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mathematics
precalculus

Thomas Calculus Early Transcendentals 13th Edition Joel R Hass, Christopher E Heil, Maurice D Weir - Solutions

What integrals lead to logarithms? Give examples.
What are the integrals of tan x and cot x? sec x and csc x?
Graph ƒ(x) = tan-1 x + tan-1(1/x) for -5 ≤ x ≤ 5. Then use calculus to explain what you see. How would you expect ƒ to behave beyond the interval [-5, 5]? Give reasons for your answer.
How is the exponential function ex defined? What are its domain, range, and derivative? What laws of exponents does it obey? Comment on its graph.
Rewrite the expression in terms of exponentials and simplify the results as much as you can.2 cosh (ln x)
Let g be a function that is differentiable throughout an open interval containing the origin. Suppose g has the following properties:i.ii.iii.a. Show that g(0) = 0.b. Show that g′(x) = 1 + [g(x)]2.c. Find g(x) by solving the differential equation in part (b). g(x) + g(y) 1 - g(x)g(y) x + y in the
Show that each function is a solution of the given initial value problem. Differential equation y' + y = 2 1+ 4e²x Initial equation y(-In 2) = TT 2 Solution candidate y = etan¹ (2¹)
Show that each function is a solution of the given initial value problem. Differential equation y' = ex² - 2xy Initial equation y(2) = 0 Solution candidate y = (x - 2)e-x²
Show that each function is a solution of the given initial value problem. Differential equation xy' + y = -sin x, x > 0 Initial equation y TT 2 = 0 Solution candidate y = COS X X
a. Suppose that g is an even function of x and h is an odd function of x. Show that if g(x) + h(x) = 0 for all x then g(x) = 0 for all x and h(x) = 0 for all x.b. Use the result in part (a) to show that if ƒ(x) = ƒE(x) + ƒO(x) is the sum of an even function ƒE(x) and an odd function ƒO(x),
Show that each function is a solution of the given initial value problem. Differential equation x²y' = xy - y², x > 1 Initial equation y(e) = e Solution candidate X In x
Rewrite the expression in terms of exponentials and simplify the results as much as you can.sinh (2 ln x)
How do you solve separable first-order differential equations?
Rewrite the expression in terms of exponentials and simplify the results as much as you can.cosh 5x + sinh 5x
What is the law of exponential change? How can it be derived from an initial value problem? What are some of the applications of the law?
Find the center of mass of a thin plate of constant density covering the region in the first and fourth quadrants enclosed by the curves y = 1/(1 + x2) and y = -1/(1 + x2) and by the lines x = 0 and x = 1.
Rewrite the expression in terms of exponentials and simplify the results as much as you can.cosh 3x - sinh 3x
What are the six basic hyperbolic functions? Comment on their domains, ranges, and graphs. What are some of the identities relating them?
A vegetable garden 50 ft wide is to be grown between two buildings, which are 500 ft apart along an east-west line. If the buildings are 200 ft and 350 ft tall, where should the garden be placed in order to receive the maximum number of hours of sunlight exposure? 200 ft tall West X 02 50 0₁ 450
The region between the curve y = 1/(2√x) and the x-axis from x = 1/4 to x = 4 is revolved about the x-axis to generate a solid.a. Find the volume of the solid.b. Find the centroid of the region
Rewrite the expression in terms of exponentials and simplify the results as much as you can.(sinh x + cosh x)4
What are the derivatives of the six basic hyperbolic functions? What are the corresponding integral formulas? What similarities do you see here with the six basic trigonometric functions?
Find the derivative of y with respect to the appropriate variable. y = sinh (2x + 1)
Rewrite the expression in terms of exponentials and simplify the results as much as you can.ln (cosh x + sinh x) + ln (cosh x - sinh x)
How are the inverse hyperbolic functions defined? Comment on their domains, ranges, and graphs. How can you find values of sech-1 x, csch-1 x, and coth-1 x using a calculator’s keys for cosh-1 x, sinh-1 x, and tanh-1 x?
What integrals lead naturally to inverse hyperbolic functions?
Use the definitions of cosh x and sinh x to show that cosh2 x - sinh2 x = 1.
How do you compare the growth rates of positive functions as x → ∞?
Find the derivative of y with respect to the appropriate variable. y = 2√ttanh Vt
Solve for y.3y = 2y+1
What roles do the functions ex and ln x play in growth comparisons?
Solve for y.4-y = 3y+2
Describe big-oh and little-oh notation. Give examples.
Find the derivative of y with respect to the appropriate variable. y =rtanh
Find the derivative of y with respect to the appropriate variable. y = 2√ttanh Vt 21
Solve for y.9e2y = x2
Which is more efficient—a sequential search or a binary search? Explain.
