Question: First make a substitution and then use
First make a substitution and then use integration by parts to evaluate the integral.
Transcribed Image Text:
| cos(In x) dx
> Determine whether the series is convergent or divergent. n? + 1 Σ n3 + 1
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C − 0). 2% dx Sxe
> A sequence is defined recursively by the equations a1 = 1, an+1 = 1 3 (a n +4). Show that {an} is increasing and an < 2 for all n. Deduce that / is convergent and find its limit.
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. {(-10)"/n!}
> Evaluate the integral. | tan?x dx
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. {(1 + 3/n)+"}
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. In n an in
> Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen. f(x) cos х, a = T /2
> Determine whether the sequence is convergent or divergent. If it is convergent, find its limit. а, — сos(пт/2)
> Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. х — cos 0, у —sec 0, 0 <0 < п/2
> Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. х —1+ е", у — е' y
> First make a substitution and then use integration by parts to evaluate the integral. arcsin(In x) dx
> Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. x = t² + 4t, y = 2 – 1, -4 < t<1
> Evaluate the integral. ax
> The astronomer Giovanni Cassini (1625–1712) studied the family of curves with polar equations where a and c are positive real numbers. These curves are called the ovals of Cassini even though they are oval shaped only for certain values
> A family of curves has polar equations Investigate how the graph changes as the number a changes. In particular, you should identify the transitional values of a for which the basic shape of the curve changes. 1 — а сos0 1+а сos0
> Use your graph in Problem 3 to estimate the value of / for which f(/) is a maximum under Planck’s Law. Data from Problem 3: Graph f as given by both laws on the same screen and comment on the similarities and differences. Use T = 5700 K (the temperatur
> Use the Maclaurin series for cos x to compute cos 58 correct to five decimal places.
> Graph f as given by both laws on the same screen and comment on the similarities and differences. Use T = 5700 K (the temperature of the sun). (You may want to change from meters to the more convenient unit of micrometers: 1 mm − 1026 m.)
> Use a Taylor polynomial to show that, for large wavelengths, Planck’s Law gives approximately the same values as the Rayleigh-Jeans Law.
> Use l’ Hospital’s Rule to show that for Planck’s Law. So this law models blackbody radiation better than the Rayleigh-Jeans Law for short wavelengths. lim f(A) = 0 and lim f(A) = 0 %3D %3D A0+
> First make a substitution and then use integration by parts to evaluate the integral. e In(1 + x) dx
> Evaluate the integral.
> Evaluate the integral. t² sin ßt dt 2
> Investigate how the graph off changes as T varies. (Use Planck’s Law.) In particular, graph f for the stars Betelgeuse (T = 3400 K), Procyon (T = 6400 K), and Sirius (T = 9200 K), as well as the sun. How does the total radiation emitted (the area under t
> Use a computer algebra system to find the exact area of the surface obtained by rotating the curve / Then approximate your result to three decimal places.
> Find the volume of the solid obtained by rotating the region of Problem 2 about the line y = x - 2. Data from Problem 2: Find the area of the region shown in the figure below. yA (2π, 2π) y=x+ sin x y=x- 2
> Find a formula (similar to the one in Problem 1) for the volume of the solid obtained by rotating 5 about the line y = mx + b. Data from Problem 1: Show that the area of / is R
> Find the area of the region shown in the figure below. yA (2π, 2π) y=x+ sin x y=x- 2
> Show that the area of R is 1 I[S«) – mx — b][1 + mf()] dx 1 + m?
> 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) = tan '(x³) S(x).
> 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)
> 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. *2 x*(In x)° dx
> 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?
