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mathematics
calculus

Questions and Answers of Calculus

Find an equation of the hyperbola with foci (±3, 0) and asymptotes 2y, ±x.
Find an equation of the ellipse with foci (3, ±2) and major axis with length 8.
Find an equation for the ellipse that shares a vertex and a focus with the parabola x2 + y = 100 and that has its other focus at the origin.
Show that if m is any real number, then there are exactly two lines of slope that are tangent to the ellipse x2/a2 + y2/b2 = 1 and their equations are y = ± √a2m2 + b2.
Find a polar equation for the ellipse with focus at the origin, eccentricity 1/3, and directrix with equation r = 4 sec θ.
Show that the angles between the polar axis and the asymptotes of the hyperbola r = ed / (1 – e cos θ), e > 1, are given by cos –1 (±1/e).
In the figure the circle of radius is stationary, and for every θ, the point P is the midpoint of the segment QR. The curve traced out by P for 0
A curve is defined by the parametric equationsFind the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line.
(a) Find the highest and lowest points on the curve x4 + y4 = x2 + y2.(b) Sketch the curve. (Notice that it is symmetric with respect to both axes and both of the lines y = ±x, so it suffices to
What is the smallest viewing rectangle that contains every member of the family of polar curves r = 1 + c sin θ, where 0 < c < 1? Illustrate your answer by graphing several members of the family
Four bugs are placed at the four corners of a square with side length a. The bugs crawl counterclockwise at the same speed and each bug crawls directly toward the next bug at all times. They approach
A curve called the folium of Descartes is defined by the parametric equations(a) Show that if (a, b) lies on the curve, then so does (b, a); that is, the curve is symmetric with respect to the line y
(a) What is a parametric curve?(b) How do you sketch a parametric curve?
(a) How do you find the slope of a tangent to a parametric curve?(b) How do you find the area under a parametric curve?
Write an expression for each of the following:(a) The length of a parametric curve(b) The area of the surface obtained by rotating a parametric curve about the x–axis
(a) Use a diagram to explain the meaning of the polar coordinates (r, θ) of a point. (b) Write equations that express the Cartesian coordinates (x, y) of a point in terms of the polar
(a) How do you find the slope of a tangent line to a polar curve?(b) How do you find the area of a region bounded by a polar curve?(c) How do you find the length of a polar curve?
(a) Give a geometric definition of a parabola. (b) Write an equation of a parabola with focus (0, p) and directrix y = – p. What if the focus is (p, 0) and the directrix is x = – p?
(a) Give a definition of an ellipse in terms of foci.(b) Write an equation for the ellipse with foci and (±c, 0) vertices (±c, 0).
(a) Give a definition of a hyperbola in terms of foci.(b) Write an equation for the hyperbola with foci (±c, 0) and vertices (±c, 0).(c) Write equations for the asymptotes of the hyperbola in part
(a) What is the eccentricity of a conic section?(b) What can you say about the eccentricity if the conic section is an ellipse A hyperbola A parabola?(c) Write a polar equation for a conic section
(a) What is a sequence? (b) What does it mean to say that lim x→∞ αn = 8? (c) What does it mean to say that lim x→∞ αn = ∞?
(a) What is a convergent sequence? Give two examples.(b) What is a divergent sequence? Give two examples.
If $1000 is invested at 6% interest, compounded annually, then after years the investment is worth an = 1000(1.06)n dollars.(a) Find the first five terms of the sequence {an}.(b) Is the sequence
Find the first 40 terms of the sequence defined by and a1 = 11.Do the same if a1 = 25. Make a conjecture about this type of sequence.
For what values of is the sequence {nrn}.convergent?
(a) If {an} is convergent, show that(b) A sequence {an} is defined by a1 = 1 and an+1 = 1/ (1 + an) for n > 1. Assuming that {an} is convergent, find its limit.
A sequence {an} is given by a1 = √2, an+1=√ 2 + an. (a) By induction or otherwise, show that {an} is increasing and bounded above by 3. Apply Theorem 11 to show that lim n→∞
Show that the sequence defined by a1 = 1, an+1 = 3 – 1/an is increasing and an < 3 for all a. Deduce that {an} is convergent and find its limit.
