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mathematics
calculus

Calculus 9th edition Dale Varberg, Edwin J. Purcell, Steven E. Rigdon - Solutions

A visitor from outer space is approaching the earth at 2 kilometers per second. How fast is the angle θ subtended by the earth at her eye increasing when she is 3000 kilometers from the surface?
In problems, verify that given equations are identities. (a) ex = cosh x + sinh x (b) e2x = cosh 2x + sinh 2x (c) e-x = cosh x - sinh x (d) e-2x = cosh 2x - sinh 2x
In problems, find Dxy. (a) y = sinh2x (b) y = cosh2x (c) y = 5sinh2x (d) y = cosh3x (e) y = cosh (3x + 1)
Find the area of the region bounded by y = cosh 2x, y = 0, x = 0 and x = ln3.
In problems, evaluate each integral.(a) ˆ«sinh (3x + 2) dx(b) ˆ«x cosh(Ï€x2 + 5) dx(c)
Find the area of the region bounded by y = cosh 2x, y = 0, x = -ln5, and x = ln5.
Find the area of region bounded by y = sinhx, y = 0, and x = ln2.
Find the area of the region bounded by y = tanhx. Y = 0, x = -8, and x = 8
The region bounded by y = coshx, y = 0, x = 0 and x = 1 is revolved about the x-axis. Find the volume of the resulting solid. cosh2 x = (1 + cosh2x)/2
The region bounded by y = sinh x, y = 0, x = 0 and x = ln 10 is revolved about the x-axis, Find the volume of the resulting solid.
The curve y = cosh x, 0 ≤ x ≤ 1, is revolved about the x-axis. Find the area of the resulting surface.
Call the graph of y = b - a cosh (x/a) an inverted catenary and imagine it to be an arch sitting on the x-axis. Show that if the width of this arch along the x-axis is 2a then each of the following is true. (a) b = a cosh 1 ≈ 1.54308a. (b) The height of the arch is approximately 0.54308a (c) The
A farmer built a large hayshed of length 100 feet and width 48 feet. A cross section has the shape of an inverted satenary with equation y = 37-24cosh(x/24) (a) Draw a picture of this shed. (b) Find the volume of the shed. (c) Find the surface area of the roof of the shed.
Show that the area under the curve y = cosht 0 ≤ t ≤ x, is numerically equal to its arc length.
Find the equation fo the Gateway Arch in St. Louis, Missouri, given that it is an inverted catenary. Assume that it stands on the x-axis, that it is symmetric with respect to the y-axis, and that it is 630 feet wide at the base and 630 feet high at the center.
Draw the graphs of y = sinh x, y = ln(x+√x2 + 1) and y = x using the same axes and scaled so that -3 ≤ x ≤ 3 and -3 ≤ y ≤ 3. What does this demonstrate?
Refer to problem 58. Derive a formula for gd-1 (x). Draw its graph and also that of gd(x) using the same axes and thereby confirm your formula. In problem 58 gd(t) = tan-1(sinht)
In problems, differentiate each function.(a) ln x4/2(b) sin2(x3)(d) log10 (x5-1)
In problems, find the anti-derivative of each function and verify your result by differentiation. (a) e3x-1 (b) 6cot 3x (c) ex sinex
Let f(x) = x5 + 2x3 + 4x, -∞ < x < ∞ (a) Prove that f has an inverse g = f-1 (b) Evaluate g(7) = f-1(7) (c) Evaluate g'(7)
A certain radioactive substance has a half-life of 10 years. How long will it take for 100 grams to decay to 1 gram?
Use Euler's Method with h = 0.2 to approximate the solution to the differential equation y' = xy with initial condition y(1) = 2 over the interval [1, 2]
The population of a town grew exponentially from 10000 in 1990 to 14000 in 2000. Assuming that the same type of growth continues what will the population be in 2010.
