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mathematics
calculus early transcendentals 9th

Questions and Answers of Calculus Early Transcendentals 9th

(a) By completing the square, show that(b) By factoring x3 + 1 as a sum of cubes, rewrite the integral in part (a). Then express 1/(x3 + 1) as the sum of a power series and use it to prove the
Give an example of a pair of series Σan and Σbn with positive terms where limn→∞ (an/bn) = 0 and Σbn diverges, but Σan converges.
Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 e n=1
Determine whether the sequence converges or diverges. If it converges, find the limit. an V2!+3n /21+3n
Use the formulaand the Maclaurin series for ln(1 + x) to show that 1 + x tanhx -1
Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1.f(x)
Use the following steps to show thatLet hn and sn be the partial sums of the harmonic and alternating harmonic series.(a) Show that s2n = h2n – hn .(b) We haveand thereforeUse these facts together
Use the Ratio Test to show that if the serieshas radius of convergence R, then each of the seriesalso has radius of convergence R. E-o Cnx" 8.
Determine whether the sequence converges or diverges. If it converges, find the limit.an = n sin(1/n)
Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1.f(x)
Show that if an > 0 and Σan is convergent, then Σ ln(1 + an) is convergent.
Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1.f(x)
Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1. f(x)
Find the Maclaurin series for f and the associated radius of convergence. You may use either the direct method (definition of a Maclaurin series) or the Maclaurin series listed in Table 11.10.1.f(x)
Let Σan and Σbn be series with positive terms. Is each of the following statements true or false? If the statement is false, give an example that disproves the statement.(a) If Σan and Σbn are
Determine whether the sequence converges or diverges. If it converges, find the limit.an = ln(2n2 + 1) − ln(n2 + 1)
Evaluateas an infinite series. e* dx
Determine whether the sequence converges or diverges. If it converges, find the limit. (In n)? an
Use series to approximatecorrect to two decimal places. I V1 + x* dx
Determine whether the sequence converges or diverges. If it converges, find the limit.an = arctan(ln n)
Determine whether the sequence converges or diverges. If it converges, find the limit.{0, 1, 0, 0, 1, 0, 0, 0, 1, . . .}
Determine whether the sequence converges or diverges. If it converges, find the limit. S1 1 1 1 1 1 1 1 lī: 3, 2» 4 3 5 4 6*
The force due to gravity on an object with mass m at a height h above the surface of the earth iswhere R is the radius of the earth and t is the acceleration due to gravity for an object on the
Find the values of x for which the series converges. Find the sum of the series for those values of x. (х - 2)" Σ in 3"
Determine whether the sequence converges or diverges. If it converges, find the limit. n! an 2"
Find the values of x for which the series converges. Find the sum of the series for those values of x. 2" Σ n=0
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. an = (-1)"
Find the values of x for which the series converges. Find the sum of the series for those values of x. 2" n=0
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess.123 sin n
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess.123 n? an
Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess.123 an 3" +
Use series to approximate the definite integral to within the indicated accuracy. r0.5 x'e* dx (| error|< 0.001)
If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly, the amount of interest accumulated after n months is given by the sequence(a) Find the
Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.y = e−x2 cos x
After injection of a dose D of insulin, the concentration of insulin in a patient’s system decays exponentially and so it can be written as De–at, where t represents time in hours and a is a
Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.y = sec x
Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.y = ex ln(1 + x)
Find the value of c if E (1 + c)" = 2. n=2
Find the value of c such that 00 Σen10. E enc
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?an = cos n
The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum
Find the function represented by the given power series. 00 4n (-1)" n! n=0
The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum
Find the function represented by the given power series. n-1
The Harmonic Series Diverges In Example 8 we proved that the harmonic series diverges. Here we outline additional methods of proving this fact. In each case, assume that the series converges with sum
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?an = n(−1)n
Find the function represented by the given power series. x2n+1 E(-1)" 22n+ (2n + 1) n+1, n=0
Graph the curves y = xn, 0 ≤ x ≤ 1, for n = 0, 1, 2, 3, 4, . . . on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example
Find the function represented by the given power series. x2a+1 E(-1)" 22n+ (2n + 1)! n=0
