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calculus with applications

Calculus With Applications 12th Edition Margaret L. Lial - Solutions

The problems in Exercises, are called self-answering problems because the answers are embedded in the question. For example, how many ways can you arrange the letters in the word “six”? The answer is six.At time t = 0, water begins pouring into an empty tank so that the volume of water is
Find each integral. -6x cos 5x dx
Find the derivatives of the functions defined as follows. y = sin x V sin 3x
Find the derivatives of the functions defined as follows. y = √ cos 4x COS X
In quadrant I, x, y, and r are all positive, so that all six trigonometric functions have positive values. In quadrant II, x is negative and y is positive (r is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises, complete the following table of
Find each function value without using a calculator. sin 6
Find each integral. fr sec x5 tan x5 dx
Find each function value without using a calculator.cot 300°
The problems in Exercises, are called self-answering problems because the answers are embedded in the question. For example, how many ways can you arrange the letters in the word “six”? The answer is six.The cost of a widget varies according to the formula C′(t) = -sin t. At time t = 0, the
Find the derivative of each function.y = ln |5 sin x|
Find the derivative of each function.y = ln |cos x|
Use a calculator to find the following function values.sin 39°
Use a calculator to find the following function values.cos 67°
Use a calculator to find the following function values.tan 123°
Use a calculator to find the following function values.tan 54°
The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2013 was modeled by the function C(t) = 80.63 sin(0.8998t + 1.154) + 416.1, where t = 1 corresponded to January.(a) Find C′(t).(b) Find and interpret C′(3).(c) Find and interpret C′(6).
Use a calculator to find the following function values.sin 0.3638
A runner’s arm swings rhythmically according to the equationwhere y denotes the angle between the actual position of the upper arm and the downward vertical position (as shown in the figure) and where t denotes time (in seconds).(a) Graph y as a function of t.(b) Calculate the velocity and the
Use a calculator to find the following function values.tan 1.0123
If a string with a fundamental frequency of 110 hertz is plucked in the middle, it will vibrate at the odd harmonics of 110, 330, 550,...hertz but not at the even harmonics of 220, 440, 660,...hertz. The resulting pressure P on the eardrum caused by the string can be approximated using the
A jogger’s arm swings according to the equationProceed as directed in parts (a)–(d) of the preceding exercise, with the following exceptions: in part (c), replace the differential equation withand in part (d), consider the times t = 1.5 seconds, t = 2.5 seconds, and t = 3.5 seconds. 1 5 y
Graph each function, considering the domain, critical points, symmetry, relative extrema, regions where the function is increasing or decreasing, inflection points, and regions where the function is concave upward or concave downward.ƒ(x) = x - sin x
Use a calculator to find the following function values.cos 1.2353
Many biological populations, both plant and animal, experience seasonal growth. For example, an animal population might flourish during the spring and summer and die back in the fall. The population, ƒ(t), at time t, is often modeled bywhere ƒ(0) is the size of the population when t = 0. Suppose
At Mauna Loa, Hawaii, atmospheric carbon dioxide levels in parts per million (ppm) have been measured regularly since 1958. The function defined by L(t) = 0.022t2 + 0.55t + 316 + 3.5 sin(2πt) can be used to model these levels, where t is in years and t = 0 corresponds to 1960.(a) Graph L(t) on [0,
Find the amplitude (a) and period (T) of each function. 1 h(x) = ——-sin(4x) 2
Graph each function, considering the domain, critical points, symmetry, relative extrema, regions where the function is increasing or decreasing, inflection points, and regions where the function is concave upward or concave downward.ƒ(x) = x - cos x
Use a calculator to find the following function values.sin 1.5359
At Barrow, Alaska, atmospheric carbon dioxide levels (in parts per million) can be modeled using the function defined by C(t) = 0.04t2 + 0.6t + 330 + 7.5 sin(2πt), where t is in years and t = 0 corresponds to 1960.(a) Graph C(t) on [0, 25].(b) Find C(25), C(35.5), and C(50.2).(c) Find C′(50.2)
Find the amplitude (a) and period (T) of each function. g(t) = -2 sin( TT -t + 2 4
Find the amplitude (a) and period (T) of each function.ƒ(x) = cos(3x)
Find dy/dx.x + cos(x + y) = y2
The grooves or tread in a tire occasionally pick up small pieces of gravel, which then are often thrown into the air as they work loose from the tire. When following behind a vehicle on a highway with loose gravel, it is possible to determine a safe distance to travel behind the vehicle so that
