Question: Evaluate the integral. /

> If the circle C rolls on the outside of the fixed circle, the curve traced out by P is called an epicycloid. Find parametric equations for the epicycloid.

> Evaluate the integral. evi

> First make a substitution and then use integration by parts to evaluate the integral. ec sin 21 dt cos! Jo

> Now try b = 1 and a = n/d, a fraction where n and d have no common factor. First let n = 1 and try to determine graphically the effect of the denominator d on the shape of the graph. Then let n vary while keeping d constant. What happens when n = d + 1?

> Use a graphing device (or the interactive graphic in TEC Module 10.1B) to draw the graphs of hypocycloids with a positive integer and b = 1. How does the value of a affect the graph? Show that if we take a = 4, then the parametric equations of the hypocy

> (a) Every elementary function has an elementary derivative. (b) Every elementary function has an elementary antiderivative.

> Determine whether the sequence converges or diverges. If it converges, find the limit. n! 2"

> 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 polynomials and f? f(a) — хе

> The Midpoint Rule is always more accurate than the Trapezoidal Rule.

> First make a substitution and then use integration by parts to evaluate the integral. 0° cos(0?) do Va/2

> 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. Ldx = | In 15 Jo x?

> If y is the solution of the initial-value problem dy y 2y( 1 5 y(0) = 1 %3D dt then lim, y = 5.

> The function f(x) = (ln x)/x is a solution of the differential equation x2y’ + xy = 1.

> A hyperbola never intersects its directrix.

> Evaluate the integral. *2m t² sin 2t dt

> A tangent line to a parabola intersects the parabola only once.

> The graph of y2 = 2y + 3x is a parabola.

> 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 polynomials and f? f(x) – In(1 + x²)

> First make a substitution and then use integration by parts to evaluate the integral. | cos(In x) dx

> The parametric equations x = t2, y = t4 have the same graph as x = t3, y = t6.

> If x = f(t) and y = t(t) are twice differentiable, then d²y _ d?y/dt² dx? d²x/dt² .2

> If the parametric curve x = f(t), y = g(t) satisfies g’(1) = 0, then it has a horizontal tangent when t = 1.

> Determine whether the sequence converges or diverges. If it converges, find the limit. an 3"7 " %3D

> Determine whether the sequence converges or diverges. If it converges, find the limit. a„ = 2 + (0.86)"

> Determine whether the sequence converges or diverges. If it converges, find the limit 3 + 5n? ,2 an n + n?

> First make a substitution and then use integration by parts to evaluate the integral. dx

> 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 polynomials and f? f(x) = cos(x²) 2

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {1, 0, – 1, 0, 1, 0, -1,0, .}

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {5, 8, 11, 14, 17, . ..}

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {-3, 2, –4. §, – 4. ..} 16 3 9

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. -1,; 16 64

> Determine whether the sequence converges or diverges. If it converges, find the limit. (In n)² an n

> Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {平学学字} か. 111 l5ふる京而, } 4> 6 8 10

> List the first five terms of the sequence. 1 an (п + 1)!

> List the first five terms of the sequence. cos 2 An

> List the first five terms of the sequence. (-1)"-' an 5"

> List the first five terms of the sequence. – 1 ,2 an n2 + 1

> Evaluate the integral. e' sin(t – s) ds

> List the first five terms of the sequence. 2" an 2n + 1

> Use a Maclaurin series to obtain the Maclaurin series for the given function. - sin x if x + 0 f(x) = if x = 0 6.

> Suppose a curve is given by the parametric equations x = f(t), y = t(t), where the range of f is [1, 4] and the range of t is [2, 3]. What can you say about the curve?

> Evaluate the integral. · dx x² – 7 -2

> Determine whether the sequence converges or diverges. If it converges, find the limit. a, = In(2n² + 1) – In(n² + 1)

> Find an equation for the conic that satisfies the given conditions. Parabola, vertex (0, 0), focus (1, 0)

> Identify the type of conic section whose equation is given and find the vertices and foci. 4x* — у + 4

> Identify the type of conic section whose equation is given and find the vertices and foci. 4x? — у? + 4 .2

> Find an equation of the ellipse. Then find its foci. 2.

> Find an equation of the ellipse. Then find its foci. 1

> Evaluate the integral. | tan?x cos'x dx .3.

> Evaluate the integral. ах dx x² – bx マ-

> Evaluate the integral. 3t – 2 -dt t + 1

> Use a Maclaurin series to obtain the Maclaurin series for the given function. f(x) = sin?x 2.

> Determine whether the sequence converges or diverges. If it converges, find the limit. а, — 2 "сos пп 'cos nT %3D

> If a water wave with length L moves with velocity v across a body of water with depth d, as in the page 782, then (a) If the water is deep, show that / (b) If the water is shallow, use the Maclaurin series for tanh to show that / (c) Use the Alternating

> Determine whether the sequence converges or diverges. If it converges, find the limit. 1 + 4n? Vi+ n?

