Figure 2.11 shows N = f(t), the number of farms in the US2 between 1930 and 2000 as a

Figure 2.11 shows N = f(t), the number of farms in the US2 between 1930 and 2000 as a function of year, t. (a) Is f(1950) positive or negative? What does this tell you about the number of farms? (b) Which is more negative: f(1960) or f(1980)? Explain. 1930 1950 1970 1990 1 2 3 4 5 6 7 t (year) millions of farms Figure 2.11

Use the graph in Figure 2.7 to decide if each of the following quantities is positive, negative or approximately zero. Illustrate your answers graphically. (a) The average rate of change of f(x) between x = 3 and x = 7. (b) The instantaneous rate of change of f(x) at x = 3.

The position s of a car at time t is given in the following table. t (sec) 0 0.2 0.4 0.6 0.8 1.0 s (ft) 0 0.5 1.8 3.8 6.5 9.6 (a) Find the average velocity over the interval 0 t 0.2. (b) Find the average velocity over the interval 0.2 t 0.4. (c) Use the previous answers to estimate the instantaneous velocity of the car at t = 0.2.

In a time of t seconds, a particle moves a distance of s meters from its starting point, where s = 4t2 + 3. (a) Find the average velocity between t = 1 and t = 1 + h if: (i) h = 0.1, (ii) h = 0.01, (iii) h = 0.001. (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time t = 1.

Figure 2.12 shows the cost, y = f(x), of manufacturing x kilograms of a chemical. (a) Is the average rate of change of the cost greater between x = 0 and x = 3, or between x = 3 and x = 5? Explain your answer graphically. (b) Is the instantaneous rate of change of the cost of producing x kilograms greater at x = 1 or at x = 4? Explain your answer graphically. (c) What are the units of these rates of change?

The distance (in feet) of an object from a point is given by s(t) = t2, where time t is in seconds. (a) What is the average velocity of the object between t = 2 and t = 5? (b) By using smaller and smaller intervals around 2, estimate the instantaneous velocity at time t = 2.

(a) Using Table 2.3, find the average rate of change in the worlds population, P, between 1975 and 2010.3 Give units. (b) If P = f(t) with t in years, estimate f(2010) and give units. Table 2.3 World population, in billions of people Year 1975 1980 1985 1990 1995 2000 2005 2010 Population 4.08 4.45 4.84 5.27 5.68 6.07 6.45 6.89

The size, S, of a tumor (in cubicmillimeters) is given by S = 2t, where t is the number of months since the tumor was discovered. Give units with your answers. (a) What is the total change in the size of the tumor during the first six months? (b) What is the average rate of change in the size of the tumor during the first six months? (c) Estimate the rate at which the tumor is growing at t = 6. (Use smaller and smaller intervals.)

Let g(x) = 4x. Use small intervals to estimate g(1). 1

(a) Let g(t) = (0.8)t. Use a graph to determinewhether g(2) is positive, negative, or zero. (b) Use a small interval to estimate g(2). 1

For the function shown in Figure 2.13, at what labeled points is the slope of the graph positive? Negative? At which labeled point does the graph have the greatest (i.e., most positive) slope? The least slope (i.e., negative and with the largest magnitude)? A B C D E F Figure 2.13 1

Match the points labeled on the curve in Figure 2.14 with the given slopes. Slope Point 3 1 0 1/2 1 2 A B C D E F Figure 2.14 1

For the function f(x) = 3x, estimate f(1). From the graph of f(x), would you expect your estimate to be greater than or less than the true value of f(1)? 1

Table 2.4 gives P = f(t), the number of households, in millions, in the US with cable television t years since 1998.4 (a) Does f(4) appear to be positive or negative? What does this tell you about the percent of households with cable television? (b) Estimate f(2). Estimate f(10). Explain what each is telling you, in terms of cable television. Table 2.4 t 0 2 4 6 8 10 12 P 64.65 66.25 66.732 65.727 65.141 64.874 60.958 1

The following table gives the percent of the US population living in urban areas as a function of year.5 Year 1800 1830 1860 1890 1920 Percent 6.0 9.0 19.8 35.1 51.2 Year 1950 1980 1990 2000 2005 Percent 64.0 73.7 75.2 79.0 79.0 (a) Find the average rate of change of the percent of the population living in urban areas between 1890 and 1990. (b) Estimate the rate at which this percent is increasing at the year 1990. (c) Estimate the rate of change of this function for the year 1830 and explain what it is telling you. 1

(a) Graph f(x) = x2 and g(x) = x2 + 3 on the same axes. What can you say about the slopes of the tangent lines to the two graphs at the point x = 0? x = 1? x = 2? x = a, where a is any value? (b) Explain why adding a constant to any function will not change the value of the derivative at any point. 1

The function in Figure 2.15 has f(4) = 25 and f(4) = 1.5. Find the coordinates of the points A, B, C. A B C x 3.9 4 4.2 f(x) Tangent line Figure 2.15 1

Use Figure 2.16 to fill in the blanks in the following statements about the function f at point A. (a) f( ) = (b) f( ) = (7, 3) A (7.2, 3.8) Tangent line f(x) Figure 2.16 1

Show how to represent the following on Figure 2.17. (a) f(4) (b) f(4) f(2) (c) f(5) f(2) 5 2 (d) f(3) 1 2 3 4 5 x f(x) Figure 2.17 2

For each of the following pairs of numbers, use Figure 2.17 to decide which is larger. Explain your answer. (a) f(3) or f(4)? (b) f(3) f(2) or f(2) f(1)? (c) f(2) f(1) 2 1 or f(3) f(1) 3 1 ? (d) f(1) or f(4)? 2

Estimate the instantaneous rate of change of the function f(x) = x ln x at x = 1 and at x = 2. What do these values suggest about the concavity of the graph between 1 and 2? 2

On October 17, 2006, in an article called US Population Reaches 300 Million, the BBC reported that the US gains 1 person every 11 seconds. If f(t) is the US population in millions t years after October 17, 2006, find f(0) and f(0). 2

The following table shows the number of hours worked in a week, f(t), hourly earnings, g(t), in dollars, and weekly earnings, h(t), in dollars, of production workers as functions of t, the year.6 (a) Indicate whether each of the following derivatives is positive, negative, or zero: f(t), g(t), h(t). Interpret each answer in terms of hours or earnings. (b) Estimate each of the following derivatives, and interpret your answers: (i) f(1970) and f(1995) (ii) g(1970) and g(1995) (iii) h(1970) and h(1995)