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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Show that the parametrization of the ellipse by the angle θ is X = y = ab cos 0 Va² sin² 0 + b² cos² 0 ab sin Va² sin² 0 + b² cos² 0
Galileo tried unsuccessfully to find the area under a cycloid. Around 1630, Gilles de Roberval proved that the area under one arch of the cycloid c(t) = (Rt − R sin t, R − R cos t) generated by a
Match the parametrizations (a)–(d) with their plots in Figure 15, and draw an arrow indicating the direction of motion.(a) c(t) = (sin t, −t) (b) c(t) = (t2 − 9, 8t − t3)(c) c(t) = (1 − t,
Apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive. 80 n=1 27² n!
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. n=1 In(n + 4) n5/2
Determine a reduced fraction that has this decimal expansion.0.808888888 . . .
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=1 n n!
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.bn = n1/n
Show, by integrating the Maclaurin series for ƒ(x) =1√1 − x2, that for |x| sin ¹ x = x + n=1 1.3.5 (2n-1) x²n+1 2.4.6 (2n) 2n + 1
Use the equalities to show that for |x − 4| 1 1 - x 1 -3-(x-4) -3-3-3 1 + (34)
Differentiate the Maclaurin series for 1/1 − x twice to find the Maclaurin series of 1/(1 − x)3.
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 Σ n=2 n? + 1 | 13.5 – 2
Let T2 be the Taylor polynomial at a = 0.5 for ƒ(x) = cos(x2). Use the error bound to find the maximum possible value of |ƒ(0.6) − T2(0.6)|. Plot ƒ(3) to find an acceptable value of K.
Use the Limit Comparison Test to prove convergence or divergence of the infinite series. Σ Σ η=1 str Hint: Compare with n²². η=1 1 - cos
Verify that 0.999999 · · · = 1 by expressing the left side as a geometric series and determining the sum of the series.
Use the first five terms of the Maclaurin series in Exercise 45 to approximate sin−1 1/2 . Compare the result with the calculator value.Data From Exercise 45Show, by integrating the Maclaurin
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.cn = 9n/n!
Use the method of Exercise 45 to expand 1/(1 − x) in power series with centers c = 2 and c = −2. Determine the interval of convergence.Data From Exercise 45Use the equalities to show that for
Calculate the Maclaurin polynomial T2 for ƒ(x) = sech x and use the Error Bound to find the maximum possible value of Ιƒ(1/2) − T2 (1/2)Ι. Plot ƒ"' to find an acceptable value of K.
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 Σ n=1 1 n - Inn
Determine convergence or divergence using any method covered in the text so far. 00 n=1 2¹/n
Use the Limit Comparison Test to prove convergence or divergence of the infinite series.Compare with the harmonic series. Σ(1-2-1/n) n=1
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.an = 82n/n!
Which of the following are not geometric series? (c) (a) M8 M8 M8 M8 -
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=1 sin n n?
How many terms of the Maclaurin series of ƒ(x) = ln(1 + x) are needed to compute ln 1.2 to within an error of at most 0.0001? Make the computation and compare the result with the calculator value.
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 80 Σ n=2 n γης + 5 +5
Use the Error Bound to find a value of n for which the given inequality is satisfied. Then verify your result using a calculator.| cos 0.1 − Tn(0.1)| ≤ 10−7, a = 0
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. an || 3n² + n + 2 2n²-3
Determine convergence or divergence using any method covered so far. n=4 1 n²-9
Show thatconverges to zero. How many terms must be computed to get within 0.01 of zero? π = 3 3! + 15 5! E 7!
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=1 n! (2n)!
