234 CHAPTER 3 Differential 19. The q throug 12. (a) Sodium chlorate crystals are easy to grow in the shape of given cubes by allowing a solution of water and sodium chlorate (a) t= to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain currer currer its meaning. (b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of 20. Newt the cube. Explain geometrically why this result is true by force arguing by analogy with Exercise 11(b). 13. (a) Find the average rate of change of the area of a circle with respect to its radius r as r changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 when (b) Find the instantaneous rate of change when r = 2. betw (c) Show that the rate of change of the area of a circle with (a) respect to its radius (at any r) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount (b) Ar. How can you approximate the resulting change in area AA if Ar is small? 14. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm / s. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, 21. The and (c) 5 s. What can you conclude? rate this 15. A spherical balloon is being inflated. Find the rate of increase in th of the surface area (S = 4Tr2) with respect to the radius r foll when r is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make? part Godw Hed sits In 16. (a) The volume of a growing spherical cell is V = 3 ur, where the radius r is measured in micrometers (1 um = 10- m). Find the average rate of change of V with respect to r when r changes from 22. Som (i) 5 to 8 um (ii) 5 to 6 um (iii) 5 to 5.1 pum Fun (b) Find the instantaneous rate of change of V with respect to r when r = 5 um. wat abo (c) Show that the rate of change of the volume of a sphere with than respect to its radius is equal to its surface area. Explain 6:43 geometrically why this result is true. Argue by analogy with Exercise 13(c). dept 17. The mass midSECTION 3.7 Rates of Change in the Natural and Social Sciences 235 24. If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the (e) In Section 1.1 we modeled P(t) with the exponential reactant B, and the initial concentrations of A and B have a function common value [A] = [B] = a moles/L, then f(t) = (1.43653 X 109) . (1.01395)' [C] = akt/(aki + 1) Use this model to find a model for the rate of population growth. where k is a constant. (f) Use your model in part (e) to estimate the rate of growth (a) Find the rate of reaction at time t. in 1920 and 1980. Compare with your estimates in parts (b) Show that if x = [C], then (a) and (d). (g) Estimate the rate of growth in 1985. dx = k(a - x)? dt 28. The table shows how the average age of first marriage of Japanese women has varied since 1950. (c) What happens to the concentration as t -> co? (d) What happens to the rate of reaction as t -> co? A(t) A(t) (e) What do the results of parts (c) and (d) mean in practical terms? 1950 23.0 1985 25.5 1955 23.8 1990 25.9 25. In Example 6 we considered a bacteria population that 1960 24.4 1995 26.3 doubles every hour. Suppose that another population of 1965 24.5 2000 27.0 bacteria triples every hour and starts with 400 bacteria. Find 1970 24.2 2005 28.0 an expression for the number n of bacteria after t hours and 1975 24.7 2010 28 .8 use it to estimate the rate of growth of the bacteria popula- 1980 25.2 tion after 2.5 hours. 26. The number of yeast cells in a laboratory culture increases (a) Use a graphing calculator or computer to model these rapidly initially but levels off eventually. The population is data with a fourth-degree polynomial. modeled by the function (b) Use part (a) to find a model for A'(t). a (c) Estimate the rate of change of marriage age for women n = f (t ) = 1 + be-0.7: in 1990. (d) Graph the data points and the models for A and A'. where t is measured in hours. At time t = 0 the population is 20 cells and is increasing at a rate of 12 cells/ hour. Find the 29. Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length values of a and b. According to this model, what happens to 3 cm, pressure difference 3000 dynes/cm, and viscosity the yeast population in the long run? m = 0.027. 27. The table gives the population of the world P(t), in millions, (a) Find the velocity of the blood along the center- where t is measured in years and t = 0 corresponds to the line r = 0, at radius r = 0.005 cm, and at the wall year 1900. r = R = 0.01 cm. (b) Find the velocity gradient at r = 0, r = 0.005, and Population Population r = 0.01. (millions) millions) (c) Where is the velocity the greatest? Where is the velocity changing most? 0 1650 60 3040 10 1750 70 3710 30. The frequency of vibrations of a vibrating violin string is 1860 80 4450 given by 20 30 2070 90 5280 2300 100 6080 25 V 40 50 2560 110 6870 where L is the length of the string, T is its tension, and p is its linear density. [See Chapter 11 in D. E. Hall, Musical (a) Estimate the rate of population growth in 1920 and in Acoustics, 3rd ed. (Pacific Grove, CA: Brooks/Cole, 2002).] 1980 by averaging the slopes of two secant lines. (a) Find the rate of change of the frequency with respect to (b) Use a graphing device to find a cubic function (a third- (i) the length (when T and p are constant), degree polynomial) that models the data. (ii) the tension (when L and p are constant), and (c) Use your model in part (b) to find a model for the rate of (iii) the linear density (when L and T are constant). population growth. (b) The pitch of a note (how high or low the note sounds) (d) Use part (c) to estimate the rates of growth in 1920 and is determined by the frequency f. (The higher the fre- 1980. Compare with your estimates in part (a). quency, the higher the pitch.) Use the signs of the