Suppose that glucose is infused into the bloodstream of a patient at the rate of 3 grams per minute, but that the patient's body converts and removes glucose from its blood at a rate proportional to the amount present. Let Q(t) be the amount present at time t, with Q(0) = 120 (a) Write the
In problems, perform the indicated integrations.(a) ˆ« (x-2)5 dx(b) ˆ« ˆš3x dx(c)
Find the length of the curve y = ln (cosx) between x = 0 and x = π/4.
In problems, use integration by parts to evaluate each integral. (a) ∫ xex dx (b) ∫ xe3xdx (c) ∫ te5t+π dt
In problems, apply integration by parts twice to evaluate each integral.(a) ˆ« x2ex dx(b)(c) ˆ« ln2 zdz
In problems, use integration by parts to derive the given formula. (a) ∫ sin x sin3x dx = -3/8 sin x cos3x + 1/8 cos x sin 3x + C (b) ∫ cos 5x sin7x dx = -7/24 cos 5x cos 7x - 5/24 sin 5x 7x + C
In problems, derive the given reduction formula using integration by parts(a)
Find the area of the region bounded by the curve y = ln x, the x-axis, and the line x = e.
Find the volume of the solid generated by revolving the region of Problem 65 about the x-axis.
Find the area of the region bounded by the curves y = 3e-x/3, y = 0, x = 0 and x = 9, make a sketch.
Find the volume of the solid generated by revolving the region described in Problem 67 about the x-axis. In problem 67 Find the area of the region bounded by the curves y = 3e-x/3, y = 0, x = 0 and x = 9, make a sketch.
Find the volume of the solid obtained by revolving the region under the graph of y = sin(x/2) from x = 0 to x = 2π about the y-axis.
Evaluate the integral ∫cot x csc2x dx by parts in two different ways. (a) By differentiating cot x (b) By differentiating csc x (c) Show that the two results are equivalent up to a constant.
If p(x) is a polynomial of degree n and G1, G2, .... Gn+1, are successive anti-derivatives of a function g, then, by repeated integration by parts. ∫ p(x) g(x) dx = p(x) G1(x) - p' (x) G2(x) + p''(x) G3(x) - ... + (-1)n p(n) (x) Gn+1 (x) + C Use this result to find each of the following: (a)
Let Gn = n√((n+1) (n+2) ... (n+n)). Show limn→∞ (Gn/n) 4/e.
Find the error in the following "proof" that 0 = 1, In ∫ (1/t) dt, set u = 1/t and dv = dt. Then du = -t-2 dt and uv = 1. Integration by parts gives ∫ (1/t)dt = 1 - ∫ (-1/t)dt or 0 = 1
Suppose that you want to evaluate the integral ∫ e5x(4cos 7x + 6sin 7x) dx And you know from experience that the result will be of the form e5x(C1 cos 7x + C2 sin 7x) + C3. Compute C1 and C2 by differentiating the result and setting it equal to the integrand.
Suppose that f(t) has the property that f'(a) = f'(b) = 0 and that f(t) has two continuous derivatives. Use integration by parts to prove that
Use the result from problem 86 to evaluate ˆ«(3x4 + 2x2) ex dxIn problem 86
In problems, perform the indicated integrations. (a) ∫ sin2 x dx (b) ∫ sin4 6x dx (c) ∫ sin3 x dx
The region bounded by y = x + sinx, y = 0, x = π, is revolved about the x-axis. Find the volume of the resulting solid.
The region bounded by y = sin2(x2), y = 0, and x = √π/2 is revolved about the y-axis. Find the volume of the resulting solid.
Use the result of Problem 34 to obtain the famous formula of Francois Viete (1540-1603)
In problems, perform the indicated integrations. (a) ∫ √x + 1 dx (b) ∫ x 3√x + π dx (c) ∫ tdt/√3t+4
In problems, use the method of completing the square, along with a trigonometric substitution if needed, to evaluate each integral. (a) ∫ dx/√(x2 + 2x + 5) (b) ∫ dx/√(x2 + 4x + 5)
The region bounded by y = 1/(x2+2x+5), y = 0, x = 0, and x = 1, is revolved about the x-axis. Find the volume of the resulting solid.
