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mathematics
calculus with applications
Calculus For Business, Economics And The Social And Life Sciences 11th Brief Edition Laurence Hoffmann, Gerald Bradley, David Sobecki, Michael Price - Solutions
Records indicate that x years after 2008, the average property tax on a three bedroom home in a certain community was T(x) = 20x2 + 40x + 600 dollars.a. At what rate was the property tax increasing with respect to time in 2008?b. By how much did the tax change between the years 2008 and 2012?
Gary’s starting salary is $45,000, and he gets a raise of $2,000 each year.a. Express the percentage rate of change of Gary’s salary as a function of time, and draw the graph.b. At what percentage rate will Gary’s salary be increasing after 1 year?c. What will happen to the percentage rate of
In Exercises 53 and 54, differentiate the given function f(x) by two different methods, first by using the general power rule and then by using the product rule. Show that the two answers are the same.f(x) = (7 − 4x)2
A toy rocket rises vertically in such a way that t seconds after liftoff, it is h(t) = −16t2 + 200t feet above ground. a. How high is the rocket after 6 seconds?b. What is the average velocity of the rocket over the first 6 seconds of flight (between t = 0 and t = 6)?c. What is the
At noon, a truck is at the intersection of two roads and is moving north at 70 km/hr. An hour later, a car passes through the same intersection, traveling east at 105 km/hr. How fast is the distance between the car and truck changing at 2 P.M.?
a. Find the derivatives of the functions y = x2 and y = x2 − 3, and account geometrically for their similarity.b. Without further computation, find the derivative of the function y = x2 + 5.
In Exercises 55 through 60, find the second derivative of the given function. f(x) = V1 + x²
It is estimated that t years from now, the circulation of a local newspaper will be C(t) = 100t2 + 400t + 5,000.a. Derive an expression for the rate at which the circulation will be changing with respect to time t years from now.b. At what rate will the circulation be changing with respect to time
In Exercises 55 through 60, find the second derivative of the given function.h(t) = (t2 + 5)8
a. Find the derivative of the linear function f(x) = 3x − 2.b. Find the equation of the tangent line to the graph of this function at the point where x = −1.c. Explain how the answers to parts (a) and (b) could have been obtained from geometric considerations with no calculation whatsoever.
In an adiabatic chemical process, there is no net change (gain or loss) of heat. Suppose a container of oxygen is subjected to such a process. Then if the pressure on the oxygen is P and its volume is V, it can be shown that PV1.4 = C, where C is a constant. At a certain time, V = 5 m3, P = 0.6
The gross domestic product of a certain country is growing at a constant rate. In 2000 the GDP was 125 billion dollars, and in 2008 it was 155 billion dollars. If this trend continues, at what percentage rate will the GDP be growing in 2015?
In Exercises 55 through 60, find the second derivative of the given function.y = (1 − 2x3)4
A car is traveling at 88 ft/sec when the driver applies the brakes to avoid hitting a child. After t seconds, the car is s = 88t − 8t2 feet from the point where the brakes were applied. How long does it take for the car to come to a stop, and how far does it travel before stopping?
