If f(x) = (1 + x3)30, what is f(58)(0)?
> (a) Use a computer algebra system to evaluate the following integrals. (b) Based on the pattern of your responses in part (a), guess the value of the integral (c) Check your guess with a CAS. Then prove it using the techniques of Section 7.2. For what va
> Determine whether the series is convergent or divergent. 1 -1 n? + 2n + 2
> Determine whether the series is convergent or divergent. 1 n2 + 4 8
> Determine whether the series is convergent or divergent. 1 + n³/2
> Determine whether the series is convergent or divergent. Vn + 4 n° ,2
> Determine whether the series is convergent or divergent. 1 + 2/2 1 + 4/4 1 + 3/3 5/5
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. S(x) — sin x, a — п/6
> Determine whether the series is convergent or divergent. 1 1 3 11 15 19
> Determine whether the series is convergent or divergent. 11 13
> Determine whether the series is convergent or divergent. Σ 0.9999 n-3
> Use the Integral Test to determine whether the series is convergent or divergent. E n’e * R-1
> (a) Use a computer algebra system to evaluate the following integrals. (b) Based on the pattern of your responses in part (a), guess the value of the integral if a ≠b. What if a = b? (c) Check your guess by asking your CAS to evaluate
> Use the Integral Test to determine whether the series is convergent or divergent. Σ A n² + 1
> Use the Integral Test to determine whether the series is convergent or divergent. 1 Σ (Зл — 1)*
> Use the Integral Test to determine whether the series is convergent or divergent. 2 Σ 5n – 1 ー1
> Use the Integral Test to determine whether the series is convergent or divergent. En 0.3 R-1 8
> Use the Integral Test to determine whether the series is convergent or divergent. En R-1
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 · dx (2х + 1)°
> Suppose f is a continuous positive decreasing function for x > 1 and an = f (n). By drawing a picture, rank the following three quantities in increasing order: 5 E ai E ai i-2 i-I
> Draw a picture to show that What can you conclude about the series? 1 Σ 1.3 -2 n 1.3
> For values of k between 3.6 and 4, compute and plot at least 100 terms and comment on the behavior of the sequence. What happens if you change p0 by 0.001? This type of behavior is called chaotic and is exhibited by insect populations under certain condi
> Evaluate the integral. tan 2y dy
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 3 — 4х 00
> (a) Show that the function defined by is not equal to its Maclaurin series. (b) Graph the function in part (a) and comment on its behavior near the origin. 1/22 if x + 0 f(x) = if x- 0
> The meaning of the decimal representation of a number 0.d1d2d3 ... (where the digit di is one of the numbers 0, 1, 2, . . . , 9) is that Show that this series always converges. di 0.d,d;dzd4 . .. 10 dz dz d4 102 103 104
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1 -1 3" + 4"
> Experiment with values of k between 3.4 and 3.5. What happens to the terms?
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. E 5 "cos’n cos'n 2.
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1/m nº
> Match the differential equation with its direction field (labeled I–IV). Give reasons for your answer. y' = x(2 – y) II yA --
> Determine whether the sequence converges or diverges. If it converges, find the limit. tan 'n an n
> Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. 1 5 + n*
> Determine whether the series converges or diverges. Σ 1+1/m
> Determine whether the series converges or diverges. Σ sin
> Determine whether the series converges or diverges. n! n"
> Determine whether the series converges or diverges. n! -1
> Explain why the Integral Test can’t be used to determine whether the series is convergent. cos'n Σ + n?
> Determine whether the series converges or diverges.
> Calculate terms of the sequence for a value of k between 3 and 3.4 and plot them. What do you notice about the behavior of the terms?
