Calculus: Early Transcendentals 7th Edition solutions

Calculus: Early Transcendentals 7
1 4 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

Calculus: Early Transcendentals 7
1 4 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

Calculus: Early Transcendentals 7
1 4 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

Calculus: Early Transcendentals 7
1 4 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

Calculus: Early Transcendentals 7
Graphs of the velocity functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

Calculus: Early Transcendentals 7
Graphs of the position functions of two particles are shown, where is measured in seconds. When is each particle speeding up? When is it slowing down? Explain.

Calculus: Early Transcendentals 7
The height (in meters) of a projectile shot vertically upward from a point 2 m above ground level with an initial velocity of is after seconds. (a) Find the velocity after 2 s and after 4 s. (b) When does the projectile reach its maximum height? (c) What is the maximum height? (d) When does it hit th

Calculus: Early Transcendentals 7
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?

Calculus: Early Transcendentals 7
If a rock is thrown vertically upward from the surface of Mars with velocity , its height after seconds is . (a) What is the velocity of the rock after 2 s? (b) What is the velocity of the rock when its height is 25 m on its way up? On its way down?

Calculus: Early Transcendentals 7
A particle moves with position function (a) At what time does the particle have a velocity of ? (b) At what time is the acceleration 0? What is the significance of this value of ?

Calculus: Early Transcendentals 7
(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 th

Calculus: Early Transcendentals 7
(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 r

Calculus: Early Transcendentals 7
(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

Calculus: Early Transcendentals 7
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?

Calculus: Early Transcendentals 7
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?

Calculus: Early Transcendentals 7
(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 r

Calculus: Early Transcendentals 7
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?

Calculus: Early Transcendentals 7
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

Calculus: Early Transcendentals 7
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?

Calculus: Early Transcendentals 7
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 the earth attrac

Calculus: Early Transcendentals 7
The force acting on a body with mass and velocity is the rate of change of momentum: . If is constant, this becomes , where is the acceleration. But in the theory of relativity the mass of a particle varies with as follows: , where is the mass of the particle at rest and is the speed of light. Show t

Calculus: Early Transcendentals 7
Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is a little more than 12 hours and on June 30, 2009, high tide occurred

Calculus: Early Transcendentals 7
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 const

Calculus: Early Transcendentals 7
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 happen

Calculus: Early Transcendentals 7
In Example 6 we considered a bacteria population that doubles every hour. Suppose that another population of bacteria triples every hour and starts with 400 bacteria. Find an expression for the number of bacteria after hours and use it to estimate the rate of growth of the bacteria population after 2

Calculus: Early Transcendentals 7
The number of yeast cells in a laboratory culture increases rapidly initially but levels off eventually. The population is modeled by the function where is measured in hours. At time the population is 20 cells and is increasing at a rate of . Find the values of and . According to this model, what hap

Calculus: Early Transcendentals 7
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. (c) Use you

Calculus: Early Transcendentals 7
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 fo

Calculus: Early Transcendentals 7
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

Calculus: Early Transcendentals 7
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 D. E. Hall, Musical Acoustics, 3rd ed. (Pacific Grove, CA, 2002).] (a) Find the rate of change of the frequency with respect to (i) the

Calculus: Early Transcendentals 7
The cost, in dollars, of producing yards of a certain fabric 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 201st yard of fabric.

Calculus: Early Transcendentals 7
The cost function for production of a commodity is (a) Find and interpret . (b) Compare with the cost of producing the 101st item.

Calculus: Early Transcendentals 7
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.

Calculus: Early Transcendentals 7
. If denotes the reaction of the body to some stimulus of strength , the x S sensitivity is defined to be the rate of change of the reaction with respect to . A particular example is that when the brightness of a light source is increased, the eye reacts by decreasing the area of the pupil. The exper

Calculus: Early Transcendentals 7
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 rat

Calculus: Early Transcendentals 7
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 pe

Calculus: Early Transcendentals 7
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 populat