1.99
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Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.

y SI sin x

** >** Use your computer algebra system to evaluate f(x) for x = 1, 0.1, 0.01, 0.001, and 0.0001. Does it appear that f has a limit as x→ 0?

** >** In view of the answers to Problems 4 and 5, how do you explain the results of Problems 1 and 2? Data from Problems 1: Use your computer algebra system to evaluate f(x) for x = 1, 0.1, 0.01, 0.001, and 0.0001. Does it appear that f has a limit as x→ 0?

** >** More complicated shapes can be represented by piecing together two or more Bézier curves. Suppose the first Bézier curve has control points P0, P1, P2, P3 and the second one has control points P3, P4, P5, P6. If we want these two pieces to join together

** >** Some laser printers use Bézier curves to represent letters and other symbols. Experiment with control points until you find a Bézier curve that gives a reasonable representation of the letter C.

** >** Find dy/dx. x = tet, y = t + sin t

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.

** >** Try to produce a Bézier curve with a loop by changing the second control point in Problem 1. Data from Problem 1: Graph the Bézier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, on the same screen, graph the line segme

** >** Use the Ratio Test to determine whether the series is convergent or divergent.

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.

** >** Evaluate the integral.

** >** From the graph in Problem 1, it appears that the tangent at P0 passes through P1 and the tangent at P3 passes through P2. Prove it. Data from Problem 1: Graph the Bézier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, o

** >** Graph the BÃ©zier curve with control points P0(4, 1), P1(28, 48), P2(50, 42), and P3(40, 5). Then, on the same screen, graph the line segments P0P1, P1P2, and P2P3. (Exercise 10.1.31 shows how to do this.) Notice that the middle control point

** >** Based on your own measurements and observations, suggest a value for h and an equation for x = f(y) and calculate the amount of coffee that each cup holds.

** >** Use Pappusâ€™s Theorem to explain your result in Problems 1 and 2. Data from Problem 1: Suppose the cups have height h, cup A is formed by rotating the curve x = fs/d about the y-axis, and cup B is formed by rotating the same curve abou

** >** Suppose the cups have height h, cup A is formed by rotating the curve x = f(y) about the y-axis, and cup B is formed by rotating the same curve about the line x = k. Find the value of k such that the two cups hold the same amount of coffee.

** >** Use the Ratio Test to determine whether the series is convergent or divergent.

** >** Evaluate the integral.

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.

** >** (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylorâ€™s Inequality to estimate the accuracy of the approximation / when x lies in the given interval. (c) Check your result in part (b) by graphing /

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.

** >** Verify Formula 62 in the Table of Integrals.

** >** Recall that the normal line to a curve at a point P on the curve is the line that passes through P and is perpendicular to the tangent line at P. Find the curve that passes through the point (3, 2) and has the property that if the normal line is drawn at

** >** Use the graphs of x = f(t) and y = t(t) to sketch the parametric curve x = f(t), y = t(t). Indicate with arrows the direction in which the curve is traced as t increases.

** >** Verify Formula 33 in the Table of Integrals (a) by differentiation and (b) by using a trigonometric substitution.

** >** 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 = x. Where does the curve intersect this line? (b) F

** >** Evaluate the integral.

** >** Show that the angles between the polar axis and the asymptotes of the hyperbola / are given by /

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.

** >** Use the Table of Integrals on the Reference Pages to evaluate the integral.

** >** Show that if m is any real number, then there are exactly two lines of slope m that are tangent to theÂ ellipse x2/a2 + y2/b2 = 1 and their equations are

** >** Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.
у —е", х — 0, у — 3;B about the х-ахis

** >** Find the curve that passes through the point (3, 2) and has the property that if the tangent line is drawn at any point P on the curve, then the part of the tangent line that lies in the first quadrant is bisected at P.

** >** 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.

** >** Find an equation of the hyperbola with foci (0, ±4) and asymptotes y = ±3x.

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.
y - e* In(1 + x)

** >** Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.
f) — In x, а — 1

** >** Find an equation of the ellipse with foci (±4, 0) and vertices (±5, 0).

** >** Find the foci and vertices and sketch the graph.
25x? + 4y? + 50x – 16y = 59

** >** Evaluate the integral.
dx X - 1

** >** Find the values of x for which the series converges. Find the sum of the series for those values of x.
E (-5)"x"

** >** Find the foci and vertices and sketch the graph.
4x? - у? — 16 %3D

** >** Find the foci and vertices and sketch the graph.
x2 y? 1 8 9

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
(-3)" an n!

