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mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
Solve the inequality.|x + 5| ≥ 2
Prove the identity.tan 2θ = 2 tan θ/1 – tan2 θ
Solve the inequality.|x + 1| ≥ 3
Prove the identity.1/1 – sin θ + 1/1 + sin θ = 2 sec2θ
Solve the inequality.|2x – 3| ≤ 0.4
Solve the inequality.1 ≤ |x| ≤ 4
Solve the inequality.0 < |x – 5| < 1/2
Prove the identity.tan x + tan y = sin(x + y)/cos x cos y
Solve the inequality.|5x – 2| < 6
Graph the function y = x3 – 150x.
Find the sum of the serieswhere |x| 00 Σ .". n=0
(a) Approximate the sum of the series ∑1/n3 by using the sum of the first 10 terms. Estimate the error involved in this approximation.(b) How many terms are required to ensure that the sum is accurate to within 0.0005?
Find the angle between the vectors a = 〈2, 2, –1〉 and b = 〈5, –3, 2〉.
Sketch the surface z = y2 – x2.
Find the extreme values of the function f(x, y) = x2 + 2y2 on the circle x2 + y2 = 1.
Evaluate the iterated integrals.(a)(b) ² f²x²y dy dx Jo
Evaluate 4-x² 2 - S²₂ S ²² (1² + y²) dz dy dx. +
If F(x, y, z) = xz i +xy z j – y2k, find curl F.
Solve the differential equation.4y" + 4y' + y = 0
Solve the equation y" – 6y' + 13y = 0.
Investigate the family of curves with parametric equationsWhat do these curves have in common? How does the shape change as a increases? x = a + cos t y = a tant + sin t
Solve the differential equation.y" – 8y' +12y = 0
Graph the curve r = sin(8θ/5).
Test the series for convergence or divergence. Σ (-1)+1, M=1 n? 3 n' + 1
Investigate the family of polar curves given by r = 1 + c sin θ. How does the shape change as changes? (These curves are called limaçons, after a French word for snail, because of the shape of the curves for certain values of c.)
Express 1/(1 + x2) as the sum of a power series and find the interval of convergence.
(a) Approximate the function f(x) = 3√x by a Taylor polynomial of degree 2 at a = 8.(b) How accurate is this approximation when 7 ≤ x ≤ 9?
Determine whether the seriesis convergent or divergent. 8 Σ n=1 cos n n² cos 1 1² + cos 2 2² + cos 3 3² +
Determine whether the series converges or diverges. 00 5 Σ n=1 2n² + 4n+ 3
Indicate which tests should be used. Σ n=1 n - 1 2n + 1
For what values of x is the series convergent? 00 Σ n.x" n=0
Find n lim - n+ 1 I + U00+u
Find the Maclaurin series of the function f(x) = ex and its radius of convergence.
Find the sum of the series correct to three decimal places. (By definition, 0! = 1.) 00 Σ n=0 (-1)" n!
Find the sum of the geometric series5 –10/3 + 20/9 – 40/27 + .....
(a) What is the maximum error possible in using the approximation when –0.3 ≤ x ≤ 0.3? Use this approximation to find sin 12° correct to six decimal places. (b) For what values of x is this approximation accurate to within 0.00005? sin x = x - EX 3! + 5!
Find a power series representation for 1/(x + 2).
For what values of p is the series convergent? Σ n=1 1 nP
Test the series for absolute convergence. 00 13 Σ (-1)". 3" n=1
Indicate which tests should be used. n³ + 1 Σ ἥ=1 3η3 + 4n? + 2
For what values of x does the series converge? 00 3 n=1 (x − 3)" n
Calculate lim 11 →∞0 In n n
Prove that ex is equal to the sum of its Maclaurin series.
Is the series convergent or divergent? Σ 22"31-n n=1
Test the series for convergence or divergence. 00 1 Σ n=1 2" - 1
In Einstein’s theory of special relativity the mass of an object moving with velocity v iswhere m0 is the mass of the object when at rest and c is the speed of light. The kinetic energy of the object is the difference between its total energy and its energy at rest:(a) Show that when v is very
Find a power series representation of x3/(x + 2).
Test the convergence of the series 00 h Σ n=1_n! 71
Indicate which tests should be used. 00 Σ new2 n=1
Find the domain of the Bessel function of order 0 defined by Jo(.x) - Σ 71-0 (-1)"x2n 22n(n!)²
Find the Taylor series for f(x) = ex at a = 2.
Determine whether the sequence an = (–1)n is convergent or divergent.