Use the Error Bound to find a value of n for which the given inequality is satisfied. Then verify your result using a calculator.|ln 1.3 − Tn(1.3)| ≤ 10−4, a = 1
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 n=1 1 3n - 2n
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. an = √n √n +4
Determine convergence or divergence using any method covered so far. 00 n=1 cos² n n²
Use the Maclaurin expansion for e−t2 to express the function F(x) = ∫x0 e−t2 dt as an alternating power series in x (Figure 3).(a) How many terms of the Maclaurin series are needed to
Apply integration to the expansion to prove that for -1 < x < 1, 00 1 1. Σ(1)"x" = 1-x+x - t 1 + x n=0 In(1 + x) = 00 Σ n=l (−1)"-1x" n x2 2 3 14 4 +
Determine convergence or divergence using any method covered in the text so far. 00 n=1 1 n+ √n
Use the Error Bound to find a value of n for which the given inequality is satisfied. Then verify your result using a calculator.|√1.3 − Tn(1.3)| ≤ 10−6, a = 1
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. an COS n n
Let F(x) = ∫x0 sin t dt/t. Show thatEvaluate F(1) to three decimal places. F(x) = x - x³ 3.3! + x5 5.5! x7 7.7! +
Prove the divergence of 00 n=0 9n + 2n 5n
Use the Error Bound to find a value of n for which the given inequality is satisfied. Then verify your result using a calculator.|e−0.1 − Tn(−0.1)| ≤ 10−6, a = 0
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=2 1 n(Inn)3
Use the Direct Comparison or Limit Comparison Test to determine whether the infinite series converges. 00 20 n +21" n21+201 Σ n=1
Use the result of Exercise 49 to prove thatUse your knowledge of alternating series to find an N such that the partial sum SN approximates ln 3/2 to within an error of at most 10−3. Confirm using
Determine convergence or divergence using any method covered so far. 8 n=1 n - cos n n³
Give a counterexample to show that each of the following statements is false.(a) If the general term an tends to zero, then (b) The Nth partial sum of the infinite series defined by {an} is aN.
Let ƒ(x) = e−x and T3(x) = 1 − x + x2/2 − x3/6.(a) Use the Error Bound to show that for all x ≥ 0,(b) Illustrate this inequality by plotting y = ƒ(x) − T3(x) and y = x4/24 together over
Express the definite integral as an infinite series and find its value to within an error of at most 10−4. So cos(x²) dx
Determine the convergence ofusing the Limit Comparison Test with bn = (2/3)n. 00 n=1 2n + n 3n - 2
Let F(x) = (x + 1) ln(1 + x) − x.(a) Apply integration to the result of Exercise 49 to prove that for −1 (b) Evaluate at x = 1/2 to prove(c) Use a calculator to verify that the partial sum
Determine convergence or divergence using any method covered so far. =1 n²-1 n²+1
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.dn = ln 5n − ln n!
Use the Error Bound with n = 4 to show that sin x-(x - 5) ≤ 120 S (for all x)
Suppose that an is an infinite series with partial sum SN = 5 − 2/N2. 00 Σ n=1 an
Express the definite integral as an infinite series and find its value to within an error of at most 10−4. S Jo 0 tan-¹(x²) dx
Determine convergence or divergence using any method covered in the text so far. 00 n=2 1 n² (Inn)³
Determine the convergence ofusing the Limit Comparison Test with bn = 1/1.4n. 00 n=1 In n 1.5n
Prove that for |x| Use the first two terms to approximate ∫01/2 dx/(x4 + 1) numerically. Use the fact that you have an alternating series to show that the error in this approximation is at most
Determine convergence or divergence using any method covered so far. 8 1 n² + sinn Σ; n=1
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.dn = ln(n2 + 4) − ln(n2 − 1)
Let Tn be the Taylor polynomial for ƒ(x) = ln x at a = 1, and let c > 1. Show thatThen find a value of n such that |ln 1.5 − Tn(1.5)| ≤ 10−2. |Inc - Tn (c)| ≤ lc - 1+1 n+1
Consider the archery competition in Example 6.(a) Assume that Nina goes first. Let pn represent the probability that Brook wins on his nth turn. Give an expression for pn. (b) Use the result from
