1. Which of the following functions have a derivative of zero? a) y=9.8 b) y = 11 5 dy=-4+x dj y = - x e) y = v7 f) y= x gly= h) y= -2.8n 2. Determine the slope of the tangent to the graph of each function at the indicated value. a) y=6, x = 12 bj f(x) = 2x', x = v3 d g(x) = - 3 -,x=4 d) bit) = -4.91, 1= 3.5 4 f) y = 1 3. Determine the point at which the slope of the tangent to each parabola is zero. i y= 6x2 - 3x + 4 in y= -x' + 5x - 1 4 iii) y = =x3 + 2x + 3x3 +2x+3 4. Simplify, and then differentiate. a) f(x) = 10x4 - 6x3 2x2 b) g(x) = (3x + 4)(2x - 1) () p(x) = = _x - 4x6 + 2x3 4x3 d) f(x) = (5x+2)25. A skydiver jumps out of a plane that is flying 2500 m above the ground. The skydiver's height above the ground, in metres, after * seconds is bir) = 2500 - 4.912. aj Determine the rate of change of the height of the skydiver after 5 s. b) The skydiver's parachute opens at 1000 m above the ground. After how many seconds does this happen? cJ What is the rate of change of the height of the skydiver at the time found in part b)?\f9) y = (5x4 - 3x2 - 1)(-5x2 + 3) 10) f(x) = (-10x2 - 7Vx2 + 9)(2x3 +4) 11) y= (5+ 3x 2)(4x5 + 6x3 + 10) 12) y = (-6.x4 + 2+6.x 4)(6x4 + 7) 13) f(x) = -7x4 + 10x5 + 8/(x2 + 10)Product Kule Practice Questions 1. Differentiate each function using the two 3. Differentiate. different methods presented below. Compare a) M(u) = (1-4#-)(# + 2) your answers in each case. b) g(x) = (-x+3)(x -10) i) Expand and simplify each binomial product and then differentiate. () p(m) = (5n +1)(-m2 +3) ii) Apply the product rule and then simplify. a) f(x) = (x +4)(2x -1) b) b(x) = (5x - 3)(1 -2x) 4. The derivative of the function b(x) = f(x)g(x) 5. Determine f'(-2) for each function. is given in the form b'(x) = f(x)g(x) + f(x)g'(x). Determine f(x) and g(x) for each derivative. a) f(x) = (x2 - 2x)(3x+1) a) b'(x) = (10x)(21 - 3x) + (5x2 +7)(-3) b) f (xx) = (1-x3)(-x2+2) b) b'(x) =(-12x2 + 8)(2x2 -4x) +(-4x3 + 8x)(4x - 4) () b'(x) =(6x2 - 1)(0.5x2 + x) +(2x3 -x)(x+1) 6. Determine the equation of the tangent to each 7. Determine the point(s) on each curve that curve at the indicated value. correspond to the given slope of the tangent. a) f(x) = (x2 -3)(x3 + 1), x=-4 al y= (-4x + 3)(x + 3), m=0 b) g(x) = (2x2 -1)(-x2 + 3), x =2 b) y= (5x + 7)(2x - 9), m = () y= (2x - 1)(-4 +x2), m= 3 14. The gas tank of a parked pickup truck develops a leak. The amount of gas, in litres, remaining in the tank after t hours is represented by the function Vin) = 90(1 - ,0515 18. a) How much gas was in the tank when the leak developed? b) How fast is the gas leaking from the tank after 12 h?