PART A: KNOWLEDGE Answers Only - 1 Mark Each Write the simplified answer in the box provided. Include units where appropriate. 1. For function f(x) = 2* , state a) the average rate of change from x =0 to x = 5 30 / 4 or 7.5 b) the instantaneous rate of change at x = 1 da = alna 2. State the derivative of the following functions a ) y = -3x5 - =-413 dx dy 4(3x ) = 129 b) y = vx c) y = cos(3x) d) y = 10* e) y = * N f) y = ex 2Full Solutions All work must be shown using proper form to receive full marks. 3. Evaluate the following limits, if they exist. If the limit does not exist, write DNE (3 marks each) a) lim = x24 4x-5 b) lim V(2-h)-3 X-3 2x2-4x-11 h-0 h 4. Differentiate using first principles. (4 marks) f(x) = 3x2 - 2x +4PART B: COMMUNICATION 5. Find the derivative of each of the following functions. Simplify where possible. (3 marks each) a ) f (x ) = 2x x +4 b ) 8 ( x ) = (4 x2 - 1 5 c) h(x) = 3x (52x) d) s(x) = esin(x) 6. If 7, 7, and k are the standard unit basic vectors in 3-space, determine the value of k . (7 - 3 k) + (7 -4k) . (i-4k) -8 7x-k|. (Simplify without using components). [3] 7 . Determine if the vectors * a = [4,1, -31, b = [-2,6,1] and c = [12,9, -5] are coplanar. [6]8 Given a = [2,6,3] and b = [1, -1,6] a. Calculate the angle between a and b. [3] b. If c = [1, -3, t], find t so that a and c are perpendicular. [3] PART C: THINKING 9. Determine the equation of the tangent to f(x) = 1 x =3 x2 -5 where (5 marks)10. Determine the intervals of increase/decrease for the function f (x ) = x2 -10 . (6 marks) 11. Calculate the area of the parallelogram whose adjacent sides are (4) a = [1,1, -2] and b= [3,3, -1]. 12. The vertices of a quadrilateral ABCD are A [0,3,5], B [-3, -1,17], C[4, -3,15] and D [7,1,3] Prove that ABCD is a parallelogram. [5] 6PART D: APPLICATION 13. The more often a person views a television commercial, the more effective it is. However, it costs a large amount of money to air the commercials. Television marketers have developed a function that relates the effectiveness per advertising dollar (called the efficiency, E) to the number of times the commercial is aired, n: E = 2ne-0.In Determine the v alue of n that maximizes the efficiency. (3 marks)14. A plastic swimming pool in the shape of a cylinder (with no top) is to be constructed such that the volume is 15m3. Find the dimensions of the pool that minimizes the amount of plastic needed (i.e. the surface area). (6 marks) 15. The speed of a plane is 360 km/hr and its heading is 120. A wind of 30km/hr is blowing on a bearing of 040. Determine the plane's resultant velocity relative to the ground. [6] 816. Find the torque produced by a cyclist exerting a force of F = [70,30,150]/ on the shaft- pedal d = [11,15,12]cm long. torque = Fxd [4] 17. Determine the following: a. Determine the vector equation of the plane that contains the following two lines L1: r =(4, -3,5) + t(2,0,3), tER and L2: r=(4, -3,5) + s(5,1, -1), SER [3] b. Determine the corresponding Cartesian equation. [5] 9