a. Find the derivative of the function y = x2 + 3x.b. Find the derivatives of the functions y = x2 and y = 3x separately.c. How is the derivative in part (a) related to those in part (b)?d. In general, if f(x) = g(x) + h(x), what would you guess is the relationship between the derivative of f and
Boyle’s law states that when gas is compressed at a constant temperature, the pressure P and volume V of a given sample satisfy the equation PV = C, where C is constant. Suppose that at a certain time the volume is 40 in3, the pressure is 70 lb /in2, and the volume is increasing at the rate of 12
An ice block used for refrigeration is modeled as a cube of side s. The block currently has volume 125,000 cm3 and is melting at the rate of 1,000 cm3 per hour.a. What is the current length s of each side of the cube? At what rate is s currently changing with respect to time t?b. What is the
It is estimated that x years from now, the average SAT mathematics score of the incoming students at an eastern liberal arts college will be f(x) = −6x + 582.a. Derive an expression for the rate at which the average SAT score will be changing with respect to time x years from now.b. What is the
It is projected that t months from now, the average price per unit for goods in a certain sector of the economy will be P dollars, wherea. At what rate will the price per unit be increasing with respect to time 5 months from now?b. At what rate will the rate of price increase be changing with
Find h'(0) ifwhere g(0) = 4 and g'(0) = 2. h(x) = V5x² + g(x),
Consider the equation x2 + y2 = 6y − 10.a. Show that there are no points (x, y) that satisfy this equation.b. Show that by applying implicit differentiation to the given equation, you obtainThe point of this exercise is to show that one must be careful when applying implicit differentiation. Just
In Exercises 61 through 64, the position s(t) of an object moving along a straight line is given. In each case:(a) Find the object’s velocity v(t) and acceleration a(t).(b) Find all times t when the acceleration is 0.s(t) = 2t4 − 5t3 + t − 3
Sand is leaking from a bag in such a way that after t seconds, there arepounds of sand left in the bag.a. How much sand was originally in the bag?b. At what rate is sand leaking from the bag after 1 second?c. How long does it take for all the sand to leak from the bag? At what rate is the sand
a. Compute the derivatives of the functions y = x2 and y = x3.b. Examine your answers in part (a). Can you detect a pattern? What do you think is the derivative of y = x4? How about the derivative of y = x27?
It is projected that x months from now, the population of a certain town will be P(x) = 2x + 4x3/2 + 5,000.a. At what rate will the population be changing with respect to time 9 months from now?b. At what percentage rate will the population be changing with respect to time 9 months from now?
Use calculus to prove that if y = mx + b, the rate of change of y with respect to x is constant.
In Exercises 55 through 60, find the second derivative of the given function. f(u) 1 2 (3u² − 1)²
In Exercises 61 through 64, the position s(t) of an object moving along a straight line is given. In each case:(a) Find the object’s velocity v(t) and acceleration a(t).(b) Find all times t when the acceleration is 0.s(t) = 3t5 − 5t3 − 7
After x weeks, the number of people using a new rapid transit system was approximately N(x) = 6x3 + 500x + 8,000.a. At what rate was the use of the system changing with respect to time after 8 weeks?b. By how much did the use of the system change during the eighth week?
Let f be the absolute value function; that is,Show thatand explain why f is not differentiable at x = 0. f(x) = x -x if x ≥ 0 if x < 0
It is estimated that t years from now, the population of a certain town will be P(t) = t2 + 200t + 10,000.a. Express the percentage rate of change of the population as a function of t, simplify this function algebraically, and draw its graph.b. What will happen to the percentage rate of change of
Find h'(−1) if g(−1) = −1 and g'(−1) = 1, where h(x) = [3g2(x) + 4g(x) + 2]5 [g(x) + x].
At a certain factory, approximately q(t) = t2 + 50t units are manufactured during the first t hours of a production run, and the total cost of manufacturing q units is C(q) = 0.1q2 + 10q + 400 dollars. Find the rate at which the manufacturing cost is changing with respect to time 2 hours after
In Exercises 61 through 64, the position s(t) of an object moving along a straight line is given. In each case:(a) Find the object’s velocity v(t) and acceleration a(t).(b) Find all times t when the acceleration is 0.s(t) = −t3 + 7t2 + t + 2
Find h'(1) ifwhere g(1) = g'(1) = 1. h(x) 3x + 1 g(x) 73/2
A medical research team determines that t days after an epidemic begins, people N(t) = 10t3 + 5t + √t people will be infected, for 0 ≤ t ≤ 20. At what rate is the infected population increasing on the ninth day?
Prove the power rule d/dx[xn] = nxn−1 for the case where n = r/s is a rational number.
Use a graphing utility to graph the curve 5x2 − 2xy + 5y2 = 8. Draw the tangent line to the curve at (1, 1). How many horizontal tangent lines does the curve have? Find the equation of each horizontal tangent.
