Procedural Skills Learning Objectives 1. Given a function and a point, sketch the corresponding tangent line. 2. Use the tangent line to estimate the value of the derivative at a point. 3. Recognize keywords in a context that correspond to the derivative. 4. What is a general procedure to sketch the derivative of the plot of a function? Interpretation Skills 1. Compare and contrast secant lines and tangent lines. 2. What is the relationship between the tangent line of a function at a point, and derivative of a function at a point? 3. Explain the difference between the derivative at a point and the AROC over an interval. The instantaneous rate of change of a function f(x) at a point x = a is defined to be the limit of the average rate of change of f(x) about some point a as x 0. Instantaneous rate of change at a point The instantaneous rate of change of a function f(x) at point x = a is defined as f f(a + h) f(a) (a) := lim. h 0 h The term f (a) is read as the derivative of f(x) at the point x = a. a a In the last lecture we solved problems of this form. This quantity represents the limit of the AROC of f(x) over the interval [a, a + h] as h approaches zero. The three dashed lines in the figure correspond to the AROC of f(x) over the intervals [0, 6], [2, 6], and [5, 6]. As our interval gets smaller and smaller we see the secant lines approach the solid line. This is a graphical interpretation of the relationship of AROC and the derivative. The solid line is known as the tangent line of f(x) at x = 6. The slope of the tangent line is the instantaneous rate of change of f(x) or, derivative of f(x) at x = 6, denoted f (6). This means we can estimate the value of the derivative of a function at a point by drawing the tangent line and determining its slope. 38
Example 1 The function g(t), given below, gives the population of a bacteria t minutes after an infection. Estimate g (5) and interpret your answer. We will use the fact that the slope of the tangent line of g(t) at t = 5 is equivalent to g (5). First we sketch the tangent line to get 1 Now we can simply choose any two points and calculate the slope. The line goes through the points (4, 0) and (7, 450) so we have g (5) 450 0 7 4 = 150. The units of our calculation are helpful in understanding what this value represents. If we include units we have g (5) (450 0) bacteria (7 4) minute = 150 bacteria minute. So this means that 5 minutes into the infection the bacteria is growing, indicated by a positive value, at a rate of 150 bacteria per minute. Our once primitive ideas about rate of change are evolving! 1 To sketch the tangent line take a very small interval containing the point of interest and sketch the corresponding secant line. 39
1. Use the figure below to answer the following questions. (a) Draw the tangent lines of f(x) at x = 2, x = 0, and x = 1.1. (b) Estimate f (0). (c) Determine the correct order for the derivatives f ( 2), f (0), and f (1.1). 2. Given the plot below, use a tangent line to approximate the derivative of g(t) at t = 2. 40
The Derivative Function If we take the derivative of a function f(x) at all points in the domain we see that the derivative is a function itself. One way to understand the derivative as a function is to plot the derivative. That is, take a sample of points in the domain, determine the derivative at those points, and plot the results. We ll assume the know that the derivative function shouldn t have any jumps or gaps in its graph so we can connect the dots so to speak. 3. Use the plot below to very roughly estimate the value of the derivative at each of the given points, then plot the graph of f (x). x 1 2 4 5.5 7 f (x) 4. What happens to the value of the derivative as the f(x) flattens out? 5. On which intervals is the the derivative positive? Negative? Tips for sketching the derivative function f (x) given the function f(x). Find where the derivatives are zero. Determine intervals where f(x) is increasing/decreasing, f (x) will be positive/negative on these intervals. Estimate the instantaneous rate of change at select points. 41
By considering the connection between a function and its derivative we can make the following generalizations.(answer increasing, decreasing, constant.) If f > 0 everywhere on an interval, then f is If f < 0 everywhere on an interval, then f is If f = 0 everywhere on an interval, then f is over that interval. over that interval. over that interval. 6. Given the plot of each of the functions below sketch their derivatives. (a) (b) (c) 42
7. Given that f(x) = x 2 to determine whether each of the quantities is positive, negative, or zero. 2 (a) f (1) (b) f ( 1) (c) f (2) (d) f (0) 8. The total acreage of farms from 1980 and 2000 is given by the following table. Estimate the instantaneous rate of change of the area of farm land in the year 1990. Explain your estimation and why it is a good approximation of the instantaneous rate of change. Year 1980 1985 1990 1995 2000 Farm land (million acres) 1039 1012 987 963 945 9. Let f(x) = x 2, estimate the value of f (2.3) numerically. 2 Hint: Plot the function! 43
10. Let g(x) = 2x 2 5x. (a) Use the definition of the limit to determine g (x). The limit definition is give below. f f(x + h) f(x) (x) = lim. h 0 h (b) Use your answer above to determine g (3). Reasoning Exercises 1. What is the derivative of a linear function? Why? 2. What is the derivative of any constant function. Why? 3. How is the derivative of a function f(x) at a x = a point related to the AROC of f(x) function containing the point x = a 4. True or False If f (x) = 0 at the point x = a, then f(x) must have either a maximum or a minimum at that point. 5. True or False If a function f(x) attains a maximum or a minimum at x = a, then f (a) = 0. 44