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mathematics
calculus

Calculus Early Transcendentals 7th edition James Stewart - Solutions

Find a function f and a number a such that
The total cost of repaying a student loan at an interest rate of r % per year is C = f(r).(a) What is the meaning of the derivative f'(r)? What are its units?(b) What does the statement f'(10) = 1200 mean?(c) Is f'(r) always positive or does it change sign?
Trace or copy the graph of the function. Then sketch a graph of its derivative directly beneath.
(a) If f(x) = √3 – 5x, use the definition of a derivative to find f’(x). (b) Find the domains of f and f’. (c) Graph f and f’ on a common screen compare the graphs to see whether your answer to part (a) is reasonable.
(a) Find the asymptotes of the graph of and use them to sketch the graph.(b) Use your graph from part (a) to sketch the graph of f€™.(c) Use the definition of a derivative to find f€™(x)(d) Use a graphing device to graph f€™ and compare with your sketch in part (b).
The graph of f is shown. State, with reasons, the numbers at which f is not differentiable.
The total fertility rate at time t, denoted by F(t), is an estimate of the average number of children born to each woman(Assuming that current birth rates remain constant) The graph of the total fertility rate in the United States shows the fluctuations from 1940 to 1990.(a) Estimate the values of
Let B(t) be the total value of U.S. banknotes in circulation at time . The table gives values of this function from 1980 to 1998, at year end, in billions of dollars. Interpret and estimate the value of B'(1990).
Graph the curve y = (x + 1) / (x – 1) and the tangent lines to this curve at the points (2, 3) and (-1, 0)
Suppose that |f (x) | < g(x) for all x, where lim x→ a g(x) = 0. Find lim x →a f(x).
Let f(x) = [x] + [-x]. (a) For what values of does lim x→ a f(x) exist? (b) At what numbers is f discontinuous?
Evaluate lim x→1 3√x – 1 / √x – 1.
Find numbers a and b such that lim x→0 √ax + b – 2 / x = 1.
Evaluate lim x→0 |2x – 1| – |2x + 1| / x.
The figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.
If [x] denotes the greatest integer function, find lim x→∞ x/[x].
Sketch the region in the plane defined by each of the following equations.(a) [x]2 + [y]2 = 1 (b) [x]2 – 1 [y]2 = 3 (c) [x + y]2 = 1 (d) [x] + [y] = 1
Find all values of a such that f is continuous on R:
A fixed point of a function f is a number in its domain such that f(c) = c. (The function doesn’t move ; it stays fixed.)(a) Sketch the graph of a continuous function with domain [0, 1] whose range also lies in [0, 1]. Locate a fixed point of f.(b) Try to draw the graph of a continuous function
If lim x →a [f(x) + g(x)] = 2 and lim x →a [f(x) – g(x)] = 1, find lim x →a f(x)g(x).
(a) The figure shows an isosceles triangle ABC with
(a) If we start from 0o latitude and proceed in a westerly direction, we can let T(x) denote the temperature at the point at any given time. Assuming that T is a continuous function of , show that at any fixed time there are at least two diametrically opposite points on the equator that have
If f is a differentiable function and g(x) = xf(x), use the definition of a derivative to show that g’(x) = xf’(x) + f(x).
Suppose f is a function that satisfies the equation f(x + y) = f (x) + f(y) + x2y + xy2 for all real numbers x and y, suppose also that lim x →O f(x) / x = 1. (a) Find f(0). (b) Find f’(0). (c) Find f’(x)
Suppose f is a function with the property that |f(x)| < x2 for all x. Show that f(0) = 0. Then show that f’(0) = 0.
(a) How is the number e defined?(b) Use a calculator to estimate the values of the limits correct to two decimal places.What can you conclude about the value of e?
(a) Sketch, by hand, the graph of the function f(x) = ex, paying particular attention to how the graph crosses the y-axis. What fact allows you to do this?(b) What types of functions are f(x) = ex and g(x) = xe?? Compare the differentiation formulas for and t.(c) Which of the two functions in part
Find f ;( x) Compare the graphs of f and f€™ and use them to explain why your answer is reasonable.
Estimate the value of f€™ (a) by zooming in on the graph of f. Then differentiate to find the exact value of f€™ (a) and compare with your estimate.
Find an equation of the tangent line to the curve at the.
Find an equation of the tangent line to the curve at the given point. Illustrate by graphing the curve and the tangent line on the same screen.
