Determining Average and Instantaneous Rates of Change

Transcription

1 MHF 4UI Unit 9 Day 1 Determining Average and Instantaneous Rates of Change From Data: During the 1997 World Championships in Athens, Greece, Maurice Greene and Donovan Bailey ran a 100 m race. The graph and table below shows Donovan Bailey s performance during this 100 m race. Donovan Bailey s Performance Time (s) Distance (m) a) Calculate Donovan Bailey s average velocity, accurate to two decimal places, for this 100 m sprint. change in distance average velocity s change in time t avg b) Draw a line from (0,0) to (9.77,100) on the graph above. A line passing through at least two different points on a curve is called a secant. c) Explain the relationship between your answer to a) and the slope of the secant. 2. a) Draw the secants from (0,0) to (5.44,50) and from (5.44,50) to (9.77,100). b) Calculate the average velocity, accurate to two decimal places, represented by each of the two secants from a). i) ii)

2 MHF 4UI Unit 9 Day 1 3. a) Estimate the instantaneous velocity at t 6 s. b) Explain why your answer to part a) is a good approximation. c) Plot a point on the curve at 6 seconds. Draw a line that passes through this point but does not pass through the curve again. This line is called a tangent to the curve Use the algebraic model s(t) = 0.28t + 8.0t to approximate the instantaneous velocity of Donavan Bailey at t 6 s by completing the chart below. Point A Point B Average Velocity t 6 t 7 s s Repeat question #4 above three more times, each time choosing a point closer to t Calculate the average velocities over these intervals. Point A Point B Average Velocity t 6 t 6 s. s s Point A Point B Average Velocity t 6 t s s Point A Point B Average Velocity t 6 t s s

3 MHF 4UI Unit 9 Day 1 5. Use the calculations from question #4 above to estimate (accurate to two decimal places) the instantaneous velocity at t 6s. 6. Draw the secants on the graph that correspond with the three average velocities calculated above in question #4. How do the secants compare to the tangent drawn in question #3 c)? 120t A Bunsen burner is used to heat water in a beaker. The equation T t, 0 t 80 t 20 expresses the temperature, T, in degrees Celsius, as a function of time, t in seconds. (from Addison-Wesley Advanced Functions and Introductory Calculus) a) Calculate the average rate of change in temperature from t 30 s to t 40 s. (accurate to four decimal places) b) Calculate the average rate of change in temperature from t 33 s to t 37 s four decimal places). (accurate to c) Estimate the instantaneous rate of change in temperature at t 35 s. (accurate three decimal places)

4 MHF 4UI Unit 9 Day 2 Estimating Instantaneous Rates of Change x 4 1. Given the function f(x) 2(3) 11, a) determine the average rate of change of the function f(x) from x = -4 to x = -3. b) estimate the instantaneous rate of change of f(x) at x = -4, accurate to 3 decimal places.

5 MHF 4UI Unit 9 Day 2 2. Given the function f(x) = sinx, a) sketch the function b) is the function increasing or decreasing over the interval x (0, ). π 2 c) determine the average rate of change (3 decimals) of the function f(x) from π x to 6 π x. 4 d) estimate the instantaneous rate of change (3 decimals) of f(x) at x. π 6

6 MHF 4UI Unit 9 Day 2 π 3. Given the function f(x) 2cos(x ) 1, a) sketch the function 3 π 4π b) is the function increasing or decreasing over the interval x,? 3 3 c) determine the average rate of change (3 decimals) of the function f(x) from 7π x to 12 2π x. 3 d) estimate the instantaneous rate of change (3 decimals) of f(x) at x. 7π 12

7 MHF 4UI Unit 9 Day 3 Secants to Tangents A secant is a line that intersects a curve. A tangent is a line that most resembles the curve near that point. It but does not the curve near that point Find the slope of the tangent line to the parabola y x at the point P(3, 9). Note: a tangent line has only one ordered pair. We need two ordered pairs to us the formula y m x 2 2 y 1 x 1 2 a) Let Q (, )be a point on y x close to P m PQ

8 MHF 4UI Unit 9 Day 3 b) Let s try another point even closer to P. Try Q (, ) m PQ c) Let s try another point close to P, on the other side. Q (, ) m PQ d) Estimate the value of the slope of tangent at x = 3. Justify your answer: In general, if we let Q be a point on then The coordinates for Q are: The slope of the secant PQ is given by: 2 y x close to P and let h be a very small number, NOTE: Point Q cannot be exactly 0 point P, or m! But the 0 closer Q is to P, the more accurate our tangent slope calculation is.