The region of Problem 27 is revolved about the y-axis. Find the volume of the resulting solid. In problem 27 The region bounded by y = 1/(x2+2x+5), y = 0, x = 0, and x = 1, is revolved about the x-axis. Find the volume of the resulting solid.
Find ∫ xdx/(x2 + 9) by (a) An algebraic substitution and (b) A trigonometric substitution. Then reconcile you answers.
Two circles of radius b intersect as shown in Figure 6 with their centers 2a apart (0 ‰¤ a ‰¤ b). Find the area of the region of their overlap.
Generalize the idea in problem 33 by finding a formula for the area of the shaded lune shown in Figure 8
In problems, use the method of partial fraction decomposition to perform the required integration. (a) ∫ 1/x(x+1) dx (b) ∫ 2/x2+3x dx (c) ∫ 3/x2 -1 dx
In problems, solve the logistic differential equation representing population growth with the given initial condition. Then use the solution to predict the population size at time t = 3. y' = y(1-y), y(0) = 0.5
Explain what happens to the solution of the logistic differential equation if the initial population size is larger than the maximum capacity.
Assuming y0 < L, for what values of t is the graph of the population size y(t) concave up?
As a model for the production of trypsin for trypsinogen in digestion, biochemists have proposed the model dy/dt = k(A - y) (B+y) Where k> 0, A is the initial amount of trypsinogen, and B is the original amount of trypsin. Solve this differential equation.
In problems, evaluate the given integral.(a) ˆ« xe-5x dx(b) ˆ« x/x2 + 9 dx(c)
In problems, use the table of integrals on the inside back cover, perhaps combined with a substitution, to evaluate the given integrals. 1. (a) ∫ x √3x + 1 dx (b) ∫ex √(3ex + 1 ex) dx 2. (a) ∫ 2t √3-4t dt (b) ∫ cos t √(3 - 4cos t ) sin t dt
Use a CAS to evaluate the definite integrals in Problems. If the CAS does not give an exact answer in term of elementary function, give a numerical approximation.(a)(b) (c) (d)
In problems, the density of a rod is given. Find c so that the mass from 0 to c is equal to 1. Whenever possible find an exact solution. If this is not possible, find an approximation for c. (a) δ(x) = 1/(x+1) (b) δ(x) = 2/(x2+1) (c) δ(x) = ln (x+1)
Find the following derivatives (a) d/dx erf (x) (b) d/dx Si(x)
Find the derivatives of the Fresnel function (a) d/dx S (x) (b) d/dx C(x)
Over what intervals (on the nonnegative side of the number line) is the error function increasing? Concave up?
Find the coordinates of the first inflection point of the Fresnel function S(x) that is to the right of the origin.
In problems, evaluate each integral.(a)(b) ˆ« cot2 (2Î¸) d Î¸ (c)
Express the partial fraction decomposition of each rational function without computing the exact coefficients. For example(a) (3-4x2)/(2x+1)3 (b) (7x-41)/(x-1)2 (2-x)3 (c) (3x+1)/(x2 + x +10)2 (d) (x+1)2/(x2-x+10)2 (1-x2)2 (e) x5/(x+3)4(x2+2x+10)2 (f) (3x2 + 2x-1)2 / (2x2 + x + 10)3 $Express the partial fraction decomposition of each rational function without$
Find the volume of the solid generated by revolving the region under the graph of y = 1/√(3x-x2) From x = 1 to x = 2 about (a) The x-axis (b) The y-axis
Find the length of the curve y = x2 / 16 from x = 0 to x = 4
The region under the curve y = 1/(x2 + 5x + 6) from x = 0 to x = 3 is rotated about the x-axis. Compute the Volume of the solid that is generated.