Show that the sequence defined by an1 = 2, an+1 = 1/3 – an satisfies 0 < an < 2and is decreasing. Deduce that the sequence is convergent and find its limit.
(a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn
(a) Let a1 = a, a2 = f(a), a3 = f(a2) = f(f(a)), . . . an+1 = f(an), where f is a continuous function. If, limn→∞ an = L show that f(L) = L. (b) Illustrate part (a) by taking f(x) = cos
(a) Use a graph to guess the value of the limit(b) Use a graph of the sequence in part (a) to find the smallest values of N that correspond to ε = 0.1 and ε = 0.001 in Definition 2.
(a) What is the difference between a sequence and a series?(b) What is a convergent series? What is a divergent series?
Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is
Find the values of for which the series converges. Find the sum of the series for those values of x.
We have seen that the harmonic series is a divergent series whose terms approach 0. Show that is another series with this property.
Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using the CAS to sum
When money is spent on goods and services, those that receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call
A certain ball has the property that each time it falls from a height onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1. Suppose that the ball is dropped from an initial height
Graph the curves y = x€, 0
The figure shows two circles C and D of radius 1 that touch at P.T is a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle
A right triangle ABC is given with
What is wrong with the following calculation?(Guido Ubaldus thought that this proved the existence of God because €œsomething has been created out of nothing.€)
The Cantor set, named after the German mathematician George Cantor (1845€“1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval (1/3, 2/3).
(a) A sequence {an} is defined recursively by the equation an = ½ (an–1 + an – 2) for n > 3, where a1 and a2 can be any real numbers. Experiment with various values of a1 and a2 and use your
Consider the series(a) Find the partial sums and s1, s2, s3. Do you recognize the denominators? Use the pattern to guess a formula for sn.(b) Use mathematical induction to prove your guess.(c) Show
In the figure there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length
(a) What is an alternating series?(b) Under what conditions does an alternating series converge?(c) If these conditions are satisfied, what can you say about the remainder after terms?
Let {Pn} be a sequence of points determined as in the figure. Thus | AP1 | = 1, | Pn Pn+1 | = 2n–1, and angle AP n Pn+1 is a right angle. Find lim n→∞ < Pn APn+1
To construct the snowflake curve, start with an equilateral triangle with sides of length . Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle
Find the sum of the series 1 + ½ + 1/3 + ¼ + 1/8 + 1/9 + 1/12 + . . . where the terms are the reciprocals of the positive integers whose only prime’s factors are 2s and 3s.
(a) Prove a formula similar to the one in Problem 7(a) but involving arccot instead of arc tan.(b) Find the sum of the series
If a0 + a1 + a2 + . . . ak = 0 show thatIf you don€™t see how to probe this, try the problem-solving strategy of using analogy (see page 80). Try the special cases k = 1 and k = 2 first. If
Suppose you have a large supply of books, all the same size, and you stack them at the edge of a table, with each book extending farther beyond the edge of the table than the one beneath it. Show
Suppose that circles of equal diameter are packed tightly in rows inside an equilateral triangle. (The figure illustrates the case n = 4.) If A is the area of the triangle and An is the total area
Starting with the vertices P1 (0, 1), P2 (1, 1), P3 (1, 0), P4 (0, 0) of a square, we construct further points as shown in the figure: P5 is the midpoint of P1 P2, P6 is the midpoint of P2P3, P7 is
Right-angled triangles are constructed as in the figure. Each triangle has height 1 and its base is the hypotenuse of the preceding triangle. Show that this sequence of triangles makes indefinitely
Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less
Calculate the first 10 partial sums of the series and graph both the sequence of terms and the sequence of partial sums on the same screen. Estimate the error in using the 10th partial sum to
Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error.
Show that the series Σ (– 1)n – 1bn, where bn = 1/n if n is odd and bn = 1/ n2if n is even, is divergent. Why does the Alternating Series Test not apply?
