2.4 Rates of Change and Tangent Lines Pages 87-93
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1 2.4 Rates of Change and Tangent Lines Pages Average rate of change the amount of change divided by the time it takes. EXAMPLE 1 Finding Average Rate of Change Page 87 Find the average rate of change of over the time interval [1, 3]. 1
2 secant line a line which intersects a graph at two points. f(x) average rate of change = slope of secant line 2
3 For example 1, the equation of the secant line: 3
4 Find the average rate of change (slope of the secant line) for the following functions for the given interval. Determine the equation of the secant line:
5 Determining instantaneous rates of change from average rates of change: P (1, 0) 5
6 x values approaching P
7 7
8 Tangent line a line that touches the curve in only one point. slope of the tangent line is the same as the instantaneous rate of change Problem: To find the slope of the tangent line, h equals 0, and, as we all know, we cannot divide by 0. Solution: Determine the limit of the slopes of the secant lines as they approach the tangent line. instantaneous rate of change = ** a represents a certain value of x; a specific point 8
9 EXAMPLE 3 Finding the Slope and the Tangent Line Page 89 Find the slope of the parabola at the point (2, 4). Write an equation for the tangent to the parabola at this point. 9
10 10
11 Example: Determine the equation of the tangent line to the curve at the point (1,0). 11
12 Slope of a curve at a point for the function y = f(x) at the point P(a, f(a)) is the slope of the tangent line at that point and is the number provided the limit exists. 12
13 All of the following mean the same thing: the slope of y = f(x) at x = a. the slope of the tangent to y = f(x) at x = a the instantaneous rate of change of f(x) with respect to x at x = a 13
14 normal line the line perpendicular to the tangent line at that point. If two lines are perpendicular, their slopes are negative reciprocals of each other. Examples: 14
15 EXAMPLE 5 Finding a Normal Line Page 91 Write an equation for the normal line to the curve at x = 1. 15
16 Application of Rates of Change: average speed found by dividing the distance covered by the elapsed time. Example: If you drive 200 km and it takes 4 hours: average speed = 16
17 More technically, if y = f(t) is a distance or a position function, then average speed is the ratio of the change in position or distance to the change in time, given by the following formula: average speed = where h = elapsed time t = initial time 17
18 EXAMPLE 1 Finding Average Speed Page 59 A rock breaks loose from the top of a tall cliff. The distance which the rock falls is given by the function where y is measured in feet and t is in seconds. What is the average speed during the fist two seconds of fall? 18
19 Instantaneous speed the speed of a moving object at a given instant in time (h=0). Issue: How to compute, since elapsed time is zero. Solution: Use a mathematical concept referred to as a limit. 19
20 EXAMPLE 2 Finding Instantaneous Speed Page 60 Investigation using average speed formula: average speed = Length of Time Interval (h) Average Speed (ft/sec)
21 Algebraically: instantaneous speed= 21
22 EXAMPLE 6 Finding Instantaneous rate of Change Page 91 For the function determine at t = 2. Interpret the answer if f(t) represents a position function in feet of an object at time t seconds. 22
23 EXAMPLE 7 Investigating Free Fall Page 92 Find the speed of the falling object at t = 1 sec for at x = 1. 23
24 More Examples: 1. For : a) Find the slope of the curve at x = 2 b) Find the equation of the tangent line. c) Find the equation of the normal line. 24
25 2. Determine the instantaneous rate of change of the position function, in meters, at 2 seconds. 25
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AP Calculus AB/IB Math SL Unit : Limits and Continuity Name: Block: Date:. A bungee jumper dives from a tower at time t = 0. Her height h (in feet) at time t (in seconds) is given by the graph below. In