Unit 1: Rates of Change Problem Set Instructions There are 10 questions in this assessment. Show all work for each question. Read each question carefully! Be sure to use the notation and methods that have been demonstrated in the course materials and to answer the question that is being asked. 1. Determine the average rate of change of y with respect to x in the function y = 3x* 5x2 + 4 over the interval [2, 5]. 2. Given the function f(x) = 3x + x 5. a) Use the secant method with at least 3 secants to determine the slope of the tangent whenx = 0. b) Use first principles (i.e. an appropriate limit) to determine the value of the derivative when x = 0. c) What did you notice? 3. Using complete sentences and your own words, explain the difference between a secant line and a tangent line. How do they relate to the rate of change of a function? Include a sketch of each type of line in your solution. 4. The path of a baseball relative to the ground can be modelled by the function d(t) = t* + 5t + 2, where d(t) represents the height of the ball in metres and represents the time in seconds. a) Find the average rate of change of the ball between 2 and 4 seconds. b) Using the secant method with at least 3 secants, find the instantaneous rate of change at 3 seconds. 5. Evaluate lim(3x> + 5x 10) and interpret the meaning of this value. x3 6. Evaluate the following limits. o x2-25 a) lim -3 X+5 . Xt Tx+1 b) Im xc0 6x4 + 3x2 + 5x 16+h -4 c) limm h0 h 3 - 27 y3 4y2 28y + 48 2 . 14+h e) lim h0 7. Find the slope of the tangent line at point (-1, 3) on the curve f(x) = 4x + x using first principles. 8. Find the derivative of the function f(x) = x*=3x+5 using first principles. 9. Using first principles, determine the equation of the tangent line at the point (1, -1) on the curve f(x) = x*> 3x + 1. 10. At what point on the parabolay = x? 3xis the tangent line parallel to the line y = 4x 3? Use first principles in your solution