> Determine whether the series converges or diverges. 1 Σ -2 n Vn? – 1
> Evaluate the integral. (In x)² 2
> Determine whether the series converges or diverges. е" + 1 -1 пе" + 1
> Determine whether the series converges or diverges. n + 3" -i n + 2"
> Determine whether the series converges or diverges. 5 + 2n Σ (1 + n²)² 2)2
> Determine whether the series converges or diverges. n + 2 Σ (n + 1)3 A-3
> Determine whether the series converges or diverges. V1 + n Σ 2 + n n-1
> Determine whether the series converges or diverges. n2 + n + 1 n* + n? 2
> Determine whether the series converges or diverges. n п +1 -i n + n
> Determine whether the series converges or diverges. Σ Vn + 2
> Use the Ratio Test to determine whether the series is convergent or divergent. (-2)" Σ n2 -1
> Evaluate the integral. x² + 2x) cos x dx
> Determine whether the series converges or diverges. 1 Σ Vn² + 1 n-1
> Determine whether the series converges or diverges. i n"
> Determine whether the series converges or diverges. 4"11 Σ 3" – 2
> Determine whether the series converges or diverges. Σ V3n4 + 1
> Determine whether the series converges or diverges. 1 + cos n e"
> Determine whether the series converges or diverges. (2k – 1)(k² – 1) Σ (k + 1)(k² + 4)²
> Determine whether the series converges or diverges. Σ k3 + 4k + 3 k-1
> Determine whether the series converges or diverges. k sin?k 1 1 + k³ WI
> Determine whether the series converges or diverges. In k k k-1
> Determine whether the series converges or diverges. 6" -i 5" - 1
> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. et dx 4 - e2*
> Show that if p is an nth-degree polynomial, then p(x + 1) = ' Σ p®(x) i!
> Determine whether the series converges or diverges. 9" Σ 3 + 10"
> Determine whether the series converges or diverges. n - 1 AI n' + 1
> Determine whether the series converges or diverges. n + 1 n/n
> Determine whether the series converges or diverges. Σ In - 1 -2
> Determine whether the series converges or diverges. 1 Σ n' + 8 n-1
> Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. x = y? – 2y, 0 < y< 2
> We arrived at Formula 6.3.2, / by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where f is one­ to­ one and therefore has an inverse fu
> For what values of p is each series convergent? (In n)" 2 (-1)* 1 n-2
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. (-9)" Σ n10*+1 n-1
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(x) — х сos x,
> For what values of p is each series convergent? (-1)" -1 n + p
> For what values of p is each series convergent? (-1)* -1 Σ n" n-1
> Approximate the sum of the series correct to four decimal places. (-1)"-1 Σ n 4"
> Approximate the sum of the series correct to four decimal places. E (-1)"ne 2n
> Determine whether the differential equation is linear. — х y tan x
> Approximate the sum of the series correct to four decimal places. i(-1)** -1 ク
> Approximate the sum of the series correct to four decimal places. (-1)" Σ (2n)!
> (a) Use integration by parts to show that (b) If f and t are inverse functions and f9 is continuous, prove that (c) In the case where f and t are positive functions and b > a > 0, draw a diagram to give a geometric interpretation of part (b). (d) U
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (lerror|< 0.00005) n-
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)* Σ n2 2" -' (lerror |< 0.0005) -1
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? Σ (-)" (|error|< 0.0005) -1 n
> Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. 2 + 3n
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)*+1 Σ n° (|error|< 0.00005) n-1
> Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.
> Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.
> Determine whether the differential equation is linear. y' + x/y = x?
> Test the series for convergence or divergence. E (-1)"(/n + 1 – Vñ)
> Test the series for convergence or divergence. E(-1)". E(-1)* "". n!
> Test the series for convergence or divergence. E (-1)" cos n-1
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C − 0). |x² sin 2x dx
> Show that cosh x ≥1+ 1 2 x2 for all x.
> Evaluate the integral. w/4 (" tan'0 sec?0 do Jo
> Find the area enclosed by the curve in Exercise 29. Data from Exercise 29: At what points does the curve have vertical or horizontal tangents? Use this information to help sketch the curve. X= 2a cos t - a cos 21 y = 2a sin t – a sin 2t
> Find the sum of the series. E [tan '(n + 1) – tan 'n] -1
> Find the sum of the series. 1 Σ п(п + 3)
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). x3/2 In x dx
> Find the sum of the series. (-3)' Σ "-1 3n n-1