> In general, it’s not easy to find t2 because it’s impossible to solve the equation y(t) = 0 explicitly. We can, however, use an indirect method to determine whether ascent or descent is faster: we determine whether y(2

> Let t2 be the time at which the ball falls back to earth. For the particular ball in Problem 3, estimate t2 by using a graph of the height function y(t). Which is faster, going up or coming down? Data from Problem 3: Let t1 be the time that the ball ta

> Let t1 be the time that the ball takes to reach its maximum height. Show that Find this time for a ball with mass 1 kg and initial velocity 20 m/s. Assume the air resistance is / m mg + pvo In mg

> Show that the height of the ball, until it hits the ground, is mg mgt (1 – e mlm) – P y(1)

> A ball with mass m is projected vertically upward from the earth’s surface with a positive initial velocity v0. We assume the forces acting on the ball are the force of gravity and a retarding force of air resistance with direction oppo

> Let {Pn} be a sequence of points determined as in the figure. Thus / / and angle APnPn+1 is a right angle. Find / |AP¡|= 1, |AP||

> (a) Show that tan 1 2 x = cot 1 2 x – 2 cot x. (b) Find the sum of the series tan in 2"

> Evaluate the integral. *2 w² In w dw

> A function f is defined by Where is f continuous? r2n - 1 f(x) = lim no xn + 1 + 1

> If f(x) = sin (x3), find f(15)(0).

> A uniform disk with radius 1 m is to be cut by a line so that the center of mass of the smaller piece lies halfway along a radius. How close to the center of the disk should the cut be made? (Express your answer correct to two decimal places.)

> Consider a flat metal plate to be placed vertically underwater with its top 2 m below the surface of the water. Determine a shape for the plate so that if the plate is divided into any number of horizontal strips of equal height, the hydrostatic force on

> Let P be a pyramid with a square base of side 2b and suppose that S is a sphere with its center on the base of P and S is tangent to all eight edges of P. Find the height of P. Then find the volume of the intersection of S and P.

> The figure shows a semicircle with radius 1, horizontal diameter PQ, and tangent lines at P and Q. At what height above the diameter should the horizontal line be placed so as to minimize the shaded area?

> (a) Show that an observer at height H above the north pole of a sphere of radius r can see a part of the sphere that has area (b) Two spheres with radii r and R are placed so that the distance between their centers is d, where d > r + R. Where should

> If a sphere of radius r is sliced by a plane whose distance from the center of the sphere is d, then the sphere is divided into two pieces called segments of one base (see the first figure). The corresponding surfaces are called spherical zones of one ba

> Find the centroid of the region enclosed by the loop of the curve y2 = x3 – x4.

> Find the centroid of the region enclosed by the ellipse x2 + (x + y + 1)2 = 1.

> If the needle in Problem 11 has length h > L, it’s possible for the needle to intersect more than one line. (a) If L = 4, find the probability that a needle of length 7 will intersect at least one line. (b) If L = 4, find the probability that a needle of

> An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and 2q and are located at a distance d from each other, then the electric field E at the point P in the figure is By expanding this expression

> A triangle with area 30 cm2 is cut from a corner of a square with side 10 cm, as shown in the figure. If the centroid of the remaining region is 4 cm from the right side of the square, how far is it from the bottom of the square? 10 cm

> Prove that if n > 1, the nth partial sum of the harmonic series is not an integer.

> (a) Show that the Maclaurin series of the function where fn is the nth Fibonacci number, that is, f1 = 1, f2 = 1, and fn=− fn1 + fn2 for n > 3. (b) By writing f(x) as a sum of partial fractions and thereby obtaining the Maclaurin ser

> Evaluate the integral. + 1)e*dx Jo

> 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 than 90.

> 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 many turns around P by showing that / divergent series.

> Find all the solutions of the equation 1 + 2!

> Carry out the following steps to show that (a) Use the formula for the sum of a finite geometric series (11.2.3) to get an expression for (b) Integrate the result of part (a) from 0 to 1 to get an expression for as an integral. (c) Deduce from part (b) t

> Find the sum of the series /

> 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 P1P2, P6 is the midpoint of P2P3, P7 is the midpoint of P3P4, and so on. The polygonal spiral path P1

> If the curve y = e-x/10 sin x, x ≥ 0 is rotated about the x-axis, the resulting solid looks like an infinite decreasing string of beads. (a) Find the exact volume of the nth bead. (Use either a table of integrals or a computer algebra system.) (b) Find t

> A sequence {an} is defined recursively by the equations Find the sum of the series / dо — ај — 1 п(n — 1)а, — (п - 1)(п — 2)а, 1 — (п — 3)а, 2

> Suppose that circles of equal diameter are packed tightly in n 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 occupied by the n rows of circles, show that An lim

> If p > 1, evaluate the expression 1 + 2" 1 + 4P 3P 1 + 4P 1 1 2" 3P + + +

> Find the sum of the series /

> 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 that it is possible to do this so that the top book extend

> Find the interval of convergence of /

> If a0 + a1 + a2 + ∙∙∙ + ak = 0, show that If you don’t see how to prove this, try the problem-solving strategy of using analogy (see page 71). Try the special cases k = 1 and k = 2

> A car is moving with speed 20 m/s and acceleration 2 m/s2 at a given instant. Using a second-degree Taylor polynomial, estimate how far the car moves in the next second. Would it be reasonable to use this polynomial to estimate the distance traveled duri

> Use the result of Problem 7(a) to find the sum of the series / Data from Problem 7: (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce the following formula of John Machin (1680–1751)

> (a) Prove a formula similar to the one in Problem 7(a) but involving arccot instead of arctan. (b) Find the sum of the series / Data from Problem 7: (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce t

> (a) Show that for xy ≠2, if the left side lies between / (b) Show that / (c) Deduce the following formula of John Machin (1680–1751): (d) Use the Maclaurin series for arctan to show that (e) Show that (f) Deduce that,

> Find the sum of the series where the terms are the reciprocals of the positive integers whose only prime factors are 2s and 3s. 1 + 6 9 12 + - l00 + | + + - l3

> Evaluate the integral. 1/2 x cos TX dx Jo

> Evaluate the integral. |(x – 1) sin 7x dx