A disease is spreading in such a way that after t weeks, the number of people infected isa. At what rate is the epidemic spreading after 3 weeks?b. Suppose health officials declare the disease to have reached epidemic proportions when the percentage rate of change of N is at least 25%. Over what
In Exercises 61 through 64, the position s(t) of an object moving along a straight line is given. In each case:(a) Find the object’s velocity v(t) and acceleration a(t).(b) Find all times t when the acceleration is 0.s(t) = 4t5/2 − 15t2 + t − 3
Find h'(0) ifwhere g(0) = 3 and g'(0) = −2. h(x) = g(x) - x 3 + g(x). 12
Show thatis not differentiable at x = 1. f(x) = |x² - 1| x-1
Use a graphing utility to graph the curve 11x2 + 4xy + 14y2 = 21. Draw the tangent line to the curve at (−1, 1). How many horizontal tangent lines does the curve have? Find the equation of each horizontal tangent.
A soccer ball made of leather 1/8 inch thick has an inner diameter of 8.5 inches. Estimate the volume of its leather shell.
An ornithologist determines that the body temperature of a certain bird species fluctuates over roughly a 17-hour period according to the cubic formulafor 0 ≤ t ≤ 0.713, where T is the temperature in degrees Celsius measured t days from the beginning of a period.a. Compute and interpret the
An object moves along a straight line so that after t minutes, its distance from its starting point isa. At what velocity is the object moving at the end of 4 minutes?b. How far does the object actually travel during the fifth minute? D(t) 10t + 5 t + 1 - 5 meters.
The gross annual earnings of a certain company arethousand dollars t years after its formation in January 2010.a. At what rate will the gross annual earnings of the company be growing in January 2015?b. At what percentage rate will the gross annual earnings be growing in January 2015? f(t) = V10²
A car traveling north at 60 mph and a truck traveling east at 45 mph leave an intersection at the same time. At what rate is the distance between them changing 2 hours later?
Answer the following questions about the curve x3 + y3 = 3xy (called the folium of Descartes).a. Find the equation of each horizontal tangent to the curve.b. The curve intersects the line y x at exactly one point other than the origin. What is the equation of the tangent line at this point?c. Try
It has been suggested that one way to reduce worldwide carbon dioxide (CO2) emissions is to impose a single tax that would apply to all nations. The accompanying graph shows the relationship between different levels of the carbon tax and the percentage of reduction in CO2 emissions.a. What tax rate
Find the x values at which the peaks and valleys of the graph of y = 2x3 − 0.8x2 + 4 occur. Use four decimal places.
After t hours of an 8-hour trip, a car has gonekilometers.a. Derive a formula expressing the acceleration of the car as a function of time.b. At what rate is the velocity of the car changing with respect to time at the end of 6 hours? Is the velocity increasing or decreasing at this time?c. By how
Find the slope of the line that is tangent to the graph of the functionat the point where x = 3.85 by filling in the following table. f(x)=√x² + 2x - √3x
At a certain factory, the total cost of manufacturing q units is C(q) = 0.2q2 + q + 900 dollars. It has been determined that approximately q(t) = t2 + 100t units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with
A child is flying a kite at a height of 80 feet above her hand. If the kite moves horizontally at a constant speed of 5 ft/sec, at what rate is string being paid out when the string is 100 feet long?
Use a graphing utility to graph the curveFind the equation of each horizontal tangent to the curve. (The curve is called a cardioid.) x² + y² = √√√x² + y² + x.
An environmental study of a certain suburban community suggests that t years from now, the average level of carbon monoxide in the air will be Q(t) = 0.05t2 + 0.1t + 3.4 parts per million.a. At what rate will the carbon monoxide level be changing with respect to time 1 year from now?b. By how much
If an object is dropped or thrown vertically, its height (in feet) after t seconds is H(t) = −16t2 + V0t + H0, where V0 is the initial speed of the object and H0 is its initial height.a. Derive an expression for the acceleration of the object.b. How does the acceleration vary with time?c. What is
After t months, the production output at a certain factory is N(t) thousand units, whereAt what rate is the production level changing after 2 months? Is production increasing or decreasing at this time? N(t) = √² + 3t+6
Our friend, the spy who escaped from the diamond smugglers in Chapter 1 (Problem 62 of Section 1.4), is on a secret mission in space. An encounter with an enemy agent leaves him with a mild concussion and temporary amnesia. Fortunately, he has a book that gives the formula for the motion of a
A person stands at the end of a pier 8 feet above the water and pulls in a rope attached to a buoy. If the rope is hauled in at the rate of 2 ft/min, how fast is the buoy moving in the water when it is 6 feet from the pier? 8 ft
According to Debye’s formula in physical chemistry, the orientation polarization P of a gas satisfieswhere μ, k, and N are positive constants, and T is the temperature of the gas. Find the rate of change of P with respect to T. 4 P = = N 3 TN м² 3КТ
Find f(4)(x) if f(x) = x5 − 2x4 + x3 − 3x2 + 5x − 6.
