2.1 Derivatives and Rates of Change

Transcription

1 2.1 Derivatives and Rates of Change In this chapter we study a special type of limit, called a derivative, that occurs when we want to find a slope of a tangent line, or a velocity, or any instantaneous rate of change. A line which is tangent to a curve is a line that touches the curve in one point, as illustrated in the figure below, and satisfies the following definition: Contrast the tangent line to a secant line, which intersects a curve more than once. Note that in order to calculate a slope, we need two points, so it is easy to calculate the slope of a secant line by finding the slope between two points, say (x, f(x)) and (a, f(a)), on the curve. If we let those two points get closer and closer together, to the point that we are taking the limit of the difference quotient as x a, that gives us the definition of the slope of the tangent line (above). This is shown in the figures below (from p. 74). 1

2 If we want to write the equation of a tangent line (or any line), we must either have the slope and the y intercept, the slope and a point, or two points. When attempting to write the equation of a tangent line, we will first calculate the slope at the given point by finding the limit stated above (or an alternate version) and then use the given point (along with the slope) to determine the equation, using either the pointslope or the slope intercept form of a linear equation. y y 1 = m(x x 1 ) or y = mx + b Ex: Find the slope of the tangent line to the graph of f(x) = x 2 at the point (3,9) using the limit below. Note in this case that a = 3. Now find the equation of the tangent line at the point (3,9). 2

3 The alternate form of the equation of the slope of the tangent line at a point (a, f(a)) is as follows: Ex: Use the alternate form to find the equation of the tangent line to the graph of f(x) = x 2 at the point ( 2,4). Note that a = 2. 3

4 Calculate the slope of the tangent line to the graph of f(x) = 2x 2 5 at the point (1, 3). Note that a = 1. 4

5 Determine the slope of the tangent line to the graph of at the point ( 1, 4). Determine the equation of the tangent line to the graph of at the point (2, 1). 5

6 Recall that average rate of change means the same thing as slope, which is. Thus, average velocity is calculated as. Specifically, the average velocity over the time interval from t = a to t = a + h is If we compute the average velocities over shorter and shorter time intervals, we can approach the instantaneous velocity, which is 6

7 Example: The displacement or position (in meters) of a particle moving in a straight line is given by s = t 2 8t + 18, where t is measured in seconds. Find the average velocity over the time intervals [3, 4] and [4, 5]. Find the instantaneous velocity when t = 4. (Note: a = 4.) 7

8 We have seen that the same type of limit arises in finding the slope of the tangent line or the velocity of an object. In fact, limits of the form arise whenever we calculate an instantaneous rate of change in many different contexts. Since this occurs so widely, it is given a special name and notation. Definition: The derivative of a function f at a number a, denoted by f '(a), is The derivative f '(a) is the instantaneous rate of change of y = f(x) with respect to x when x = a. Example: Find f '(a) if f(x) = x 2 5x

9 Thinking of the derivative f '(a) as the slope of the tangent line to the curve y = f(x) at a makes it clear that the derivative takes on different values at different values of a along the curve. Where the curve is steeper, the derivative is larger, where the curve is less steep, the derivative is smaller, and where the curve has turning points (maxima or minima), the derivative is 0. Use the graph of f below to compare the sizes of the following derivatives: f '( 4), f '( 2), f '(0), f '(2), f '(4) 9

10 In the context of y = f(t) as a position function, note that f '(a) is the velocity of the particle or object at time t = a. The speed of the particle or object is the absolute value of the velocity, or f '(a). Thus where the curve of the position function is steeper, the derivative (velocity) is greater, where the curve is less steep, the derivative (velocity) is smaller, and where the curve has turning points (maxima or minima), the derivative (velocity) is 0. Use the graph of the position function f(t) below to answer the following questions: 1) What was the initial velocity? 2) Was the object moving faster at 4 or 6 seconds? 3) What is occurring between 7 and 8 seconds? 4) Is the object always moving in the same direction? 10

11 The next example relies not on a formula or graph, but on a table of values. The table given shows the number N of locations of a popular coffeehouse chain for particular years. 11