(a) Use a graphing calculator or computer to graph the function f(x) = x4 – 3x3 – 6x2 + 7x + 30 in the viewing rectangle [– 3, 5] by [– 10, 50].(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of f’. (See Example 1 in Section 2.9)(c) Calculate
(a) Use a graphing calculator or computer to graph the function g(x) = ex – 3x2 in the viewing rectangle [– 1, 4] by.(b) Using the graph in part (a) to estimate slopes, make a rough sketch, by hand, of the graph of g’. (See Example 1 in Section 2.9)(c) Calculate g’(x) and use this
Find the points on the curve y = 2x3 + 3x2 – 12x + 1 where the tangent is horizontal.
For what values of does the graph of f(x) = x3 + 3x2 + x + 3 have a horizontal tangent?
Show that the curve y = 6x3 + 5x – 3 has no tangent line with slope 4.
At what point on the curve y = 1 + 2ex – 3x is the tangent line parallel to the line 3x – y = 5? Illustrate by graphing the curve and both lines.
Draw a diagram to show that there are two tangent lines to the parabola y = x2 that pass through the point (0, - 4). Find the coordinates of the points where these tangent lines intersect the parabola.
Find equations of both lines through the point (2, – 3) that are tangent to the parabola y = x2 + x.
A particle moves according to a law of motion s = f (t) t > 0, where is measured in seconds and in feet.(a) Find the velocity at time t.(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
The position function of a particle is given byWhen does the particle reach a velocity of 5 m/s?
If a ball is given a push so that it has an initial velocity of 5 m/s down a certain inclined plane, then the distance it has rolled after seconds is s = 5t + 3t2.(a) Find the velocity after 2 s.(b) How long does it take for the velocity to reach 35 m/s?
If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 m/s, its height (in meters) after t seconds is h = 10t – 0.83t2.(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 ft/s, then its height after seconds is s = 80t – 16t2.(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 A(x) of a wafer changes when the side length x changes. Find A` (15) and explain its meaning in this situation. (b) Show that the rate
(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 dV/dx when x = 3 mm and explain its meaning.(b) Show that the rate of change of the volume of
(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 r = 2. (c) Show that the rate of change of the area of a circle with respect to its radius (at any r) is
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, and(c) 5 s. What can you conclude?
A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr2 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 V = 4/3πr, where the radius is measured in micrometers (1 μ m = 10-6 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 5.1m. (b) Find the instantaneous rate of V change of with
The mass of the part of a metal rod that lies between its left end and a point meters to the right is 3x2 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 Torricelli€™s Law gives the volume V of water remaining in the tank after minutes asFind the rate at which water is draining from the tank after(a) 5 min,(b) 10 min,(c) 20 min, and(d) 40
The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (measured in seconds) is given by Q(t) = t3 – 2t2 + 6t + 2. Find the current when(a) t = 0.5 s and (b) t = 1 s. [See Example 3. The unit of current is an ampere (1 A = 1 C/s.)] At what time is the
Newton€™s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is where G is the gravitational constant and is the distance between the bodies.(a) Find df/dr and explain its meaning. What does the minus sign indicate?(b) Suppose it is
Boyle’s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV = C. (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
The data in the table concern the lactonization of hydroxyvaleric acid at 25oC. They give the concentration C(t)of this acid in moles per liter after minutes.(a) Find the average rate of reaction for the following time intervals:(i) 2 (b) Plot the points from the table and draw a smooth curve
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
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 A' (t).(c) Estimate the rate of change of marriage
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 [A] = [B] = a moles/L, then [C] = a2kt/(akt + 1) where is a constant.(a) Find the rate of reaction at time
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 3000 dynes/cm2 , and viscosity η = 0.027. (a) Find the velocity of the blood along the centerline r = 0, at radius r = 0.005 cm, and at the wall r =R = 0.01
The frequency of vibrations of a vibrating violin string is given by where L is the length of the string,T is its tension, and is its linear density. [See Chapter 11 in Donald E. Hall, MusicalAcoustics, 3d ed. (Pacific Grove, CA: Brooks/Cole, 2002).](a) Find the rate of change of the frequency with
Suppose that the cost (in dollars) for a company to produce x pairs of a new line of jeans is(a) Find the marginal cost function.(b) Find C'(100) and explain its meaning. What does it predict?(c) Compare C'(100) with the cost of manufacturing the 101st pair of jeans.
The cost function for a certain commodity is C(x) = 84 + 0.16X - 0.0006X2 + 0.000003X3(a) Find and interpret C'(100).(b) Compare C'(100) with the cost of producing the 101st item.
If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is(a) Find A'(x). Why does the company want to hire more workers if A'(x) > 0?(b) Show that A'(x) > 0 if p'(x) is greater than the average productivity.