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 thatShow that this series always converges.
Suppose Σ an and Σ bn are series with positive terms and Σ bn is known to be convergent. (a) If an > bn for all n, what can you say about Σ an? Why? (b) If an < bn for all n,
What can you say about the series Σ an in each of the following cases?
For which of the following series is the Ratio Test inconclusive (that is, it fails to give a definite answer)?
(a) Find the partial sum s5 of the series Σn–1 1/n2n. Use Exercise 34 to estimate the error in using as an approximation to the sum of the series. (b) Find a value of so that sn is within
Use the sum of the first 10 terms to approximate the sum of the series. Use Exercise 34 to estimate the error.
Prove that if Σ an is absolutely convergent, then
Given any series Σ an we define a series Σ an+ whose terms are all the positive terms of Σ an and a series Σ an€“ whose terms are all the negative terms of Σ
Prove that if Σ an is a conditionally convergent series and r is any real number, then there is a rearrangement of Σ an whose sum is
What is a power series?
(a) What is the radius of convergence of a power series? How do you find it?(b) What is the interval of convergence of a power series? How do you find it?
The function J1 defined by is called the Bessel function of order 1.(a) Find its domain.(b) Graph the first several partial sums on a common screen.(c) If your CAS has built-in Bessel functions,
The function A defined byis called the Airy function after the English mathematician and astronomer Sir George Airy (1801€“1892).(a) Find the domain of the Airy function.(b) Graph the first
Show that if lim n→∞ n√ |Cn| = c, where c ≠ 0, then the radius of convergence of the power series Σ cn xn is R = 1/c.
Suppose that the power series Σ cn (x – a)n satisfies cn ≠ 0 for all n. Show that if lim n→∞ | Cn / Cn+1 | = c exists, then it is equal to the radius of convergence of the
Suppose the series Σ cn xn has radius of convergence 2 and the series Σ dn xn has radius of convergence 3. What is the radius of convergence of the series Σ (cn + dn)xn?
Suppose that the radius of convergence of the power series is Σ cn xn. What is the radius of convergence of the power series Σ cn x2n?
Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence.
(a) Find a power series representation for f(x) = in (1 + x).What is the radius of convergence?(b) Use part (a) to find a power series for f(x) = x in (1 + x).(c) Use part (a) to find a power series
Find a power series representation for the function and determine the radius of convergence.
Find a power series representation for f, and graph and several partial sums sn (x) on the same screen. What happens as n increases?
Evaluate the indefinite integral as a power series. What is the radius of convergence?
Use a power series to approximate the definite integral to six decimal places.
Use the result of Example 6 to compute correct to five decimal places.
Show that the function is a solution of the differential equation f€(x) + f(x) = 0
(a) Show that J0 (the Bessel function of order 0 given in Example 4) satisfies the differential equation(b) Evaluate f J0(x) dx x1 correct to three decimal places
Let fn(x) = (sin nx) / n2. Show that the series Σ fn(x) converges for all values of but the series of derivatives Σ f’ n(x) diverges when x = 2nπ, an integer. For what values of does
Use the power series for tan €“1x to prove the following expression for π as the sum of an infinite series:
Find the Maclaurin series of f (by any method) and its radius of convergence. Graph f and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these
Use the Maclaurin series for ex to calculate e–0.2 correct to five decimal places.
Use the Maclaurin series for sin x to compute sin 3o correct to five decimal places.
Evaluate the indefinite integral as an infinite series.
Use series to approximate the definite integral to within the indicated accuracy.
Use the series in Example 10(b) to evaluateWe found this limit in Example 4 in Section 4.4 using l€™ Hospital€™s Rule three times. Which method do you prefer?
Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function.
(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.
Use the binomial series to expand the function as a Maclaurin series and to find the first three Taylor polynomials T1, T2, and T3. Graph the function and these Taylor polynomials in the interval of
(a) Use the binomial series to expand 1/√1 – x2. (b) Use part (a) to find the Maclaurin series for sin –1x.

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