If the region given in problem 46 is rotated about the y-axis find the volume of the solid.In problem 46
Find the volume of the solid created by revolving the region bounded by the x-axis and the curve y = 4x√2 - x about the y-axis.
Find the volume when the region created by the x-axis, y-axis, the curve y = 2(ex-1) and the curve x = ln3 is revolved about the line x = ln3.
Find the area of the region bounded by the x-axis, the curve y = 18/(x2√x2+9), and the lines x = √3 and x = 3√3.
Find the area of the region bounded by the curve s = t/(t-1)2, s = 0, t = -6, and t = 0
Find the length of the segment of the curve y = ln (sinx) form x = π/6 to x = π/3.
Use the table of integrals to evaluate the following integrals. (a) ∫ √(81-4x2)/x dx (b) ∫ ex(9-e2x)3/2 dx
Use the table of integrals to evaluate the following integrals (a) ∫ cosx √(sin2x + 4) dx (b) ∫ 1/(1-4x2) dx
Evaluate the first two derivatives of the sine integral
A rod has density δ(x) = 1/(1+x3). Use Newtwon's method to find the value of c so that mass of the road from 0 to c is 0.5
1. L'Hpital's Rule in useful in findingWhere both and are zero. 2. L'Hpital's Rule says that under appropriate conditions 3. From l'Hpital's Rule, we can conclude that = , but l'Hpital's Rule gives us no information about Because. 4. The proof of l'Hpital's Rule depends on Theorem?
Find each limit in Problem 1-5?1.2. 3. 4. 5.
In Problems 19-38, evaluate the given improper integral or show that it diverges?1.2. 3. 4. 5.
For what values of p does the integralConverge and for what values does it diverge?
For what values of p does the integralConverge and for what values does it diverge?
In Problem 41-44, use a comparison test (see Problem 46 of Section 8.4) to decide whether each integral converges or diverges.1.2. 3.4.
In problem 1-24, find the indicated limit Make sure that the have an indeterminate form before you apply I'Hpital's Rule?1.2. 3. 4. 5.
Find limx → 0 x2 sin (1/x) / tan x. Begin by deciding why l'Hôpital's Rule is not applicable. Then find the limit by other means?
For Figure 2, compute the following limits?(a) (b)
In Figure 3, CD = DE = DF = t. Find each limit?(a) (b)
Using the concepts of Section 5.4, you can show that he surface area of the prolate spheroid gotten by rotating the ellipse x2/a2 + y2/b2 = 1 (a > b) about the x-axis isWhat should A approach as a †’ b+? Use I'Hpital's Rule to show that this does happen?
Determine constants a, b, and c so that
L'Hpital's Rule in its 1696 form said this: if lim x → a f(x) = lim x → a g(x) = 0, then lim x → a f(x) /g(x) = f'(a) /g' (a). provided that f' (a) and g' (a) both exist and g'(a) ( 0. Prove this result whiteout recourse to Cauchy's Mean Value Theorem?
Use a CAS to evaluate the limits in Problems 34-37?1.2. 3. 4.
For Problems 38-41, plot the numerator f(x) and the denominator g(x) in the same graph window for each of these domains: -1 ( x ( 1, -0.1 ( x ( 0.1, and -0.01 ( x ( 0.01. From the plot, estimate the values of f'(x) and g' (x) and use these to approximate the given limit?1.2.3.4.
Use the concept of the linear approximation to a function (Section 2.9) to explain the geometric interpretation of l'Hopital's Rule in the marginal box next to Theorem A?
Find each limit in Problem 1-40, be sure you have an indeterminate form before applying I'Hpital's Rule?1.2. 3. 4. 5.
Find each limit. Transform to problems involving a continuous variable x. Assume that a > 0?a.b.c.d.
Find each limit?(a)(b) (c)(d)(e)

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