Suppose that a bacteria population starts with 500 bacteria and triples every hour. (a)

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

A particle moves according to a law of motion , , where is measured in seconds and in feet. (a) Find the velocity at time . (b) What is the velocity after 3 s? (c) When is the particle at rest? (d) When is the particle moving in the positive direction? (e) Find the total distance traveled during the first 8 s. (f) Draw a diagram like Figure 2 to illustrate the motion of the particle.

The position function of a particle is given by When does the particle reach a velocity of ?

If a ball is given a push so that it has an initial velocity of down a certain inclined plane, then the distance it has rolled after seconds is . (a) Find the velocity after 2 s. (b) How long does it take for the velocity to reach ?

If a stone is thrown vertically upward from the surface of the moon with a velocity of , its height (in meters) after seconds is . (a) What is the velocity of the stone after 3 s? (b) What is the velocity of the stone after it has risen 25 m?

If a ball is thrown vertically upward with a velocity of 80 fts, then its height after seconds is . (a) What is the maximum height reached by the ball? (b) What is the velocity of the ball when it is 96 ft above the ground on its way up? On its way down?

(a) A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 15 mm and it wants to know how the area of a wafer changes when the side length x changes. Find and explain its meaning in this situation. (b) Show that the rate of change of the area of a square with respect to its side length is half its perimeter. Try to explain geometrically why this is true by drawing a square whose side length x is increased by an amount . How can you approximate the resulting change in area if is small?

(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate when mm and explain 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 the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b).

(a) Find the average rate of change of the area of a circle with respect to its radius as changes from (i) 2 to 3 (ii) 2 to 2.5 (iii) 2 to 2.1 (b) Find the instantaneous rate of change when . (c) Show that the rate of change of the area of a circle with respect to its radius (at any ) 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 . How can you approximate the resulting change in area if is small?

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cms. Find the rate at which the area within the circle is increasing after (a) 1 s, (b) 3 s, and (c) 5 s. What can you conclude?

A spherical balloon is being inflated. Find the rate of increase of the surface area with respect to the radius when is (a) 1 ft, (b) 2 ft, and (c) 3 ft. What conclusion can you make?

(a) The volume of a growing spherical cell is , where the radius is measured in micrometers (1 m ). Find the average rate of change of with respect to when changes from (i) 5 to 8 m (ii) 5 to 6 m (iii) 5 to 5.1 m (b) Find the instantaneous rate of change of with respect to when m. (c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c).

The mass of the part of a metal rod that lies between its left end and a point meters to the right is kg. Find the linear density (see Example 2) when is (a) 1 m, (b) 2 m, and (c) 3 m. Where is the density the highest? The lowest?

If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricellis Law gives the volume of water remaining in the tank after minutes as Find the rate at which water is draining from the tank after (a) 5 min, (b) 10 min, (c) 20 min, and (d) 40 min. At what time is the water flowing out the fastest? The slowest? Summarize your findings.

The quantity of charge in coulombs (C) that has passed through a point in a wire up to time (measured in seconds) is given by . Find the current when (a) s and (b) s. [See Example 3. The unit of current is an ampere ( A Cs).] At what time is the current lowest?

Newtons Law of Gravitation says that the magnitude of the force exerted by a body of mass on a body of mass is where is the gravitational constant and is the distance between the bodies. (a) Find and explain its meaning. What does the minus sign indicate? (b) Suppose it is known that Earth attracts an object with a force that decreases at the rate of 2 Nkm when r 20,000 km. How fast does this force change when r 10,000 km?

Boyles Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: . (a) Find the rate of change of volume with respect to pressure. (b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain. (c) Prove that the isothermal compressibility (see Example 5) is given by .

The data in the table concern the lactonization of hydroxyvaleric acid at . They give the concentration of this acid in moles per liter after minutes. (a) Find the average rate of reaction for the following time intervals: (i) (ii) (iii) (b) Plot the points from the table and draw a smooth curve through them as an approximation to the graph of the concentration function. Then draw the tangent at and use it to estimate the instantaneous rate of reaction when .