9 MHF 4UI Unit 9 Day 3 New notation: The limiting function f(x) is value as h approaches 0 of the written as: The slope of the tangent line at point P is the slope of the secant line PQ as Q moves closer and closer to P. m tangent So, the slope of the tangent at x = 3 on the graph 2 y x is: m = (because as h 0, then 6 h, )

10 MHF 4UI Unit 9 Day 3 2. Find the slope of the tangent to y = x 2 at P (3, 9) using proper form. Now, let s go over expected proper form using the new notation and concepts. 3. a) Calculate the slope of the tangent line to y = x 2 + 3x at P (2, 10). b) Determine the equation of the tangent line, in standard form.

11 MHF 4UI Unit 9 Day 4 Using Limits to Find Slopes of Tangents As Q approaches P (as Q P), the slope of the secant is a very good estimate of the slope of the tangent. lim (slope of secant) = slope of tangent Q P Let s create a formula to recap the work we did yesterday. slope of tangent at x = a m = y x m = Formula 1 (first principles formula) OR, alternate way to approach the same question: y m = x Formula 2

12 MHF 4UI Unit 9 Day 4 1. Determine the equation of the tangent to y = x 2 + 3x + 4 at x = -2. (*always standard form) Formula 1: Formula 2:

13 MHF 4UI Unit 9 Day 4 2. Determine the equation of the tangent to 1 y at x = 3. x Formula 1: Formula 2:

14 MHF 4UI Unit 9 Day 5 More Tangent Lines 1. Determine the equation of the tangent to the curve y x 2 at the point P(-1, 1). Formula 1: Formula 2:

15 MHF 4UI Unit 9 Day 5 2. Determine the equation of the tangent to 1 y at x = 4. x 3 Formula 1: Formula 2:

16 MHF 4UI Unit 9 Day 6 Rates of Change Formula 1: Formula 2: f(a h) f(a) m lim h 0 h f(x) f(a) m lim x a x - a The rate of change in position, s(t), with respect to time = velocity. v(t ) 0 s(t lim Δt 0 0 Δt) s(t Δt 0 ) s(t) s(t0 ) v(t0 ) lim t t0 t - t 0 t = change in time, as time approaches 0 1. A pebble is dropped from a cliff. After t seconds, it is s meters above the ground, where s(t) = 80 5t 2. Determine the instantaneous velocity when t = 1 s. Formula 1: Formula 2:

17 MHF 4UI Unit 9 Day 6 2. The height, s, in meters, of a toy rocket is given by s(t) = -4.9t t + 1, where t is time in seconds. Determine the velocity at 4 seconds. Formula 1: Formula 2:

18 MHF 4UI Unit 9 Day 7 General Expressions For Slope 1. For each curve, i) determine and expression for the slope of the tangent at the general point whose x-coordinate is a. ii) determine the slopes of the tangents at the points whose x-coordinates are: -2, -1, 0, 1, 2 a) y = x 2 b) y = x 3 Solutions 1a) Formula 1: Formula 2:

19 MHF 4UI Unit 9 Day 7 1b) Formula 1: Formula 2: 2. At what point on the curve y = x 2 is the tangent parallel to the line y = -14x + 5?

20 MHF 4UI Unit 9 Day 8 Rates of Change Recap 2 For questions 1 to 3, f x 4x 5x 7 : 1. Determine the average rate of change of fx from x = 4 to x = 6. 2 For questions 4 to 6, s t 0.28t 8t 2.54 : 4. Find the average rate of change in position with respect to time from t = 1 to t = Estimate the instantaneous rate of change at x = Estimate the instantaneous rate of change in position with respect to time at t = Determine the slope of the tangent to fx at x = Determine the velocity at t = 3.