Mathematical models of ground temperature variation usually involve Fourier series or other sophisticated methods. However, the elementary modelhas been developed for temperature 0(x, t) at a given location at a variable time t (in months) and a variable depth x (in centimeters) beneath Earth’s
Find dy/dx.tan(xy) + x3 = y
Find each integral. cos 5x dx
Graph each function defined in Exercises over a twoperiod interval. y 1 2 COS X
Find the amplitude (a) and period (T) of each function.s(t) = 3 sin(880πt - 7)
Find each integral. sin 2x dx
Find dy/dt.y = sin x; dx/dt = -1, x = π/3
Graph each function defined in Exercises over a twoperiod interval.y = 2 cos x
Find each integral. sec² 5x dx
Find dy/dt.y = tan x; dx/dt = 3, x = π/4
As shown in Exercise 68, a formula that can be used to determine the distance of a piston with respect to the crankshaft for a 1937 John Deere B engine is s(θ) = 2.625 cos(θ) + 2.625(15 + cos2 θ)1/2, where s is measured in inches and θ in radians. θ(a) Given that the angle θ is changing with
A television camera on a tripod 60 ft from a road is filming a car carrying the president of the United States. (See the figure.) The car is moving along the road at 600 ft per minute.(a) How fast is the camera rotating (in revolutions per minute) when the car is at the point on the road closest to
Graph each function defined in Exercises over a twoperiod interval.y = 2 sin x
The beacon on a lighthouse 50 m from a straight shoreline rotates twice per minute. (See the figure.)(a) How fast is the beam moving along the shoreline at the moment when the light beam and the shoreline are at right angles?(b) In part (a), how fast is the beam moving along the shoreline when the
Find each integral. tan 7x dx
Graph each function defined in Exercises over a twoperiod interval. y = 2 cos 3x 77 4 + 1
A particle moves along a straight line. The distance of the particle from the origin at time t is given by s(t) = sin t + 2 cos t. Find the velocity at the following times.(a) t = 0 (b) t = π/4 (c) t = 3π/2Find the acceleration at the following times.(d) t = 0 (e) t = π4 (f)
Graph each function defined in Exercises over a twoperiod interval. y 4 sin x + π = √(√2/₁ TT + 2
Find each integral. 4 csc² x dx
Graph each function defined in Exercises over a twoperiod interval.y = -sin x
A janitor in a hospital needs to carry a ladder around a corner connecting a 10-ft-wide corridor and a 5-ft-wide corridor. (See the figure below.) What is the longest such ladder that can make it around the corner? -10 ft- 0 5 ft
A thief tries to enter a building by placing a ladder over a 9-ft-high fence so it rests against the building, which is 2 ft back from the fence. (See the following figure.) What is the length of the shortest ladder that can be used? 0 19A 9 ft 2 ft
Find each integral. 5x sec 2x² tan 2x² dx
Consider the triangle shown on the next page, in which the three angles 0 are equal and all sides have length 2.(a) Using the fact that the sum of the angles in a triangle is 180°, what are the measures of the three equal angles θ?(b) Suppose the triangle is cut in half as shown by a vertical
Find each integral. 8 sec² x dx
Find each integral. x² sin 4x³ dx
Graph each function defined in Exercises over a twoperiod interval. y || 1 2 tan x
Consider the right triangle shown, in which the two sides have length 1.(a) Using the Pythagorean Theorem, what is the length of the hypotenuse?(b) Using the fact that the sum of the angles in a triangle is 180°, what are the measures of the three angles? 1 1
Find each integral. [cos cos x sin x dx
Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated bywhere t is time in months, with t = 0 corresponding to November. Find the sales for (a)–(e).(a) November (b) January (c) February(d) May (e) August (f) Graph y =
Graph each function defined in Exercises over a twoperiod interval.y = -3 tan x
Find each integral. Vcos x sin x dx
The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2019 is given in the following table.(a) Plot the data, letting t = 1 correspond to January, t = 2 to February, and so on. Is it reasonable to assume that electrical consumption is periodic?(b) Use a calculator
Find each integral. x tan 11x² dx
Find each integral. (cos x)-4/3 sin x dx
Find each integral. x² cot 8x³ dx
A study on the circadian rhythms of patients with Alzheimer’s disease found that the body temperature of patients could be described by a function of the formwhere t is the time in hours since midnight. For the patients without Alzheimer’s, the average values of T0 (the MESOR), a (the
The “Transylvania hypothesis” claims that the full moon has an effect on health-related behavior. A study investigating this effect found a significant relationship between the phase of the moon and the number of general practice consultations nationwide, given bywhere y is the number of
Find each integral. sec² 5x tan 5x dx
The amount of pollution in the air fluctuates with the seasons. It is lower after heavy spring rains and higher after periods of little rain. In addition to this seasonal fluctuation, the long-term trend in many areas is upward. An idealized graph of this situation is shown in the figure below.