Determine convergence or divergence using any method covered in the text so far. 00 Σ n=2 1 3 – n? –
Let an = 1 − √1 − 1. Show that lim an = n→∞0 O and that Σan an diverges. Show that an > n=1 2n
Use the result of Example 7 to show thatis an antiderivative of ƒ(x) = tan−1 x satisfying F(0) = 0. What is the radius of convergence of this power series? EXAMPLE 7 Power Series for Arctangent
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. an = 2+ 4 n² 1/3
Determine convergence or divergence using any method covered so far. 00 Σ(4/5), " n=5
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. tan-¹1 b₁ = tan SIN n
Let n ≥ 1. Show that if |x| is small, thenUse this approximation with n = 6 to estimate 1.51/6. (x + 1)¹/n 1+ X n + 1-n 2n² 2 ²x²
Consider the archery competition in Example 6. Assume that Nina’s probability of hitting the bull’s-eye on a turn is 0.45 and that Brook’s probability is p. Assume that Nina goes first. For
Express the definite integral as an infinite series and find its value to within an error of at most 10−4. dx Jo Vx4 + 1 مة
Determine convergence or divergence using any method covered in the text so far. 8 Σ n=1 n? + 4n - 3n4 + 9
Determine whether converges. 00 n=2 |-| n²
Verify that function F(x) = x tan−1 x − 1/2 ln(x2 + 1) is an antiderivative of ƒ(x) = tan−1 x satisfying F(0) = 0. Then use the result of Exercise 53 with x = √1/3 to show thatUse a
Determine convergence or divergence using any method covered so far. 00 n=1 1 3n²
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. Cn = In 2n + 1 3n + 4
Express the integral as an infinite series. 10 1 - cost t - dt, for all x
Verify that the third Maclaurin polynomial for ƒ(x) = ex sin x is equal to the product of the third Maclaurin polynomials of ƒ(x) = ex and ƒ(x) = sin x (after discarding terms of degree greater
Determine convergence or divergence using any method covered in the text so far. 00 Ση h=1 -0.8 η
Evaluate Use differentiation to show that 00 Σ n=1 n 2n
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. Cn = n n+n¹/n
The winner of a lottery receives m dollars at the end of each year for N years. The present value (PV) of this prize in today’s dollars is where r is the interest rate. Calculate PV if m =
Find the fourth Maclaurin polynomial for ƒ(x) = sin x cos x by multiplying the fourth Maclaurin polynomials for ƒ(x) = sin x and ƒ(x) = cos x.
Express the integral as an infinite series. So t - sint t - dt, for all x
Determine convergence or divergence using any method covered in the text so far. 00 Σ(0.8)="n n=1 -n₂-0.8
Determine whether the series converges absolutely. If it does not, determine whether it converges conditionally. 00 n=1 (-1)" Vn+ 2n
Use the power series for (1 + x2)−1 and differentiation to prove that for |x| 2x (x² + 1)² = Σ(-1)"-1(2n)x21-1 n=1
Determine convergence or divergence using any method covered so far. 00 n=2 (Inn)¹2 12 n⁹/8
Determine convergence or divergence using any method covered so far. 00 k=1 Alk
Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. Уп п 2n
Find the Maclaurin polynomials Tn for ƒ(x) = cos(x2). You may use the fact that Tn(x) is equal to the sum of the terms up to degree n obtained by substituting x2 for x in the nth Maclaurin
Express the integral as an infinite series. So In(1+²) dt, for x| < 1
If a patient takes a dose of D units of a particular drug, the amount of the dosage that remains in the patient’s bloodstream after t days is De−kt, where k is a positive constant depending on
Determine convergence or divergence using any method covered in the text so far. 00 Σ4-2+1 n=1
Determine whether the series converges absolutely. If it does not, determine whether it converges conditionally. 00 Σ n=1 (-1)" n1.1 In(n + 1)
Show that the following series converges absolutely for |x| Write F(x) as a sum of three geometric series with common ratio x3. F(x) = 1- x-x² + x³ = x² = x³ + xº - x² - x² + ... x°- - -
Determine convergence or divergence using any method covered so far. 00 n=1 4n 5n - 2n

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