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calculus

Questions and Answers of Calculus

Use the limit definition to prove that
Give an example of a divergent sequence {an} such thatconverges.
Let bn = an+1. Use the limit definition to prove that if {an} converges, then {bn} also converges and
Using the limit definition, prove that if {an} converges and {bn} diverges, then {an + bn} diverges.
Calculate S3, S4, and S5 and then find the sumusing the identity
Calculate S3, S4, and S5 and then find the sum of the telescoping seriesIn Exercises, use a computer algebra system to compute S10, S100, S500, and S1000 for the series. Do these values suggest
In Exercises, use a computer algebra system to compute S10, S100, S500, and S1000 for the series. Do these values suggest convergence to the given value?
The seriesconverges to 5/4. Calculate SN for N = 1, 2, . . . until you find an SN that approximates 5/4 with an error less than 0.0001.
In Exercises 1-2, compute the partial sums S2, S4, and S6.1.2.
In Exercises 1-2, calculate the first four terms of the sequence, starting with n = 1. 1. cn = 3n / n! 2. a1 = 2, an+1 = 2a2n - 3
Find a formula for the general term an (not the partial sum) of the infinite series.a.b. c. d.
Which of the following are not geometric series?a.b. c. d.
In Exercises 1-2, use the formula for the sum of a geometric series to find the sum or state that the series diverges.1.2.
In Exercises 17-22, use Theorem 3 to prove that the following series diverge.1.2.
Find the sum of
Compute the total area of the (infinitely many) triangles in Figure 4.
Find the total length of the infinite zigzag path in Figure 5 (each zag occurs at an angle of Ï€/7).
Let {bn} be a sequence and let an = bn ˆ’ bnˆ’1. Show that
Show that if a is a positive integer, then
Find a formula for the nth term of each sequence.(a)(b)
Professor George Andrews of Pennsylvania State University observed that we can use Eq. (7) to calculate the derivative of f (x) = xN (for N ‰¥ 0). Assume that a = 0 and let x = ra. Show
In Exercises 1-2, use Theorem 4 to determine the limit of the sequence1.2.
In Exercises 1-2, use Theorem 1 to determine the limit of the sequence or state that the sequence diverges. 1. An = 12. 2. cn = - 2-n
Show thatis increasing. Find an upper bound.
In Exercises use the Squeeze Theorem to evaluateby verifying the given inequality.
In Exercises 1-2, find the limit of the sequence using L'HÃ´pital's Rule.1.2.
In Exercises 35-62, use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges.1.2. cn = 1.01n
What happens if you apply the formula for the sum of a geometric series to the following series? Is the formula valid?
Find an N such that SN > 25 for the series
Colleen claims thatconverges because Is this valid reasoning?
Arvind asserts thattends to zero. Is this valid reasoning?
What is the sum of the following infinite series?
Give an example of a divergent infinite series whose general term tends to zero.
Does there exist an N such that SN > 25 for the seriesExplain.
In Exercises 1-2, use the Integral Test to determine whether the infinite series is convergent.1.2.
Apply Eq. (4) with M = 40,000 to show thatIs this consistent with Euler's result, according to which this infinite series has sum Ï€2/6?
Using a CAS and Eq. (5), determine the value ofto within an error less than 10ˆ’4.
Show thatconverges by using the Comparison Test with
In Exercises 19-30, use the Comparison Test to determine whether the infinite series is convergent.1.2.
LetVerify that for n ‰¥ 1, Can either inequality be used to show that S diverges? Show that and conclude that S diverges.
If the partial sums SN are increasing, then (choose the correct conclusion):{an} is a positive sequence
Exercise 1-2: For all a > 0 and b > 1, the inequalitiesare true for n sufficiently large (this can be proved using L'Hopital's Rule). Use this, together with the Comparison Theorem, to
In Exercises 1-2, use the Limit Comparison Test to prove convergence or divergence of the infinite series.1.2.
In Exercises 1-2, determine convergence or divergence using any method covered so far.1.2.
Let an = f (n), where f (x) is a continuous, decreasing function such that f (x) ‰¥ 0 andShow that
Give an example of a series such thatconverges but
In Exercises 3-10, determine whether the series converges absolutely, conditionally, or not at all.1.2.
Let
Show thatconverges absolutely.
Find a value of N such that SN approximates the series with an error of at most 10ˆ’5. If you have a CAS, compute this value of SN.
Show by counterexample that the Leibniz Test does not remain true if the sequence an tends to zero but is not assumed non increasing.
Use Exercise 38 to show that the following series converges:
Show that the following series diverges:
In Exercises 1-2, apply the Ratio Test to determine convergence or divergence, or state that the Ratio Test is inconclusive.1.2.
Is the Ratio test conclusive for the p-series
Show thatConverges if
Show thatconverges for all exponents k.
In Exercises 36-41, use the Root Test to determine convergence or divergence (or state that the test is inconclusive).1.2.
In Exercises 43-56, determine convergence or divergence using any method covered in the text so far.1.2.