If R 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 x. A particular example is that when the brightness of a lightsource is increased, the eye reacts by decreasing the area R of the pupil. The
The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT, where is the number of moles of the gas and R = 0.0821 is the gas constant. Suppose that, at a certain instant, P = 8.0 atm and is increasing at a rate of 0.10
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 r0 is the birth rate of the fish,Pc is the maximum population that the pond can sustain (called the carrying capacity), and
In the study of ecosystems, predator-prey models are often used to study the interaction between species. Consider populations of tundra wolves, given by W(t), and caribou, given by C(t), in northern Canada. The interaction has been modeled by the equations(a) What values of dC/dt and dW/dt
Prove, using the definition of derivative, that if f(x) = cos x, then f’(x) = – 1 sin x.
Find an equation of the tangent line to the curve at the given point.
(a) Find an equation of the tangent line to the curve y = x cos x at the point (π, – π). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) Find an equation of the tangent line to the curve y = x sec x – 2 cos x at the point (π/3, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) If f(x) = 2x + cot x, find f’ (x). (b) Check to see that your answer to part (a) is reasonable by graphing both f and f’ for 0 < x < π.
(a) If f(x) = √x sin x, find f’ (x). (b) Check to see that your answer to part (a) is reasonable by graphing both f and f’ for 0 < x < 2π.
For what values of does the graph of f (x) = x + 2 sin x have a horizontal tangent?
Find the points on the curve y = (cos x) / (2 + sin x) at which the tangent is horizontal.
A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t) = 8 sin t, where is in seconds and in centimeters.(a) Find the velocity at time t.(b) Find the position and velocity of the mass at time t = 2π/3. In what direction is it moving at
An elastic band is hung on a hook and a mass is hung on the lower end of the band. When the mass is pulled downward and then released, it vibrates vertically. The equation of motion is s = 2 cos t + 3 sin t, t > 0, where is measured in centimeters and in seconds. (We take the positive direction to
A ladder 10 ft long rests against a vertical wall. Let θ be the angle between the top of the ladder and the wall and let be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to θ when θ =
An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is where μ is a constant called the coefficient of friction. (a) Find the rate of change of F with
Differentiate each trigonometric identity to obtain a new (or familiar) identity.
A semicircle with diameter PQ sits on an isosceles triangle to PQR form a region shaped like an ice-cream cone, as shown in the figure. If A(θ) is the area of the semicircle and B(θ)is the area of the triangle, find
The figure shows a circular arc of length and a chord of length d, both subtended by a central angle θ. Find
Write the composite function in the form f(g(x)). [Identify the inner function u = g(x) and the outer function y = f(u).] Then find the derivative dy/dx.
Find an equation of the tangent line to the curve at the given point.
(a) Find an equation of the tangent line to the curve y = 2 / (1 + e–x) at the point (0, 1).(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) The curve y = |x|/√2 – x2 is called a bullet-nose curve. Find an equation of the tangent line to this curve at the point (1, 1). (b) Illustrate part (a) by graphing the curve and the tangent line on the same screen.
(a) If f(x) = √1 – x2/x, find f’(x). (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of f and f’.
The function f(x) = sin (x + sin 2x), 0 < x < π, arises in applications to frequency modulation (FM) synthesis. (a) Use a graph of f produced by a graphing device to make a rough sketch of the graph of f’. (b) Calculate f’(x) and use this expression, with a graphing device, to graph .
Find all points on the graph of the function f(x) = 2 sin x + sin2x at which the tangent line is horizontal.
Find the x-coordinates of all points on the curve y = sin 2x – 2 sin x at which the tangent line is horizontal.
Suppose that F(x) = f(g(x)) and g(3) = 6, g’(3) = 4, f’(3) = 2, and f’(6) = 7, Find F’(3).
Suppose that w = u o v and u(0) = 1, v(0) = 2, u’(0) = 3, u’(2) =4, v’(0) = 5, and v’(2) = 6. Find w’(0).
A table of values for f, g, f€™, and g€™ is given.(a) If h(x) = f(g(x)), find h€™(1).(b) If H(x) = g(f(x)), find H€™(1).
Let f and g be the functions in Exercise 55.(a) If F(x) = f(f(x)), find F’(2).(b) If G(x) = g(g(x)), find G’(3)
If f and g are the functions whose graphs are shown, let u(x) = f (g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why.(a) u€™ (1) (b) v€™ (1) (c) w€™ (1)
If f is the function whose graph is shown, let h(x) = f(f(x)) and g(x) = f(x2). Use the graph of f to estimate the value of each derivative.(a) h€™ (2)(b) g€™(2)
Use the table to estimate the value of h€™ (0, 5), where h(x) = f (g(x)).
If g(x) = f(f(x)), use the table to estimate the value of g€™(1)

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