The table gives the population of the world in the 20th century. (a) Estimate the rate of population growth in 1920 and in 1980 by averaging the slopes of two secant lines. (b) Use a graphing calculator or computer to find a cubic function (a third-degree polynomial) that models the data. (See Section 1.2.) (c) Use your model in part (b) to find a model for the rate of population growth in the 20th century. (d) Use part (c) to estimate the rates of growth in 1920 and 1980. Compare with your estimates in part (a). (e) Estimate the rate of growth in 1985.

The table shows how the average age of first marriage of Japanese women varied in the last half of the 20th century. (a) Use a graphing calculator or computer to model these data with a fourth-degree polynomial. (b) Use part (a) to find a model for . (c) Estimate the rate of change of marriage age for women in 1990. (d) Graph the data points and the models for .

If, in Example 4, one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B, and the initial concentrations of A and B have a common value , then where is a constant. (a) Find the rate of reaction at time . (b) Show that if C , then (c) What happens to the concentration as ? (d) What happens to the rate of reaction as ? (e) What do the results of parts (c) and (d) mean in practical terms?

Suppose that a bacteria population starts with 500 bacteria and triples every hour. (a) What is the population after 3 hours? After 4 hours? After hours? (b) Use (5) in Section 3.1 to estimate the rate of increase of the bacteria population after 6 hours.

Refer to the law of laminar flow given in Example 7. Consider a blood vessel with radius 0.01 cm, length 3 cm, pressure difference , and viscosity . (a) Find the velocity of the blood along the centerline , at radius cm, and at the wall . r 0.005 r R 0.01 cm r (b) Find the velocity gradient at , , and x. (c) Where is the velocity the greatest? Where is the velocity changing most?

The frequency of vibrations of a vibrating violin string is given by where is the length of the string, is its tension, and is its linear density. [See Chapter 11 in Donald E. Hall, Musical Acoustics, 3d ed. (Pacific Grove, CA: Brooks/Cole, 2002).] (a) Find the rate of change of the frequency with respect to (i) the length (when and are constant), (ii) the tension (when and are constant), and (iii) the linear density (when and are constant). (b) The pitch of a note (how high or low the note sounds) is determined by the frequency . (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in part (a) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates, (ii) when the tension is increased by turning a tuning peg, (iii) when the linear density is increased by switching to another string.

Suppose that the cost (in dollars) for a company to produce pairs of a new line of jeans is (a) Find the marginal cost function. (b) Find and explain its meaning. What does it predict? (c) Compare with the cost of manufacturing the 101st pair of jeans.

The cost function for a certain commodity is (a) Find and interpret (b) Compare with the cost of producing the 101st item.

If is the total value of the production when there are workers in a plant, then the average productivity of the workforce at the plant is (a) Find . Why does the company want to hire more workers if ? (b) Show that if is greater than the average productivity.

If denotes the reaction of the body to some stimulus of strength , the sensitivity is defined to be the rate of change of the reaction with respect to . A particular example is that x when the brightness of a light source is increased, the eye reacts by decreasing the area of the pupil. The experimental formula has been used to model the dependence of on when is measured in square millimeters and is measured in appropriate units of brightness. (a) Find the sensitivity. ; (b) Illustrate part (a) by graphing both and as functions of . Comment on the values of and at low levels of brightness. Is this what you would expect?

The gas law for an ideal gas at absolute temperature (in kelvins), pressure (in atmospheres), and volume (in liters) is , where is the number of moles of the gas and is the gas constant. Suppose that, at a certain instant, atm and is increasing at a rate of 0.10 atmmin and and is decreasing at a rate of 0.15 Lmin. Find the rate of change of with respect to time at that instant if mol.

In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation where is the birth rate of the fish, is the maximum population that the pond can sustain (called the carrying capacity), and is the percentage of the population that is harvested. (a) What value of corresponds to a stable population? (b) If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. (c) What happens if is raised to 5%?

In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by , and caribou, given by , in northern Canada. The interaction has been modeled by the equations (a) What values of and correspond to stable populations? (b) How would the statement The caribou go extinct be represented mathematically? (c) Suppose that , , , and . Find all population pairs that lead to stable populations. According to this model, is it possible for the two species to live in balance or will one or both species become extinct?