** >** The curves defined by the parametric equations are called strophoids (from a Greek word meaning â€œto turn or twistâ€). Investigate how these curves vary as c varies.
(? — с) y = 1? + 1 1? - c 1? + 1

** >** Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.
у —е", у— 0, х — —1, х — 0; about x— 1 %3D

** >** Find the area of the surface obtained by rotating the given curve about the x-axis.
x = 2 + 3t, y = cosh 31, 0<i<1

** >** Find the area of the surface obtained by rotating the given curve about the x-axis.
x = 4 T, y=+, x = 4 /t, y=- 3 212"

** >** Find the length of the curve.
r= sin'(0/3), 0< 0<

** >** Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.
x = 2 cos 0, y =1 + sin0 = 1 + sin@

** >** Express the number as a ratio of integers.
5.71358

** >** Evaluate the integral.

** >** Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1 Σ 1 + (?)"

** >** Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
10" a, = 1 + 9"

** >** Find the length of the curve.
r= 1/0, 7< 0 < 27

** >** Find the length of the curve.
x = 2 + 3t, y = cosh 31, 0< t<1

** >** Find the length of the curve.
x = 312, y = 21°, 0<t<2

** >** Find the area of the region that lies inside both of the circles /

** >** Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.
у —е", у—е", х — 1;B about the y-аxis

** >** 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 area enclosed by the loop of the curve in Exercise 27. Data from Exercise 27: Use a graph to estimate the coordinates of the lowest point on the curve x = t3 - 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.

** >** Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
3 – 4 + – 4 - +

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
a„ = In(n + 1) – In n

** >** Use a graph to estimate the coordinates of the lowest point on the curve x = t3 - 3t, y = t2 + t + 1. Then use calculus to find the exact coordinates.

** >** Find the volume obtained by rotating the region bounded by the curves about the given axis.
у — sin x, у — 0, п/2 <x< п; about the x-axis

** >** Find dy/dx and d2y/dx2. x = 1 + t2, y = t – t3

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.
y - sec x

** >** Find dy/dx and d2y/dx2. x = t + sint. y = t – cos t

** >** Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.
r = e e "; 0 = T

** >** Populations of birds and insects are modeled by the equations (a) Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. (b) Find the equilibrium solutions and explain their significance. (c) F

** >** Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.
x = In 1, y = 1 + t²; t=1

** >** Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves about the given axis.
y = cos(Tx/2), y= 0, 0<x< 1; about the y-axis

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
{{东春 !!!!!!! ..} 3> 4> 6

** >** Express the number as a ratio of integers.
1.234567

** >** A tank contains 100 L of pure water. Brine that contains 0.1 kg of salt per liter enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes?

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
(2n – 1)! (2n + 1)!

** >** The von Bertalanffy growth model is used to predict the length L(t) of a fish over a period of time. If L is the largest length for a species, then the hypothesis is that the rate of growth in length is proportional to L` 2 L, the length yet to be achiev

** >** Use multiplication or division of power series to find the firstÂ three nonzero terms in the Maclaurin series for each function.
y = e * cos x

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
{0, 1, 0, 0, 1, 0, 0, 0, 1, . }

** >** Express the number as a ratio of integers.
10.135 = 10.135353535 ...

** >** Use a graph to find approximate x coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
y = x у — х Iп(x + 1), у3 Зх — .2

** >** Evaluate the integral.
( In Va dx

** >** Solve the initial-value problem.
dr + 2tr = r, r(0) = 5 dt

** >** Determine whether the series converges or diverges.
1/m Σ 1 п

** >** Solve the differential equation.
x?y' – y = 2x'e-1/½

** >** Solve the differential equation.
2ye "'y' = 2x + 3 /x %3D

** >** Solve the differential equation.
dx 1 - t + x - tx dt

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
а, — п — уп +1 ул + 3

** >** Use the Table of Integrals on the Reference Pages to evaluate the integral.
cot x - dx 1 + 2 sin x

** >** Use the Table of Integrals on the Reference Pages to evaluate the integral.
cos x /4 + sin²x dx

** >** Express the number as a ratio of integers.
2.516 = 2.516516516 ...

** >** Use the Table of Integrals on the Reference Pages to evaluate the integral.
| csc't dt

** >** Use the series in Example 13(b) to evaluate We found this limit in Example 4.4.4 using lâ€™ Hospitalâ€™s Rule three times. Which method do you prefer?
tan x - x lim

** >** Graph the function f(x) = cos2x sin3x and use the graph to guess the value of the integral / . Then evaluate the integral to confirm your guess.

** >** Use a graph to find approximate xÂ­ coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.
y = arcsin n(4x), у — 2 — х?

** >** Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0).
dx /x² + 1

** >** Determine whether the sequence converges or diverges. If it converges, find the limit.
an arctan(In n)

** >** Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0).
In(x² + 2x + 2) dx

** >** Evaluate the integral or show that it is divergent.
tan 'x - dx x?

** >** Solve the differential equation.
sin x y' = xe y cos x

** >** Test the series for convergence or divergence.
E n’e E n²e n-