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calculus

Questions and Answers of Calculus

If the pattern shown in Figure 3 is continued indefinitely what fraction of the original square will eventually be painted?
Each triangle in the descending chain (Figure 4) has its vertices at the midpoints of the sides of the next larger one. If the indicated pattern of painting is continued indefinitely, what fraction
Circles are inscribed in the triangles of Problem 31 as indicated in Figure 5. If the original triangle is equilateral, what fraction of the area is eventually painted?
The Koch snowflake is formed as follows. Begin with an equilateral triangle, which we'll assume has sides of length 9. On each side, replace the middle third with two sides of an equilateral triangle
Consider the right triangle ABC as shown in Figure 7. Point A1 is determined by drawing a perpendicular to line AB through C; B1 is formed by drawing a line parallel to AC through A1. This process is
In one version of Zeno's paradox, Achilles can run ten times as fast as the tortoise, but the tortoise has a 100-yard head start. Achilles cannot catch the tortoise, says Zeno, because when Achilles
Tom and Joel are good runners, both able to run at a constant speed of 10 miles per hour. Their amazing dog Trot can do even better; he runs at 20 miles per hour. Starting from towns 60 miles apart,
Suppose that Peter and Paul alternate tossing a coin for which the probability of a head is 1/3 and the probability of a tail is 2/3. If they toss until someone gets a head, and Peter goes first,
Suppose that Peter and Paul alternate tossing a coin for which the probability of a head is 1/3 and the probability of a tail is 2/3. If they toss until someone gets a head, and Peter goes first,
Suppose that Mary rolls a fair die until a "6" occurs. Let X denote the random variable that is the number of tosses needed for this "6" to occur. Find the probability distribution for X and verify
Use the fact that(Which we will derive in Section 9.7) to find the expected value of the random variable X in Problem 39?
Use Problem 41 to conclude thatDiverges?
Suppose that one has an unlimited supply of identical blocks each 1 unit long.(a) Show that they may be stacked as in Figure 8 without toppling. Consider centers of mass.(b) How far can one make the
Prove that if (an diverges and (bn converges, then ((an + bn) diverges?
Show that it is possible for (an and (bn both to diverge and yet for ((an + bn) to converge?
By looking at the region in Figure 9 first vertically and then horizontally, conclude thatAnd use this fact to calculate: (a) (b) xÌ…, the horizontal coordinate of the centroid of the
Let r be a fixed number with | r |Converges, say with sum S. Use the properties ( of to show that And then obtain a formula for S, thus generalizing Problem 47a?
Many drugs are eliminated from the body in an exponential manner. Thus, if a drug is given in dosages of size C at time intervals of length t, the amount An of the drug in the body just after the (n
Find the sum of the series?
Use the Integral Test to determine the convergence or divergence of each of the following series?1.2.3.4.
In Problems 1-4, use any test developed so far, including any from Section 9.2, to decide about the convergence or divergence of the series. Give a reason for your conclusion?1.2. 3. 4.
In Problems 1-4, estimate the error that is made by approximating the sum of the given series by the sum of the first five terms?1.2.3.4.
For the series given in Problems 1-4, determine how large mast be so that using the nth partial sum to approximate the series gives an error of no more than 0.0002?1.2. 3. 4.
For what values of p doesConverge? Explain.
Use diagrams, as in Figure 1, to show that
Use diagrams, as in Figure 1, to show thatShow that the sequence In increasing and bounded above by 1?
Use the result of Problem 35 to prove that in Problem 36 exists. (The limit, denoted (, is called Euler's constant and is approximately 0.5772. It is currently not known whether y is rational or
Use Problem 35 to get good upper and lower bounds for the sum of the first 10 million terms of the harmonic series?
From Problem 37, we infer thatUse this to estimate the number of terms of the harmonic series that are needed to get a sum greater than 20 and compare with the result reported in Problem 44 of
Now that we have shown the existence of Euler's constant the hard way we will solve a much more general problem the easy way and watch y appear out of thin air, so to speak. Let f be continuous and
Let f be continuous, increasing, and concave down on [1, () as in Figure 4. Furthermore, let An be the area of the shaded region. Show that An is increasing with n, that An ( T where T is the area of
Specialize f of Problem 41 to f(x) = In x?(a) Show that(b) Conclude from part (a) and Problem 41 thatExists. It can be shown that k = (2(.(c) This means that n! ( (2(n(n/e)n, which is called
Show that the error used in approximating S by Sn satisfiesWhere the notation is the same as in the discussion preceding Example 5?