Find each definite integral. -TT/2 0 cos x dx
Find each definite integral. -2π/3 IT -sin x dx
When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light changes, and the direction that the ray is traveling changes. (This is why a fish under water is in a different position from the place at which it appears to be.) These changes
Find each definite integral. -2π (10 + 10 cos x) dx
In a study of how monkeys’ eyes pursue a moving object, an image was moved sinusoidally through a monkey’s field of vision with an amplitude of 2° and a period of 0.350 seconds.(a) Find an equation giving the position of the image in degrees as a function of time in seconds.(b) After how many
Using a computer or a graphing calculator, sketch the function for air pollution given in Exercise 86 over the interval [0, 6].Exercise 86The amount of pollution in the air fluctuates with the seasons. It is lower after heavy spring rains and higher after periods of little rain. In addition to this
Find each definite integral. 7/3 (3 - 3 sin x) dx
Pure sounds produce single sine waves on an oscilloscope. Find the period of each sine wave in the photographs in Exercises. On the vertical scale each square represents 0.5, and on the horizontal scale each square represents 30°. 豆
Pure sounds produce single sine waves on an oscilloscope. Find the period of each sine wave in the photographs in Exercises. On the vertical scale each square represents 0.5, and on the horizontal scale each square represents 30°.
When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light changes, and the direction that the ray is traveling changes. (This is why a fish under water is in a different position from the place at which it appears to be.) These changes
The monthly residential consumption of natural gas in Pennsylvania for 2019 is found in the following table.(a) Plot the data, letting t = 1 correspond to January, t = 2 to February, and so on. Is it reasonable to assume that the monthly consumption of energy is periodic?(b) Find the trigonometric
A mathematical model for the temperature in Fairbanks iswhere T(t) is the temperature (in degrees Fahrenheit) on day t, with t = 0 corresponding to January 1 and t = 364 corresponding to December 31. Use a calculator to estimate the temperature for (a)–(d).(a) March 16 (Day 74)(b) May 2 (Day
In the Kodak Customer Service Pamphlet AA-26, Optical Formulas and Their Applications, the near and far limits of the depth of field (how close or how far away an object can be placed and still be in focus) are given byIn these equations, θ represents the angle between the lens and the “circle
A surveyor standing 65 m from the base of a building measures the angle to the top of the building and finds it to be 42.8°. (See the figure.) Use trigonometry to find the height of the building. 42.8° 65 m
The body’s system of blood vessels is made up of arteries, arterioles, capillaries, and veins. The transport of blood from the heart through all organs of the body and back to the heart should be as efficient as possible. One way this can be done is by having large enough blood vessels to avoid
The number of minutes after noon, Eastern Standard Time, that the sun sets in Boston for specific days of the year is approximated in the following table.(a) Plot the data. Is it reasonable to assume that the times of sunset are periodic?(b) Use a calculator with trigonometric regression to find a
Suppose the A key above Middle C is played as a pure tone. For this tone, P(t) = 0.002 sin9880πt), where P(t) is the change of pressure (in pounds per square foot) on a person’s eardrum at time t (in seconds).(a) Graph this function on [0, 0.003].(b) Determine analytically the values of t for
The table lists the average monthly temperatures in Vancouver, CanadaThese average temperatures cycle yearly and change only slightly over many years. Because of the repetitive nature of temperatures from year to year, they can be modeled with a sine function. Some graphing calculators have a sine
The maximum afternoon temperature (in degrees Fahrenheit) in a given city is approximated by T(t) = 60 - 30 cos(t/2), where t represents the month, with t = 0 representing January, t = 1 representing February, and so on. Use a calculator to find the maximum afternoon temperature for the following
A person’s blood pressure at time t (in seconds) is given by P(t) = 90 + 15 sin 144πt. Find the maximum and minimum values of P on the interval [0, 1/72]. Graph one period of y = P(t).
It is possible to model the flight of a tennis ball that has just been served down the center of the court by the equationwhere y is the height (in feet) of a tennis ball that is being served at an angle α relative to the horizontal axis, x is the horizontal distance (in feet) that the ball has
Marisol stands on a cliff at the edge of a canyon. On the opposite side of the canyon is another cliff equal in height to the one she is on. (See the figure.) By dropping a rock and timing its fall, she determines that it is 105 ft to the bottom of the canyon. She also determines that the angle to
A mathematics textbook author rafting down the Colorado River was told by a guide that the river dropped an average of 26 ft per mile as it ran through Cataract Canyon. Find the average angle of the river with the horizontal in degrees.
A mathematics textbook author has determined that her monthly gas usage y approximately follows the sine curvewhere y is measured in thousands of cubic feet (MCF) and t is the month of the year ranging from 1 to 12.(a) Graph this function on a graphing calculator.(b) Find the approximate gas usage
A mathematics professor wanted to use a computer drawing program to draw a picture of a regular pentagon (a five-sided figure with sides of equal length and with equal angles). He first made a 1-in. base by drawing a line from (0, 0) to (1, 0). (See the figure.) He then needed to find the
A 6-ft board is placed against a wall as shown in the figure below, forming a triangle-shaped area beneath it. At what angle θ should the board be placed to make the triangular area as large as possible? 6 ft 01
A proud father is attempting to take a picture of his daughters while they are riding on a merry-goround. Horses on this particular ride move up and down as the ride progresses according to the functionwhere h(t) represents the height (in feet) of the horse’s nose at time t, relative to the

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