Solve for y.3y = 3 ln x
Find the derivative of y with respect to the appropriate variable.y = ln (sinh z)
Solve for y.ln (y - 1) = x + ln y
True, or false? Give reasons for your answers.a.b.c. x = o(x + ln x) d. ln (ln x) = o(ln x)e. tan-1 x = O(1) f. cosh x = O(ex) (²) 0 = DX + 1 x² 11-
Find the derivative of y with respect to the appropriate variable.y = ln (cosh z)
Find the derivative of y with respect to the appropriate variable. y = Incosh v - tanh² v
True, or false? Give reasons for your answers.a.b.c. ln x = o(x + 1) d. ln 2x = O(ln x)e. sec-1 x = O(1) f. sinh x = O(ex) 1 x4 = 0 (1/2/2 x² 1 + x 4,
Solve for y.ln (10 ln y) = ln 5x
Find the derivative of y with respect to the appropriate variable.y = sech θ(1 - ln sech θ)
Find the derivative of y with respect to the appropriate variable. y = In sinh v 1 coth² v
Does ƒ grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.a. ƒ(x) = log2 x, g(x) = log3 xb. ƒ(x) = x, g(x) = x + 1/xc. ƒ(x) = x/100, g(x) = xe-xd. ƒ(x) = x, g(x) = tan-1 xe. ƒ(x) = csc-1 x, g(x) = 1/xf. ƒ(x) = sinh x, g(x) = ex
Find the derivative of y with respect to the appropriate variable.y = csch θ(1 - ln csch θ)
Does ƒ grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.a. ƒ(x) = 3-x, g(x) = 2-xb. ƒ(x) = ln 2x, g(x) = ln x2c. ƒ(x) = 10x3 + 2x2, g(x) = exd. ƒ(x) = tan-1(1/x), g(x) = 1/xe. ƒ(x) = sin-1(1/x), g(x) = 1/x2f. ƒ(x) = sech x, g(x) = e-x
Repeat Exercise 23 for the functionsData from in Exercise 23a. Suppose you have three different algorithms for solving the same problem and each algorithm takes a number of steps that is of the order of one of the functions listed here:Which of the algorithms is the most efficient in the long run?
a. Suppose you have three different algorithms for solving the same problem and each algorithm takes a number of steps that is of the order of one of the functions listed here:Which of the algorithms is the most efficient in the long run? Give reasons for your answer.b. Graph the functions in part
The earth’s atmospheric pressure p is often modeled by assuming that the rate dp/dh at which p changes with the altitude h above sea level is proportional to p. Suppose that the pressure at sea level is 1013 millibars (about 14.7 pounds per square inch) and that the pressure at an altitude of 20
In some chemical reactions, the rate at which the amount of a substance changes with time is proportional to the amount present. For the change of δ-glucono lactone into gluconic acid, for example, when t is measured in hours. If there are 100 grams of δ-glucono lactone present when t = 0, how
Find the inverse of the function ƒ(x) = 1 + (1/x), x ≠ 0. Then show that ƒ-1(ƒ(x)) = ƒ(ƒ-1(x)) = x and that df-1 dx f(x) 1 f'(x)*
Show that ln x grows slower as x → ∞ than any nonconstant polynomial.
The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process did not come to a halt some 30,000 years ago, as many scientists contend.In northern
Find the derivative of y with respect to the appropriate variable.y = (x2 + 1) sech (ln x)
The function ƒ(x) = ex + x, being differentiable and one-to-one, has a differentiable inverse ƒ-1(x). Find the value of dƒ-1/dx at the point ƒ(ln 2).
Find the derivative of y with respect to the appropriate variable. -1 y = sinh ! Và
Find the derivative of y with respect to the appropriate variable.y = (4x2 - 1) csch (ln 2x)
Find the derivative of y with respect to the appropriate variable. y cosh ¹2√x + 1 =
The intensity L(x) of light x feet beneath the surface of the ocean satisfies the differential equationAs a diver, you know from experience that diving to 18 ft in the Caribbean Sea cuts the intensity in half. You cannot work without artificial light when the intensity falls below one-tenth of the
Find the derivative of y with respect to the appropriate variable. y = (1 - 0) tanh¯¹ 0
A girl is sliding down a slide shaped like the curve y = 9e-x/3. Her y-coordinate is changing at the rateft/sec. At approximately what rate is her x-coordinate changing when she reaches the bottom of the slide at x = 9 ft? (Take e3 to be 20 and round your answer to the nearest ft/sec.) dy/dt =
A particle is traveling upward and to the right along the curve y = ln x. Its x-coordinate is increasing at the rate (dx/dt) = √x m/sec. At what rate is the y-coordinate changing at the point (e2, 2)?
Suppose you are looking for an item in an ordered list one million items long. How many steps might it take to find that item with a sequential search? A binary search?
