PART A: Answer in the space provided. Each correct answer is worth one mark each.

Transcription

1 PART A: Answer in the space provided. Each correct answer is worth one mark each. 1. Find the slope of the tangent to the curve at the point (,6). =. If the tangent line to the curve k( ) = is horizontal, what is the -coordinate of one of it s points of tangenc?. Given f ( ) = g( ) h( ), find f '() given g () =, g '() = 1, h () = 5, and h '() = 0. 1 r. Given q '( r) =, then what is a possible epression for q( r )? a) 5. Given V = π r h, then dv dt = 6. Sketch a graph of the derivative of each of the following functions. b) 7. State the initial velocit given that s( t) ( ) = t. 8. Given the graph below, determine: a) an interval where f '( ) > 0 and f "( ) < 0. b) one point where f '( ) = 0. c) one point where f "( ) = 0.

2 9. Given f ( ) = e π, find f '( ). 10. State the slope of the tangent to the curve = e at =. 11. Determine the horizontal asmptote for + 1 g( ) =. 1. If f ''( ) < 0, f ( ) is concave: up / down (circle one) 1. Evaluate the following limit. 1 lim If a = (-,,0), b = (,1,) and c = (5,-1,), find a) b c = b) c i a = c) a unit vector opposite to c = 15. Show a sketch of two vectors in R that illustrates. p i q > State whether the epression ( a b ) is a vector, scalar or neither. a c 17. Find a vector equation of a line through the point (p,q) and parallel to the -ais. 18. If a and b are linearl independent and ( - ) a + ( - )b = 0 then =, = 19. A direction vector of the line = - 1 is 0. If = 1 + t, = 1 - t, t ε R; are the parametric equations of a line, the smmetric equations are 1. The line r = t(a,b,c) and the plane a + b + cz = 1 are parallel / coincident / intersecting. The sstem is dependent / independent / inconsistent.. A bearing of 7, is equivalent to, using a compass bearing.

3 PART B: Answer in the space provided. Show all work for full marks. 5. Evaluate (algebraicall) each of the following limits, if the eist. a) lim b) lim [,] 6. The function: d( t) = 8t + 16, models the distance in km that a motorccle travels in t hours. Determine the motorccle s average speed from a time of hours to a time of hours and compare that to its instantaneous speed at hours. [5] 7. Given 1, f 1, 1 ( ) =, < < 1 [1,1,1] (i) a) Find the following limits if the eist. lim f ( ) + (ii) lim f ( ) b) Sketch the graph on the grid below. (iii) lim f ( ) 1 [] [1] c) Where is f ( ) discontinuous?

4 8. Differentiate each of the following. Simplif where appropriate. a) = + log ( ) b) 5 7 ( 1) g( ) = + 1 [,] [] c) = 9. Find an equation of the tangent line to + 7 = 1 at the point (1,). [5] 0. The position function of an object is: seconds. 5 d( t) = t ( t 7), t 0, where d is in metres and t is in d) When is the object stopped. Show our work. [] e) When is the acceleration positive? []

5 1. An Austin Mini and a Nissan Maima leave an intersection in Kingston at the same time. The Austin Mini, driven b Jackie, travels south at 80 km/h, and the Maima, driven b Josh, travels east at 90 km/h. At what rate is the distance between the two cars changing 0 minutes later, assuming the can continue to travel at the given initial speeds? []. Priscilla and Graham want to build a to George in the Bo. The have a square piece of cardboard that is 10 cm wide. Describe how the can build an open top bo with the cardboard and determine the dimensions of the bo that will have the largest volume. [5]. The dimensions of a conical tank is of radius m and height 6 m. Water is added to it at a rate of π m /min. Find the rate of change of the water level when the height is m. [5] 1 V = π r h

6 1. Find the points of tangenc to the curve = ln + 1 where the slope of the tangent is 0.5. [5] t 5. The percentage of population affected b a virus is given b P( t) = 100te, where t is time in weeks and P is in percent. Find the maimum percent affected, correct to decimal places. [] 5 6. Sketch the curve = and label all intercepts, maimums, minimums, and points of inflection. Evaluation will be based on the documentation of each step of our analsis. SHOW ALL WORK! [6] 7. The Side -Splitting Theorem states : The line joining the midpoints of two sides of a triangle is parallel to the third side and one-half as long the third side. Prove this Theorem using vectors. A B C

7 8. For the diagram of the rectangular prism shown, Z a) State the co-ordinates of : G F C H H O J 7 C Y A B b) WriteOJ in terms of basis vectors. X c) Calculate the angle between two bod diagonals d) State the vector projection of CJ on OA. 9. forces, 5N, 6N and 7N act on a particle which remains in a state of equilibrium. What is the angle between the 5N and 6N force?

8 5 0. A wind surfer sails from Kingston sailing club with the intention of landing due south of the club, on Wolfe Island. The normal summer wind is blowing from the S W at 0 knots. If the wind surfer can maintain a velocit of 50 knots, what direction must she sail, to guarantee arriving at her planned destination? Z X Y 1. a) A 10 kg block lies on a smooth plane inclined at 51. What force parallel to the incline would prevent the block from slipping? b) An object is dragged 6 m up a ramp under a constant force of 7 N applied at an angle of to the ramp. Calculate the work done.. Given the points A(-1,,), B(0,,5) and C(,,-) a) Find the parametric equations for the line through AB b) Show that A, B, C are not collinear. c) Calculate the area of triangle ABC.. Given the plane π 1 ; r = (, -, 1) + s(-,, 0) + t(, -6, 1) ; s,t ε R. a) Determine the Cartesian equation of the plane.

9 b) Draw a sketch of the plane.. Investigate the relationship of the lines l 1 and l given l 1 : r = (-1,1,0) + t(,,-) and l : 5 9 = = z + in terms of parallel, intersecting, coincident, or skew. 5. Use matrices to find a solution to the following sstem of equations z = + = z = 11