1. Which one of the following does not parametrize a line? (a) r1(t) = (8 − t, 2t, 3t) (b) r2(t) = t3i − 7t3j + t3k (c) r3(t) = (8 − 4t3, 2 + 5t2, 9t3)
Evaluate r(2) and r(-1) for r(t) = (sin (π/2)t, t2,(t2 + 1) 1 j.
Find a vector parametrization of the line through P = (3, -5, 7) in the direction v = (3, 0, 1).
The function r(t) traces a circle. Determine the radius, center, and plane containing the circle. r(t) = (9 cos t)i + (9 sin t)j
The function r(t) traces a circle. Determine the radius, center, and plane containing the circle. r(t) = (sin t, 0, 4 + cos t)
Let C be the curve r(t) = (t cos t, t sin t, t). Show that C lies on the cone x2 + y2 = z2. Sketch the cone and make a rough sketch of C on the cone.
Find the points where r(t) intersects the xy-plane.
What is the projection of r(t) = ti + 14j + e1k onto the xz-plane?
Parametrize the intersection of the surfaces y2 − z2 = x − 2, y2 + z2 = 9 Using t = y as the parameter (two vector functions are needed as in Example 3).
Viviani's Curve C is the intersection of the surfacesx2 + y2 = z2, y = z2(a) Parametrize each of the two parts of C corresponding to x ‰¥ 0 and x ‰¤ 0, taking t = z as
Use sine and cosine to parametrize the intersection of the cylinders x2 + y2 = 1 and x2 + z2 = 1 (use two vector-valued functions). Then describe the projections ofjjthis curve onto the three
Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = 1 and z = 4x2 (Figure 14).
Here are two paths r1(t) and r2(t) intersect if there is a point P lying on both curves. We say that r1(t) and r2(t) collide if r1(t0) = r2(t0) at some time t0. Determine whether rj and r2 collide or
Which projection of (cos t, cos2t, sin t) is a circle?
The line through the origin whose projection on the xy-plane is a line of slope 3 and whose projection on the yz-plane is a line of slope 5 (i.e., ∆z/∆y = 5)
The circle of radius 2 with center (1, 2, 5) in a plane parallel to the yz-plane. Find a parametrization of the curve
The intersection of the plane y = 1 with the sphere x2 + y2 + z2 = 1 Find a parametrization of the curve
The ellipseIn t he xz-plane, translated to have center (3, 1, 5) [Figure 15(A)]
What is the center of the circle with parametrization? r(t) = (-2 + cos t)i + 2j + (3 - sint)k?
Sketch the curve parametrized by r(t) = (|t| + t, |t| − t).
Let C be the curve obtained by intersecting a cylinder of radius r and a plane. Insert two spheres of radius r into the cylinder above and below the plane, and let F1 and F2 be the points where the
How do the paths ri (t) = (cos t, sin t) and r2 (t) = (sin t, cos t) around the unit circle differ?
Which three of the following vector-valued functions parametrize the same space curve? (a) (-2 + cos t)i + 9j + (3 - sin t)k (b) (2 + cos t)i - 9j + (-3 - sin t)k (c) (-2 + cos 3t)i + 9j + (3 - sin
Match the space curves in Figure 8 with their projections onto the xy-plane in Figure 9.
What is the domain of r(t) = eli + (1/t)j + (t + 1)-3k?
Match the vector-valued functions (a)-(f) with the space curves (i)-(vi) in Figure 10.(a) r(t) = (t + 15,e0.08t cost,e0.08t sin t)(b) r(t) = (cos t, sin t, sin 12t)(c) r(t) =(t, t, (25t/1+t2))(d)
Which vector field F is being integrated in the line integral
Draw a domain in the shape of an ellipse and indicate with an arrow the boundary orientation of the boundary curve. Do the same for the annulus (the region between two concentric circles)?
The circulation of a conservative vector field around a closed curve is zero. Is this fact consistent with Green's Theorem? Explain.
Indicate which of the following vector fields possess the following property: For every simple closed curve C,Is equal to the area enclosed by C. (a) F = (ˆ’y, 0) (b) F = (x, y) (c) F =
Verify Green's Theorem for the line integralWhere C is the unit circle, oriented counterclockwise?
The region between the x-axis and the cycloid parametrized by c(t) = (t ˆ’ sin t, 1 ˆ’ cos t) for 0 ‰¤ t ‰¤ 2Ï€ (Figure 20)
The Centroid via Boundary Measurements The centroid (see Section 15.5) of a domain D enclosed by a simple closed curve C is the point with coordinates (xÌ…, yÌ…) =
Let CR be the circle of radius R centered at the origin. Use the general form of Green's Theorem to determineWhere F is a vector field such that And For (x, y) in the annulus 1 ‰¤ x2 + y2
For the vector fields (A)-(D) in Figure 26, state whether the curlz at the origin appears to be positive, negative, or zero.
And C encloses a small region of area 0.25 containing the point P = (1, 1)?
Let CR be the circle of radius R centered at the origin. Use Green's Theorem to find the value of R that maximizes
Use the result of Exercise 32 to compute the areas of the polygons in Figure 27. Check your result for the area of the triangle in (A) using geometry?
Define div (F)Use Green's Theorem to prove that for any simple closed curve C, Flux across Where D is the region enclosed by C. This is a two-dimensional version of the Divergence Theorem discussed
Use Eq. (12) to compute the flux of F = (cos y, sin y) across the square 0 ≤ x ≤ 2, 0 ≤ y ≤ π/2?
In Exercises 3-10, use Green's Theorem to evaluate the line integral. Orient the curve counterclockwise unless otherwise indicated.Where C is the boundary of the unit square 0 ( x ( 1, 0 ( y ( 1

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