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C 19. Prove that the tangent to the curve y x 2x 2 x 4 at point (1, 0) is also tangent to the curve at (1, 2). 20. For any point (x, y) on the graph of y 3x 2 15x 3, determine the coordinates of the point on y 2x 2 5x 4 such that the tangents at the two points are parallel. 21. Show that f (x) 8x 3 7x 5 has no tangent line with slope 5. 22. Show that the x- and y-intercepts for any tangent to the curve y 16 8x x have a sum of 16. ADDITIONAL ACHIEVEMENT CHART QUESTIONS Knowledge and Understanding: Consider the function f (x) x 4 7x 12. (a) Write the equation of the tangent line at x 1. (b) Find the coordinates of the point on the function so that the tangent line is perpendicular to x 25y 175 0. Application: An ant colony was treated with an insecticide and the number of survivors, A, in hundreds at t hours is A(t) t 3 5t 750. (a) Find A′(t). (b) Find the rate of change of the number of living ants in the colony at 10 h. (c) How many ants were in the colony before it was treated with the insecticide? (d) How many hours after the insecticide was applied were no ants remaining in the colony? Thinking, Inquiry, Problem Solving: Suppose that f (x) is a quadratic polynomial function. Where would you find a horizontal tangent line? a vertical tangent line? Communication: You are writing a self-help manual in calculus. Write the steps for helping readers find the equation of the line that is tangent to the graph of g(x) x 4 2x 3 5x 2 where x 2. Explain each step. The Chapter Problem Average Salaries in Professional Sports Apply what you learned to answer this question about The Chapter Problem on page 168. CP9. Use the cubic regression equations you created for question CP7. Then apply the differentiation rules to verify the derivatives you found for question CP7. 232 CHAPTER 3 RATES OF CHANGE IN POLYNOMIAL FUNCTION MODELS
3.7 Polynomial Function Models and the First Derivative SETTING THE STAGE EXAMINING THE CONCEPT Displacement f(t) secant s = f(t) tangent Time (s) Displacement versus time This chapter has discussed using the derivative to determine the instantaneous rate of change of a quantity. The derivative has wide application in fields such as economics, science, engineering, and the social sciences. Here is an example: Suppose a rock is thrown into the air from a bridge 15 m above the water. As it rises and then falls, its height above the water is a function of the time since it was thrown. The height of the rock, in metres, above the water at t seconds can be modelled by the function h(t) 4.9t 2 12t 15. What is the velocity of the rock when it enters the water? In this section we will examine applications of the first derivative of a polynomial model. Polynomial Models Involving Velocity and Speed In many situations, an object’s position, s, can be described by a function of time, s f (t). Average velocity is defined as the rate of change of displacement over an interval of time. Instantaneous velocity is the rate of change of displacement at a specific point in time. On a displacement-time graph, the slope of a secant represents average velocity, while the slope of a tangent represents instantaneous velocity. t average velocity ∆ ∆t instantaneous velocity lim s ∆ s ∆t → 0 ∆t lim ∆t → 0 lim h → 0 f ′(t) As a result, the derivative of the position function, s′ f ′(t), represents the instantaneous velocity of the object at time t. So s v(t) d dt s′ f ′(t) f (t ∆t) f (t) ∆t f (t h) f (t) h 3.7 POLYNOMIAL FUNCTION MODELS AND THE FIRST DERIVATIVE 233
Page 1 and 2: 3.6 Finding Some Shortcuts — The
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