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?an = 3 − 2ne−n
Find the sum of the series. –1)" п! n=0
A right triangle ABC is given with ∠A = θ and |AC| = b. CD is drawn perpendicular to AB, DE is drawn perpendicular to BC, EF⊥ AB, and this process is continued indefinitely, as shown in the
Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?an = n3 − 3n + 3
Find the sum of the series. (-1)" 72 Σ 62" (2n)! n=0
What is wrong with the following calculation? 0 = 0 +0 +0 + ... = (1 – 1) + (1 - 1) + (1 – 1) + ... = 1-1+1- 1 + 1 – 1+ ... = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ... = 1 + 0 +0 + 0 + ...= 1
Find the sum of the series. 3" Σ 5"n!
Find the sum of the series. (In 2)? 1- In 2 + 2! (In 2) + 3!
Find the sum of the series. 81 9 + 2! 27 3 + 3! 4! + +
Find the sum of the series. 1 1 + 3· 23 1 1 1. 2 5. 25 7. 27
Use the Maclaurin series for f(x) = x/(1 + x2) to find f(101)(0).
Use the Maclaurin series for f(x) = x sin(x2) to find f(203)(0).
Use Theorem 7 to prove the Power Law: lim ak [lim a" if p > 0 and an >0
If f(x) = (1 + x3)30, what is f(58)(0)?
Prove that if limn→∞an = 0 and {bn} is bounded, then limn→∞(anbn) = 0.
Prove Taylor’s Inequality for n = 2, that is, prove that if|f"'(x)| ≤ M for |x − a| ≤ d, then M | R:(x)| x- a for |x – a|
Let an = (1 + 1/n)n.(a) Show that if 0 ≤ a < b, then(b) Deduce that bn[(n + 1)a − nb], an+1.(c) Use a = 1 + 1/(n + 1) and b = 1 + 1/n in part (b) to show that {an} is increasing.(d) Use a = 1
(a) Show that if limn→∞a2n = L and limn→∞a2n+1 =− L, then {an} is convergent and limn→∞an = L.(b) If a1 1 andfind the first eight terms of the sequence {an}. Then use part (a) to show
(a) What does the equation y − x2 represent as a curve in R2 ?(b) What does it represent as a surface in R3?(c) What does the equation z − y2 represent?
Find the cross product a × b and verify that it is orthogonal to both a and b.a = (2, 3, 0), b = (1, 0, 5)
Is each of the following quantities a vector or a scalar? Explain.(a) The cost of a theater ticket(b) The current in a river(c) The initial flight path from Houston to Dallas(d) The population of the
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If u = (u1, u2) and v = (ν1, ν2), then u ·
Determine whether each statement is true or false in R3.(a) Two lines parallel to a third line are parallel.(b) Two lines perpendicular to a third line are parallel.(c) Two planes parallel to a third
(a) Sketch the graph of y = ex as a curve in R2.(b) Sketch the graph of y = ex as a surface in R3.(c) Describe and sketch the surface z = ey.
Find the cross product a × b and verify that it is orthogonal to both a and b.a = (4, 3, –2), b = (2, –1, 1)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3 , |u + v| = |u|
Describe and sketch the surface.x2 + z2 = 4
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3, |u · v| =
Find a vector equation and parametric equations for the line.The line through the point (−1, 8, 7) and parallel to the vector 1 1 1 2 3 4
Find a · b.a = (1.5, 0.4), b = (−4, 6)
Describe and sketch the surface.x2 + y + 1 = 0
Find the cross product a × b and verify that it is orthogonal to both a and b. a = }i +j+k, b= i+ 2j – 3k
Copy the vectors in the figure and use them to draw the following vectors.(a) a + b(b) b + c(c) a + c(d) a − c(e) b + a + c(f) a − b − c a b
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3, u · v = v ·
Find a vector equation and parametric equations for the line.The line through the point (5, 7, 1) and perpendicular to the plane 3x − 2y + 2z = 8
Find parametric equations and symmetric equations for the line.The line through the points (−5, 2, 5) and (1, 6, −2)
Find the cross product a × b and verify that it is orthogonal to both a and b.a = (t3, t2, t), b = (t, 2t, 3t)
Write expressions for the scalar and vector projections of b onto a. Illustrate with diagrams.
Suppose that u · (v × w) = 2. Find the value of each of the following.(a) (u × v) · w(b) u · (w × v)(c) v · (u × w)(d) (u × v) · v
Find parametric equations and symmetric equations for the line.The line through the origin and the point (8, −1, 3)
If a = i – 2k and b = j + k, find a × b. Sketch a, b, and a × b as vectors starting at the origin.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3 and any scalar
Show that if a, b, and c are in V3, then(a × b) · [(b × c) × (c × a)] = [a · (b × c)]2
Find parametric equations and symmetric equations for the line.The line through the points (0.4, −0.2, 1.1) and (1.3, 0.8, −2.3)
Write an equation whose graph could be the surface shown. ZA y
Find the distance between the given points.(3, 5, −2), (−1, 1, −4)
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.For any vectors u and v in V3 and any scalar
Find parametric equations and symmetric equations for the line.The line through the points (12, 9, −13) and (−7, 9, 11)
Find a · b.|a| = 7, |b| = 4, the angle between a and b is 30°
Write an equation whose graph could be the surface shown. ZA
Find the distance between the given points.(−6, −3, 0), (2, 4, 5)

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