Write the numberas a ratio of integers. 2.317 2.3171717... =
Determine whether the series converges or diverges. In n Σ n=1 n
Determine whether the series converges or diverges. 00 2n² + 3n Σ n=1 15 + ης
Test the convergence of the series Σ n=1 2n + 3 3η + 2 Τ
Indicate which tests should be used. η3 ht + 1 00 Σ (-1)" - n=1
Find the radius of convergence and interval of convergence of the series (-3)"x" Σ n=o Vn + 1 n+
Find the Maclaurin series for sin x and prove that it represents sin x for all x.
Evaluateif it exists. lim (-1)" n
What is wrong with the following calculation? 3 (²-4) --- 1 --4 dx = = -1 X 1 3 3
Use Property 8 to estimate 10 e-x² dx.
A ball is thrown upward with a speed of 48 ft/s from the edge of a cliff 432 ft above the ground. Find its height above the ground t seconds later. When does it reach its maximum height? When does it hit the ground?
(a) How much work is done in lifting a 1.2-kg book off the floor to put it on a desk that is 0.7 m high? Use the fact that the acceleration due to gravity is g = 9.8 m/s2. (b) How much work is done in lifting a 20-lb weight 6 ft off the ground?
Find the average value of the function f(x) = 1 + x2 on the interval [–1, 2].
Evaluate Si dx (3 – 5x)
Calculate JI In x -dx.
Find the area of the region bounded above by y = ex, bounded below by y = x, and bounded on the sides by x = 0 and x = 1.
Find the volume of the solid obtained by rotating about the y-axis the region bounded by y = 2x2 – x3 and y = 0.
Show that the volume of a sphere of radius r is V = 4/3πr3.
When a particle is located a distance x feet from the origin, a force of x2 + 2x pounds acts on it. How much work is done in moving it from x = 1 to x = 3?
Show that the average velocity of a car over a time interval [t1, t2] is the same as the average of its velocities during the trip.
Find the area of the region enclosed by the parabolas y = x2 and y = 2x – x2.
Find the volume of the solid obtained by rotating about the y-axis the region between y = x and y = x2.
Find the volume of the solid obtained by rotating about the x-axis the region under the curve y = √x from 0 to 1. Illustrate the definition of volume by sketching a typical approximating cylinder.
A force of 40 N is required to hold a spring that has been stretched from its natural length of 10 cm to a length of 15 cm. How much work is done in stretching the spring from 15 cm to 18 cm?
Find the approximate area of the region bounded by the curves y = x/√x2 + 1 and y = x4 – x.
Use cylindrical shells to find the volume of the solid obtained by rotating about the x-axis the region under the curve y = √x from 0 to 1.
If f(x) = x2 and g(x) = x – 3, find the composite functions f ∘ g and g ∘ f.
Expressas an integral on the interval [0, π]. Μ lim Σ (x + x, sinx;) Δι i-1
Evaluate *x sin t t X lim x-3 x - 3√3 - dt
Find ∫x3 cos(x4 + 2) dx.
If f is the function whose graph is shown in Figure 2 and g(x) = ∫0x f(t) dt, find the values of g(0), g(1), g(2), g(3), g(4), and g(5). Then sketch a rough graph of g.Figure 2 y -2. 1 0 1 2 y = f(t) 4
Evaluate cos 0 sin²0 - do.
For the region S in Example 1, show that the sum of the areas of the upper approximating rectangles approaches 1/3, that is,Data from Example 1Use rectangles to estimate the area under the parabola y = x2 from 0 to 1 (the parabolic region S illustrated in Figure 3). lim Rn 11-00 3
(a) Evaluate the Riemann sum for f(x) = x3 – 6x taking the sample points to be right endpoints and a = 0, b = 3, and n = 6. (b) Evaluate £³ (x³ - (x³ - 6x) dx.
Find the derivative of the function g(x) = f* √1 + 1² dt.
Evaluate ∫ √2x + 1 dx.
Evaluate ∫03(x3 – 6x) dx.
Let A be the area of the region that lies under the graph of f(x) = e–x between x = 0 and x = 2. (a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit. (b) Estimate the area by taking the sample points to be midpoints and using four sub- intervals and then
Find √1 - 4x² dx.
(a) Set up an expression for ∫13 ex dx as a limit of sums. (b) Use a computer algebra system to evaluate the expression.
Find d dx Ji sec t dt.
Find and interpret the result in terms of areas. (21 ੴ 2x³ - 6x + 3 x² + 1, dx
Evaluate the following integrals by interpreting each in terms of areas. (a) √1 - x² dx (b) f(x - 1) a dx
Calculate ∫e5x dx.
Evaluate 9 21² + 1²√1 - 1 dt.
Evaluate the integral ∫13 ex dx.
Use the Midpoint Rule with n = 5 to approximate 2 1 J1 X - dx.
Find ∫√1 + x2 x5 dx.
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