The processing of raw sugar has a step called “inversion” that changes the sugar’s molecular structure. Once the process has begun, the rate of change of the amount of raw sugar is proportional to the amount of raw sugar remaining. If 1000 kg of raw sugar reduces to 800 kg of raw sugar during
Suppose that electricity is draining from a capacitor at a rate that is proportional to the voltage V across its terminals and that, if t is measured in seconds,Solve this equation for V, using V0 to denote the value of V when t = 0. How long will it take the voltage to drop to 10% of its original
Find the derivative of y with respect to the appropriate variable. y = (0² +20) tanh¯¹ (0+1)
Find the derivative of y with respect to the appropriate variable. y = (1 t) coth¯¹ √t
Find the derivative of y with respect to the appropriate variable. y = cos¹x - x sech¯¹x
Find the derivative of y with respect to the appropriate variable. y = (1 - 1²) coth-¹ t
The functions ƒ(x) = ln 5x and g(x) = ln 3x differ by a constant. What constant? Give reasons for your answer.
An antibiotic is administered intravenously into the bloodstream at a constant rate r. As the drug flows through the patient’s system and acts on the infection that is present, it is removed from the bloodstream at a rate proportional to the amount in the bloodstream at that time. Since the
Find the derivative of y with respect to the appropriate variable. y = ln x + V1 - x² sech ¹x -1
a. If (ln x)/x = (ln 2)/2, must x = 2?b. If (ln x)/x = -2 ln 2, must x = 1/2?Give reasons for your answers.
Find the derivative of y with respect to the appropriate variable. y csch 1 = 0 (2)
Find the derivative of y with respect to the appropriate variable. y = cosh¯¹ (sec x), 0 < x < π/2
The quotient (log4 x)/(log2 x) has a constant value. What value? Give reasons for your answer.
Verify the integration formula.a.b. Is sech x dx = tan`'(sinh x) +C
Find the derivative of y with respect to the appropriate variable. y = csch ¹20
Find the derivative of y with respect to the appropriate variable. y = sinh '(tan x)
A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 3 hours there are 10,000 bacteria. At the end of 5 hours there are 40,000. How many bacteria were present initially?
Verify the integration formula. [₁ x sech ¹x dx -sech ¹x-V1-x² + C VI-R+C 2 2 =
To encourage buyers to place 100-unit orders, your firm’s sales department applies a continuous discount that makes the unit price a function p(x) of the number of units x ordered. The discount decreases the price at the rate of $0.01 per unit ordered. The price per unit for a 100-unit order is
How does ƒ(x) = logx (2) compare with g(x) = log2 (x)? Here is one way to find out.a. Use the equation loga b = (ln b)/(ln a) to express ƒ(x) and g(x) in terms of natural logarithms.b. Graph ƒ and g together. Comment on the behavior of ƒ in relation to the signs and values of g.
Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.a. How long will it take to reduce the number of cases to 1000?b. How long will it take to eradicate the disease, that is, reduce the number of cases to less than 1?
Verify the integration formula. S x coth ¹x dx = x² = 10 2 coth¹ x + + C
Verify the integration formula. s tanh-1x dx = xtanh-1x + In (1 - x²) + C
Biologists consider a species of animal or plant to be endangered if it is expected to become extinct within 20 years. If a certain species of wildlife is counted to have 1147 members at the present time, and the population has been steadily declining exponentially at an annual rate averaging 39%
The U.S. Census Bureau keeps a running clock totaling the U.S. population. On September 20, 2012, the total was increasing at the rate of 1 person every 12 sec. The population figure for 8:11 p.m. EST on that day was 314,419,198.a. Assuming exponential growth at a constant rate, find the rate
The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
Physicists using the radioactivity equation y = y0e-kt call the number 1/k the mean life of a radioactive nucleus. The mean life of a radon nucleus is about 1/0.18 = 5.6 days. The mean life of a carbon-14 nucleus is more than 8000 years. Show that 95% of the radioactive nuclei originally present in
Show that for any number a > 1as suggested by the accompanying figure. Ina a ["im xdx + ["" w dy In 0 e dy = a lna,
a. Show that the graph of ex is concave up over every interval of x-values.b. Show, by reference to the accompanying figure, that if 0 c. Use the inequality in part (b) to conclude thatThis inequality says that the geometric mean of two positive numbers is less than their logarithmic mean, which in
What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
What costs $27 million per gram and can be used to treat brain cancer, analyze coal for its sulfur content, and detect explosives in luggage? The answer is californium-252, a radioactive isotope so rare that only 8 g of it have been made in the Western world since its discovery by Glenn Seaborg in
A deep-dish apple pie, whose internal temperature was 220°F when removed from the oven, was set out on a breezy 40°F porch to cool. Fifteen minutes later, the pie’s internal temperature was 180°F. How long did it take the pie to cool from there to 70°F?
When hyperbolic function keys are not available on a calculator, it is still possible to evaluate the inverse hyperbolic functions by expressing them as logarithms, as shown here.Use the formulas in the box here to express the numbers in terms of natural logarithms.sinh-1 (-5/12) sinh'x = In(x +

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