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 smoothly, then the tangents at P3 should match and so the points P2, P3, and P4 all have to lie on this common tangent line. Using this principle, find control points for a pair of Bézier curves that represent the letter S.
> Test the series for convergence or divergence. Σ 3"
> Find the sum of the series. (-1)" 7" 32"(2n)!
> Express the number as a ratio of integers. 0.46 = 0.46464646 . . .
> Determine whether the sequence converges or diverges. If it converges, find the limit 2 3 + 5n? 1 +n
> In Section 4.8 we considered Newton’s method for approximating a root r of the equation f(x) = 0, and from an initial approximation x1 we obtained successive approximations x2, x3, . . . , where Use Taylor’s Inequality
> Determine whether the sequence converges or diverges. If it converges, find the limit. In n In 2n
> Evaluate the integral. dx J31+ |x|
> Evaluate the integral. /2 cos'x sin 2x dx
> Determine whether the series is conditionally convergent, absolutely convergent, or divergent. E(-1)" 'n-3
> Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter. x = t³ + 6t + 1, y=2t – t2; t=-1
> Find a polar equation for the curve represented by the given Cartesian equation. x2 + y2 = 2
> Use the Root Test to determine whether the series is convergent or divergent. (-1)“ 1 Σ (In n)" R-2
> Find a polar equation for the curve represented by the given Cartesian equation. x + y = 2
> Find the values of x for which the series converges. Find the sum of the series for those values of x. sin"x o 3"
> Express the number as a ratio of integers. 0.8 = 0.8888...
> Evaluate the integral. x? + 2 dx x + 2
> Determine whether the series is convergent or divergent. Σ 5"
> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. rV4 - x² dx
> Evaluate the integral. /arctan x dx 1 + x?
> Determine whether the series is convergent or divergent. Σ i n' + 1 R-1
> Find the area of the surface obtained by rotating the curve in Exercise 9 about the y-axis. Data from Exercise 9: Find the length of the curve y - i VJi - I di 1<x< 16
> Evaluate the integral. sin(In t) dt
> Evaluate the integral. dt J 21? + 3t + 1
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. n sin n an n2 + 1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ
> Find the values of x for which the series converges. Find the sum of the series for those values of x. 2" Σ o x"
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 3. n an 1 + .2 re
> Use the Root Test to determine whether the series is convergent or divergent. п? + 1 2n? + 1
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 9n+1 an 10"
> Calculate the average value of f(x) = x sec2x on the interval /
> Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral. (4x? – 3 dx; entry 39
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. 2 + n3 an 1+ 2n3
> What does your result from Problem 1 say about the areas A1 and A2 shown in the figure? 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 curv
> All solutions of the differential equation y’ = -1 – y4 are decreasing functions.
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 2 arctan n n-1
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. x(x? + 4) x² – 4 A can be put in the form x + 2 B x - 2
> Find the values of x for which the series converges. Find the sum of the series for those values of x. E (-4)"(x – 5)"
> Find a formula for the area of the surface obtained by rotating C about the line y = mx + b.
> If f is continuous, then /
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. - dx is convergent.
> Use the Ratio Test to determine whether the series is convergent or divergent. 2" n! E (-1)" - 5. 8· 11 · · (3n + 2)
> Evaluate the integral. M dM eM
> Calculate the volume generated by rotating the region bounded by the curves y = ln x, y = 0, and x = 2 about each axis. (a) The y axis (b) The x axis
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. x? - 4 B A + x? + 4' can be put in the form x(x² + 4)
> Determine whether the series is absolutely convergent or conditionally convergent.
> Show that any tangent line to a hyperbola touches the hyperbola halfway between the points of intersection of the tangent and the asymptotes.
> Use the CAS to graph f near x = 0. Does it appear that f has a limit as x→ 0?
> 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?
> 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 first 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 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 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 Pappus’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 Taylor’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 first 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 the 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 first 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 “to turn or twist”). 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@
> Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for each function. y SI sin x
> 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"
