DO ALL QUESTIONS (DUE DATE: APRIL 2, 2024) [Question 1] a) Evaluate the average rate of change for each function over the given interval: i. f(x) = 4x+5, from x = 3 to x = 5. (2 marks) ii. f(x) = x2 +3x+2 from x =2 to x =4 (2 marks) b) Find the gradient of the curve f (x) = x2 -4x+3 at x =3 (2 marks) c) Use the quotient rule to differentiate the function f(x) = 2x-3 (2 marks) d) Rewrite the function as f (x) =(2x-3)x 3 and differentiate using the Product rule. (2 marks) e) Show that your answer in parts (a) and (b) are the same. (2 marks) f) Given the function f(x) = 12x3 - 2x, find f' (x) when x = 3 using First Principles and use the result to indicate if f(x) is increasing, decreasing or neither. (8 marks) TOTAL: 20 MARKS Note: Average rate of change Ay _ 12 - VI _ f(x2) - f(x, ) Ax X2 - X1 *2 - X 1 [Question 2] Differentiate each of the following functions with respect to x 2 a) y= (1 mark) 3x3 b ) f (x ) = 2x (2 marks) 5 -+5x2 + c) f (x) = (2x-3)3 (use Chain Rule) (3 marks) d) f(x) = (1+x)(2-3x2) (use Product Rule) (4 marks) e) f(x) = (3-5x) (use Quotient Rule) (4 marks) (7x-18)f) Given the curve f(x) = x3 -12x2 +36x : Find the coordinates of the stationary points and determine the nature of them. (6 marks) TOTAL: 20 MARKS [Question 3] Given that the demand function for a monopolist's product is p =2x2 +10x+1000 and the average-cost function is C =2x2 +36x-1600-20 -, where x is number of units, and both p and X C are expressed in dollars per unit. i. Determine the level output at which profit is maximized. (5 marks) ii. Determine price at which maximum profit occurs. (5 marks) iii. Determine the maximum profit. (2 marks) If the average-cost function for a manufacturer is given by C =4x-10+ 4000 X where q is the number of pens manufactured and C is expressed in dollars per pen. (a) Find the marginal-cost function. (4 marks) (b ) Find the marginal cost when x =20 and interpret your result. (4 marks) TOTAL: 20 MARKS