1. Differentiate each of the following using the quotient or the chain rule. 2x - 1 Vxx a. f(x) = (5-3x)' b. f (x) = - x2 + 3x +1 C. y = - x5-2x3-7 d. g(x) = Vx7 - 3x4 + 15x e . Vx + 2 y= V Vx - 2 f. y = (t + Vtz + +4) 20 g. h (x ) = - x 6 x5-10 h . f ( x ) = -x x +1 i. y = * 4 - 2 x 3 + 1 2. Use implicit differentiation for each of the following. a. x2 - y2 = 1 b. x2 + xy + y2 = 1 Vx + Vy =5 d. 2x - = y xty 3. Find the equation of the tangent to the curve at the given point. 3x a. y = = 5 x2 + 1 1 X = 2 b. y = (x2 - 3)8, x = 2 C. y2 = 2xy - 3, (2, 3) d. x4 + y4 = 17, (2, 1) 4. Given y = (x2+3x) 5' a determine the derivative by using the: quotient rule chain rule Which method in part a) is more efficient? Justify your answer. C. Which method in part a) do you prefer? Explain. 5. The red squirrel population, p, in a neighbourhood park can be modelled by the function p(t) = 420t - t3 , where t is time, in years. Determine the rate of growth of the squirrel population at t = 5 years. b . When will the population reach 60 squirrels? 2 ? What is the instantaneous rate of change of the population at the time in b)? When is the instantaneous rate of change of the squirrel population approximately 7 squirrels per year? 6. The curve with the equation = x3 + =y3 = 5 is called an asteroid and is shown in the figure below. Calculus and Vectors Lesson 5 a. Find y'. b. Find the equation of the tangent line to the asteroid at the point (27, c. Find the points on the asteroid where the tangent line has a slope of 1