In Problems 1-4, use the Limit Comparison Test to determine convergence or divergence?1.2.3.4.
In Problem 1-4, determine convergence or divergence for each of the series. Indicate the test you use?1.2.3.4.
Let an > 0 and suppose that (an converges. Prove that (an2 converges?
Prove that limn→( (n! / nn) = 0 by considering the series (n! / nn?
Prove that if an ( 0, bn > 0, and (bn converges then (an converges?
Prove that if an ( 0, bn > 0, limn→( and (bn diverges then (an diverges?
Suppose that limn→( nan = 1. Prove that (an diverges?
Prove that if (an is convergent series of positive terms then (In (1 + an) converges?
Root Rest Prove that if an > 0 and limn→( (an)1 / n = R then (an converges if R < 1 and diverges if R > 1?
Test for convergence or divergence using the Root Test.a.b. c.
Test for convergence or divergence. In some cases, a clever manipulation using the properties of logarithms will simplify the problem?(a)(b) (c) (d) (e) (f)
Let p(n) and q(n) be polynomials in n with non-negative coefficients. Give simple conditions that determine the convergence or divergence of
Give conditions on p that determine the convergence or divergence of
1Test for convergence or divergence?a.b. c.
In Problem 1-4, Use the Ratio Test to determine convergence or divergence?1.2. 3. 4.
In Problems 1-4, show that each alternating series converges, and then estimate the error made by using the partial sum S9 as an approximation to the sum S of the series (see Examples 1-3)?1.2.3.4.
In Problems 1-4, classify each series as absolutely convergent, conditionally convergent, or divergent?1.2. 3. 4.
Prove that if (an diverges, so does (|an|?
Show that the positive terms of the alternating harmonic series form a divergent series. Show the same for the negative terms?
Show that the results in Problem 33 hold for any conditionally convergent series? The positive terms of the alternating harmonic series form a divergent series. Show the same for the negative terms?
Show that the alternating harmonic series1 - 1 / 2 + 1 / 3 - 1 / 4 + 1 / 5 - 1 / 6 + ...(Whose sum is actually In 2 ( 0.69) can be rearranged to converge to 1.3 by using the following steps.(a) Take
Use your calculator to help you find the first 20 terms of the series described in Problem 35. Calculate S20?
Explain why a conditionally convergent series can be rearranged to converge to any given number?
Show that a conditionally convergent series can be rearranged so to diverge?
Is not sufficient to guarantee the convergence of the alternating series ((-1)n+1an. Alternate the terms of (1/n and ((-1 / n2)?
Discuss the convergence or divergence of1 / (2 - 1 - 1 / (2 + 1 + 1 / (3 - 1 - 1 / (3 - 1 + 1 (4 - 1 - 1(4 + 1 + ... ?
Prove that ifAndBoth converge thenConverges absolutely. First show that 2|akbk| ( a2k + b2k?
Sketch the graph of y = (sin x) / x and then show thatConverges?
Show thatDiverges?
Show thatDiverges?
That 1 - 1 / 2 + 1 / 3 - 1 / 4 + ... - 1 / 2( = 1 + 1 / 2 + 1 / 3 + ... + 1 / 2( - (1 + 1 / 2 + 1 / 3 + ... + 1 / n) = 1 / n + 1 + 1 / n + 2 + ... + 1 / 2n Recognize the latter expression as a
In Problems 1 - 4, show that each series converges absolutely?1.2. 3. 4.
In Problems 1-4, find the convergence of set for the given power series?1.2.3.4.
Find the convergence set forWe know that (xn / n! converges for all x. Why can we conclude that limn†’( xn / n! = 0 for all x?
Let k be an arbitrary number and -1See Problem 29?
Find the radius of convergence of?
Find the radius of convergence of?Where p is a positive integer?
Find the sum s(x) ofWhat is the convergence set?