In Exercises 70 through 73, s(t) is the position of a particle moving along a straight line at time t.(a) Find the velocity and acceleration of the particle.(b) Find all times in the given interval when the particle is stationary.s(t) = t2 − 2t + 6 for 0 ≤ t ≤ 2
A lantern falls from the top of a building in such a way that after t seconds, it is h(t) = 150 − 16t2 feet above ground. A woman 5 feet tall originally standing directly under the lantern sees it start to fall and walks away at the constant rate of 5 ft/sec. How fast is the length of the
Find d3y/dx3 if y = √x - 1 Vx X + 2x V2
a. Show thatb. Find dy/dx where y = (2x + 1)(x − 3)(1 − 4x). dh d dx dg -[fgh] = fg7x + fh- +fh+gh- dx dx df dx
After t weeks, a factory is producing N(t) thousand DVD players, whereAt what rate is the production level changing after 4 weeks? Is production increasing or decreasing at this time? N(t) 21 t + 3t + 12
a. By combining the product rule and the quotient rule, find an expression forb. Find dy/dx, where dfg dx h
The value V (in thousands of dollars) of an industrial machine is modeled bywhere N is the number of hours the machine is used each day. Suppose further that usage varies with time in such a way thatwhere t is the number of months the machine has been in operation.a. How many hours per day will the
The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled bySuppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure, wherea. What will be
The number of units Q of a particular commodity that will be produced when L worker-hours of labor are employed is modeled bySuppose that the labor level varies with time in such a way that t months from now L(t) worker-hours will be employed, wherefor 0 ≤ t ≤ 12.a. How many worker-hours will
If $10,000 is invested at an annual rate of r% compounded monthly, then the total amount (principal and interest) accumulated after 10 years is given by the formulaa. Find the instantaneous rate of change of A with respect to r. What is A'(5)? Include units.b. How much does the value of the account
Suppose the total manufacturing cost C at a certain factory is a function of the number q of units produced, which in turn is a function of the number t of hours during which the factory has been operating.a. What quantity is represented by the derivative dC/dq? In what units is this quantity
The product rule tells you how to differentiate the product of any two functions, while the constant multiple rule tells you how to differentiate products in which one of the factors is constant. Show that the two rules are consistent. In particular, use the product rule to show thatif c is a
Prove the power rulefor the case where n = −p is a negative integer. d dx [x²] = nx"²² -1
An object moves along a straight line in such a way that its position at time t is given bya. What are the object’s velocity v(t) and acceleration a(t) at time t?b. When is the object stationary for 0 ≤ t ≤ 2? Where is the object and what is its acceleration at each such time?c. When is the
You are standing on the top of a building and throw a ball vertically upward. After 2 seconds, the ball passes you on the way down, and 2 seconds after that, it hits the ground below.a. What is the initial velocity of the ball?b. How high is the building?c. What is the velocity of the ball when it
In Exercises 70 through 73, s(t) is the position of a particle moving along a straight line at time t.(a) Find the velocity and acceleration of the particle.(b) Find all times in the given interval when the particle is stationary.s(t) = t4 − 4t3 + 8t for 0 ≤ t ≤ 4
Derive the quotient rule. Show that the difference quotient for f/g is Before letting h approach zero, rewrite this quotient using the trick of subtracting and adding g(x) f (x) in the numerator. 1 f(x + h) h[g(x + h) - f(x) g(x) = g(x)f(x +h)-f(x) g(x + h) g(x + h)g(x)h
In Exercises 70 through 73, s(t) is the position of a particle moving along a straight line at time t.(a) Find the velocity and acceleration of the particle.(b) Find all times in the given interval when the particle is stationary.s(t) = t3 − 9t2 + 15t + 25 for 0 ≤ t ≤ 6
In Exercises 70 through 73, s(t) is the position of a particle moving along a straight line at time t.(a) Find the velocity and acceleration of the particle.(b) Find all times in the given interval when the particle is stationary.s(t) = 3t2 + 2t − 5 for 0 ≤ t ≤ 1
Find all the points (x, y) on the graph of the function y = 4x2 with the property that the tangent to the graph at (x, y) passes through the point (2, 0).
