km[ S] Find the sum vecto s a b the diffeence vecto d a b Two vectos ae defined b a m[ E] and b m[ ] Find the sum vecto s a b the diffeence vecto d a b 6 Given a i j k, b i j k, simplif the following epessions a) a b a b c) a b 7 Find a unit vecto paallel to the sum between a m[ E] and b m[ N] 8 Given u 8m[ W ] and v m[ S W ], detemine the magnitude and the diection of the vecto u v 9 Adam can swim at the ate of km / h in still wate At what angle to the bank of a ive must he head if he wants to swim diectl acoss the ive and the cuent in the ive moves at the ate of km / h? A plane is heading due noth with an ai speed of km / h when it is blown off couse b a wind of km / h fom the notheast Detemine the esultant gound velocit of the aiplane (magnitude and diection) A ca is tavelling at v ca km / h[ E], a motoccle is tavelling at v moto 8km / h[ W ], a tuck is tavelling at v tuck km / h[ N] and an SUV is tavelling at v SUV km / h[ SW ] Find the elative velocit of the ca elative to a) motoccle tuck c) SUV Page 7 of Designed b Teodou Gugoiu

8 Find the algebaic vecto AB in odeed tiplet notation and unit vecto notation whee A(,,) and B (,,) Find the magnitude of the vecto v i j k Given a (,, ), b i j k, and c i j do the equied opeations a) a b c ( a ( a c) Given A (,,), B (,, ), and C (,, ), find the coodinates of a point D(,, ) such that ABCD is a paallelogam The magnitudes of two vectos a and b ae a and b espectivel, and the angle between them is α 6 Find the value of the dot poduct of these vectos 6 Find the dot poduct of the vectos a and b whee a (,,) and b i j k 7 Fo what values of k ae the vectos a ( 6,, ) and b (, k, ) a) pependicula (othogonal)? paallel (collinea)? 8 Find the angle between the vectos a and b whee a (,, ) and b j k 9 A tiangle is defined b thee points A (,, ), B (,, ), and C (,,) Find the angles A, B, and C of this tiangle Given the vecto a (,,), find the scala pojection a) of a onto the unit vecto i of a onto the vecto i j c) of a onto the vecto b i j k d) of the unit vecto i onto the vecto a Given two vectos a (,, ) and b (,, ), find a) the vecto pojection of the vecto a onto the vecto b the vecto pojection of the vecto b onto the vecto a c) the vecto pojection of the vecto a onto the unit vecto k d) the vecto pojection of the vecto i onto the vecto a The magnitudes of two vectos a and b ae a and b espectivel, and the angle between them is α 6 Find the magnitude of the coss poduct of these vectos Page 8 of Designed b Teodou Gugoiu

9 Fo each case, find the coss poduct of the vectos a and b a) a (,,), b (,,) a i j, b i j k Use the coss poduct popeties to pove the following elations a) ( a ( a ( a ( a ( a ( a ( a ( a a)( b Find an unit vecto pependicula to both a (,, ) b (,,) and 6 Find the aea of the paallelogam defined b the vectos a (,,) and b (,, ) 7 Find the aea of the tiangle defined b the vectos a (,,) and b (,, ) 8 Find the volume of the paallelepiped defined b the vectos a (,, ), b (,, ) and c (,, ) 9 Conside the following vectos a i j k, b i j, and c i k Compute the equied opeations in tems of the unit vectos i, j, and k a) a b a b c) a b d) b c e) ( a b ) c f) ( a c g) Poj( a onto Find the equation of a D line which a) passes though the points A (, ) and B (, ) passes though the point A (, ) and is paallel to the vecto v (, ) c) passes though the point A (,) and is pependicula to the vecto v (, ) d) passes though the point A (, ) and is paallel to the line e) passes though the point A (, ) and is pependicula to the line Find the point(s) of intesection between the two given lines s a) (,) t(, ) and s and c) and 6 Page 9 of Designed b Teodou Gugoiu

10 Find the equation of the pependicula line to the given line though the given point a) (,) t(,), B (, ), B (,) C), B (, ) Find the distance fom the given point to the given line a) (, ) t(,), B (,), B (, ) c), B (,) Find the vecto equation of a line that a) passes though the points A (,, ) and B (,, ) passes though the point A(,,) and is pependicula on the plane c) passes tough the point A (,,) and is paallel to the - ais d) passes though the point A (,,) and is paallel to the vecto u (,,) e) passes though the oigin O and is paallel to the vecto i j 6 Convet the equation(s) of the line fom the vecto fom to the paametic fom o convesel a) (,,) t(,,) t t 7 Convet each fom of the equation(s) of the line to the othe two equivalent foms a) (,,) t(,, ) t t c) 8 Find the -int, -int, and -int fo the line (,,) t(,,) if the eist 9 Find the -int, -int, and -int fo the line (,,) t(,, ) if the eist Find if the lines ae paallel o not a) (,,) t(,,), (,,) s(,, 6) (,,) t(,, ), (,,) s(,, 6) c) (,,) t(,, 6), (,,) s(,,) In the case the lines ae paallel and distinct, find the distance between the lines a) (,,) t(,,), (,,) s(,, 6) (,,) t(,, ), (,,) s(,, 6) c) (,,) t(,, 6), (,,) s(,,) Page of Designed b Teodou Gugoiu

11 Fo each case, find the distance between the given line and the given point a) (,, ) t(,, ), M (,,) t t, E (,, ) t c), B (,, ) Find the point of intesection if it eists a) (,,) t(,,), (,,) s(,,) (,,) t(,, ), (,,) s(,, ) Find the vecto equation of a plane a) passing though the point A (,, ) and paallel to the vectos u (,, ) and v (,, ) passing though the points A(,,) and B (,, ) and C (,,) c) passing though the oigin and containing the line (,,) t(,,) d) passing though the point A (,,) and is paallel to the -plane Convet the vecto equation fo a plane to the paametic equations o convesel a) (,,) t(,,) s(,,) t s t s t Find the scala equation of a plane that a) passes though the point (,,) and is pependicula to the -ais passes though the point (,, ) and is paallel to the plane c) passes though the oigin and is pependicula to the vecto (,,) Find the intesection with the coodinate aes fo the plane π 6 Fo each case, find the distance between the given plane and the given point a) (,,) t(,,) s(,,), B (,,) 6, R (,,) Page of Designed b Teodou Gugoiu

12 Page of Designed b Teodou Gugoiu 6 Find the intesection between the given line and the given plane a) 9 9 π, t t t L π, ) (,,,,) ( t L 7 Find the equation of the line of intesection fo each pai of planes (if it eists) a) π, 6 π 9 6 π, 6 π c) π, π 8 Find the angle between each pai of planes a) π, π π, π 9 Solve the following sstem of equations Give a geometic intepetation of the esult ) ) 6 ) )