Suppose thatConverges at x = - 1. Why can you conclude that it converges at x = 6? Can you be sure that it converges at x = 7? Explain
Find the convergence set for each series?(a)(b)
Refer to Problem 52 of Section 9.1, where the Fibonacci sequence f1, f2, f3,.... Was defined. Find the radius of convergence of
Suppose that an+3 = an and let
Suppose that an+3 = an and let
In Problems 1-4, find the convergence set for the given power series. Find a formula for the nth term; then use the Absolute Ratio Test?1. x / 1 ( 2 - x2 / 2 ( 3 + x3 / 3 ( 4 - x4 / 4 ( 5 + x5 / 5 (
In Problems 1-4, find the power series representation for f(x) and specify the radius of convergence. Each is somehow related to a geometric series? 1. f(x) = 1 / 1 + x 2. f(x) = 1 / (1 + x)2
Obtain the power series in x for In[(1 + x) / (1 - x)] and specify its radius of convergence.In[(1 + x) / (1 - x)] = 1n(1 + x) - 1n(1 - x)?
Show that any positive number M can be represented by (1 + x) / (1 - x), where x lies within the interval of convergence of the series of Problem 11. Hence conclude that the natural logarithm of any
In Problem 1-4, use the result of Example 3 to find the power series in x for the given functions? 1. f(x) = e-x 2. f(x) = xex2 3. f(x) = ex + e-x 4. f(x) = e2x - 1 - 2x
In Problems 1-4, use the methods of Example 5 to find power series in x for each function f? 1. f(x) = e-x ( 1 / 1 - x 2. f(x) = ex tan-1 x 3. f(x) = tan - 1 x / ex 4. f(x) = ex / 1 + 1n (1 + x)
Find the sum of each of the following series by recognizing how it is related to something familiar? (a) x - x2 + x3 - x4 + x5 - ... (b) 1 / 2! + x / 3! + x2 / 4! + x3 / 5! + ... (c) 2x + 4x2 / 2 +
Follow the directions of Problem 25. (a) 1 + x2 + x4 + x6 + x8 + ... (b) cos x + cos2 x + cos3 x + cos4 x + ... (c) x2 / 2 + x4 / 4 + x6 / 6 + x8 / 8 + ...
Find the sum of
Find the sum of n(n + 1)xn?
Use the method of substitution to find power series through terms of degree 3?(a) tan-1 (ex - 1)
Suppose that f(x) =For | x | Let x = 0; then differentiate and let x = 0 again. Continue.?
Find the power series representation of x / (x2 - 3x + 2)? Use partial fractions?
Let y = y(x) = x - x3 / 3! + x5 / 5! - x7 / 7! + .... Show that y satisfies that differential equation y" + y = 0 with the conditions y(0) = 0 and y'(0) = 1. From this guess a simple formula for y?
Let {fn} be the Fibonacci sequence defined byf0 = 0, f1 = 1, fn+2 = fn+1 + fnIf F(x) =
Let y = y(x) =Where fn is as in Problem 33. Show that differential equation yn - y' -y = 0?
Did you ever wonder how people find the decimal expansion of it to a large number of places? One method depends on the following identity?( = 16 tan-1 (1/5) - 4 tan-1 (1/239)Find the first 6 digits
The number e is readily calculated to as many digits as desired using the rapidly converging seriese = 1 + 1 + 1 / 2! + 1 / 3! + 1 / 4! + ...This series can also be used to show that e is irrational.
In Problems 1-4, find the terms through x5 in the Maclaurin series for f(x). It may be easiest to use known Maclaurin series and then perform multiplications divisions, and so on. For example, tan x
In Problems 1-4, find the Taylor series in x - a through the term (x - a)3.1. ex, a = 12. Sin x, a = ( / 63. Cos x, a = ( / 34. Tan x, a = ( / 4
Let f(x) = (anxn be an even function (f(-x) = f(x)) for x in (-R, R). Prove that an = 0 if n is odd. Use the Uniqueness Theorem?
State and prove a theorem analogous to that in Problem 25 for odd functions?
Recall thatFind the first four nonzero terms in the Maclaurin series for sin-1 x?
Given thatFind the first four nonzero terms in the Maclaurin series for sinh-1 x?

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