An object projected from a point P moves along a straight line. It is known that the velocity of the object is directly proportional to the product of the time the object has been moving and the distance it has moved from P. It is also known that at the end of 5 seconds, the object is 20 feet from
Use a graphing utility to sketch the curveand on the same set of coordinate axes, draw the tangent lines to the graph of f(x) at x = −2 and at x = 0. Use TRACE and ZOOM to find where f'(x) = 0 f(x) 3x² - 4x + 1 x + 1
a. If f(x) = x4, show thatb. If f(x) = xn for positive integer n, show thatc. Use the result in part (b) in the definition of the derivative to prove the power rule: f(x +h)-f(x) h = 4x² + 6x²h + 4xh² + h³.
In a research paper,* V. A. Tucker and K. Schmidt-Koenig demonstrated that a species of Australian parakeet (the Budgerigar) expends energy (calories per gram of mass per kilometer) according to the formulawhere v is the bird’s velocity (in km/hr). Find a formula for the rate of change of E with
A stone is dropped from a height of 144 feet.a. When will the stone hit the ground?b. With what velocity does it hit the ground?
Use a graphing utility to sketch the curve f(x) = x2(x − 1), and on the same set of coordinate axes, draw the tangent line to the graph of f (x) at x = 1. Use TRACE and ZOOM to find where f'(x) = 0.
Graph f(x) = x4 + 2x3 − x + 1 using a graphing utility with a viewing rectangle of [−5, 5]1 by [0, 2]0.5. Use TRACE and ZOOM, or the maximum and minimum commands, to find the minima and maxima of this function. Find the derivative function f'(x) algebraically, and graph f(x) and f'(x) on the
Find the equations of all the tangents to the graph of the function f(x) = x2 − 4x + 25 that pass through the origin (0, 0).
Suppose y is a linear function of x; that is, y = mx + b. What will happen to the percentage rate of change of y with respect to x as x increases without bound? Explain.
Let f(x) = (3x + 5)(2x3 − 5x + 4). Use a graphing utility to graph f (x) and f'(x) on the same set of coordinate axes. Use TRACE and ZOOM to find where f'(x) = 0.
The growth of certain insects varies with temperature. Suppose a particular species of insect grows in such a way that the volume of an individual iswhen the temperature is TºC, and that its mass is m grams, wherea. Find the rate of change of the insect’s volume with respect to temperature.b.
When organic matter is introduced into a body of water, the oxygen content of the water is temporarily reduced by oxidation. Suppose that t days after untreated sewage is dumped into a particular lake, the proportion of the usual oxygen content in the water of the lake that remains is given by the
Suppose L(x) is a function with the property that L'(x) = 1/x Use the chain rule to find the derivatives of the following functions and simplify your answers. a. f(x) = L(x²) c. f(x) = L 2 3√x b. f(x) = L (²) d. f(x) = L 1 ( ²x + 1) 1-
Find numbers a, b, and c such that the graph of the function f(x) = ax2 + bx + c will have x intercepts at (0, 0) and (5, 0), and a tangent with slope 1 when x = 2.
Observations show that the length L in millimeters (mm) from nose to tip of tail of a Siberian tiger can be estimated using the function L = 0.25w2.6, where w is the weight of the tiger in kilograms (kg). Furthermore, when a tiger is less than 6 months old, its weight (kg) can be estimated in terms
Use a graphing utility to graphand f'(x) on the same set of coordinate axes. Use TRACE and ZOOM to find where f'(x) = 0. f(x) || 2x + 3 1 - 3x
When you first begin to study a topic or practice a skill, you may not be very good at it, but in time, you will approach the limits of your ability. One model for describing this behavior involves the functionwhere T is the time required for a particular person to learn the items on a list of L
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