Limits, Rates of Change, and Tangent Lines

Transcription

1 Limits, Rates of Change, and Tangent Lines jensenrj July 2, 2018 Contents 1 What is Calculus? 1 2 Velocity Average Velocity Instantaneous Velocity Other Rates of Change 7 1 What is Calculus? Calculus is the mathematics of change. A usual first semester Calculus course will include ideas such as limits, differentiation, the derivative and its applications, and parametric equations. Calculus enables us to solve more problems than we could using only algebra. Without Calculus: Value of f(x) at x = c The slope of a line Secant line to a curve Average rate of change Curvature of a circle Height of a curve at a point Direction of motion along a strait line 1

2 Area of a rectangle Work done by a constant force Center of a rectangle Length of a line With Calculus: Limit of f(x) as x goes to c The slope of a curve Tangent line to a curve Instantaneous rate of change Curvature of a curve Maximum height of a curve on an interval Direction of motion along a curve Area of a region Work done by a variable force Center of a region Length of a curve 2 Velocity A familiar example of when calculus can be used is with velocity. When we speak of the velocity of an object, we usually mean instantaneous velocity, or the speed and direction of the object at a given moment in time. Before we work instantaneous velocity however, we should talk about average velocity. 2

3 2.1 Average Velocity Definition 1. Average Velocity is the change in position divided by the change in time. Average velocity works well when something is moving in a strait line, at a constant rate. Consider the following example. [Trip to Dalls] Suppose you drive your car from Nacogdoches to Dallas. It takes you 3 hours to get there, and it is 165 miles from Nacogdoches to Dallas. What was your average velocity? 165 miles = 55 mph 3 hours So the average velocity is 55 mph (Actually this is the speed. Recall that velocity is speed with a direction, so the velocity would be 55 mph northwest). At any point in time you might be going faster or slower, but the average velocity is still 55 mph. We generally let t denote time, and s(t) the position of a moving object at the time t. So in the above example, t = 1.5 would mean the time was 1.5 hours, and s(1.5) would be the position at 1.5 hours. We use to denote a change in something, so t = t f t i = 3 0 = 3 = the change in the time interval (t f is the final time, and t i is the initial time). Definition 2. With this new notation, we can define the average velocity to be average velocity = s t = s(t f ) s(t i ). (1) t f t i Galileo found that the position of a falling object is given by s(t) = 4.9t 2 m. (2) Suppose you drop a rock from a cliff. Use equation (2) and the definition of average velocity to find the average velocity over the time interval [1.5, 2.0]. 3

4 average velocity = s t = s(t f ) s(t i ) t f t i s(2.0) s(1.5) = = 4.9(2.0)2 4.9(1.5) 2 =.5 4.9(4 2.25).5 average velocity = m/s. Sketch a graph of the position of the rock given in the falling rock example. Then graph the average velocity you found in the falling rock example. We first make a table of points: (t, s(t)) (0.0, 0.0) (0.5, 1.225) (1.0, 4.9) (1.5, ) (2.0, 19.6) (2.5, ) (3.0, 44.1) Next we plot these points, and sketch in the parabola. Finally we plot the line with slope m = through the points (1.5, s(1.5)) and (2.0, s(2.0)). We can then conclude that the average velocity over a time interval is the slope of the secant line through the corresponding points. 4

5 2.2 Instantaneous Velocity Instantaneous velocity or simply velocity is the speed and direction of an object at a given moment in time. Use the information in falling rock to find the average velocity for the [1.5, 2.0] time intervals: [1.5, 1.7] [1.5, 1.6] [1.5, 1.55] [1.5, 1.51] [1.5, ] Using the definition of average velocity and s(t) as given in the falling rock example, we can compute the following table: 5

6 Time Interval Average Velocity [1.5, 2.0] [1.5, 1.7] [1.5, 1.6] [1.5, 1.55] [1.5, 1.51] [1.5, ] We then can estimate that the instantaneous velocity at t = 1.5 seconds is 14.7 m/s. We say that the average velocity converges to the instantaneous velocity, or that the instantaneous velocity is the limit of the average velocity as the time interval goes to zero. Graph the secant lines defined by the average velocities found in the average velocities example. As before, we graph lines whose slopes are given by the average velocity, and go through the respective points. We can then see that the instantaneous velocity at time t is the slope of the tangent line to the position graph at time t. 6

7 3 Other Rates of Change Velocity is an example of a rate of change. We can extend the idea of average velocity to average rate of change. Definition 3. If f(x) is a function, we define the average rate of change of f(x) with respect to x over [x 0, x 1 ] by average rate of change = f x = f(x 1) f(x 0 ). (3) Definition 4. The instantaneous rate of change of a function f(x) is the limit of the average rate of change as x 1 goes to x 0. We estimate the instantaneous rate of change by smaller and smaller intervals. Let f(x) = x 3 + 5x 2 + x. Estimate the instantaneous rate of change of f(x) at x = 1. We start by graphing the function f(x). Next we make a table of the aver- Interval Average Rate of Change age rates of change for smaller and smaller intervals. [1, 1.1] [1, 1.01] [1, 1.001] [1, ] We then estimate that the instantaneous rate of change of f(x) at x = 1 is 14. Let y = f(x) = mx + b. Find the average rate of change on the interval [x 0, x 1 ]. average rate of change = f x = f(x 1) f(x 0 ) = (mx 1 + b) (mx 0 + b) = mx 1 + b mx 0 b = m( ) average rate of change = m 7