Find f. f"(x) = 6x + sin x
> Find the values of x for which the series converges. Find the sum of the series for those values of x. (х + 3)" Σ 24 A-0
> Find the values of x for which the series converges. Find the sum of the series for those values of x. 3" n-
> Express the number as a ratio of integers. 7. ¯12345
> Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain w
> Express the number as a ratio of integers. 1. ¯5342
> Find the radius of convergence and interval of convergence of the series. - 4)ª n(x – 4)" n + 1 A-1
> Express the number as a ratio of integers. 6. ¯254 = 6.2545454…
> Express the number as a ratio of integers. 0. ¯2 = 0.2222…
> Express the number as a ratio of integers. 0. ¯73 = 0.73737373…
> Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. a, = 1 + (-2/e)"
> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 00 E In- п+1 A-1
> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 3 Σ п(n + 3) A-1
> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 00 2 Σ A-1 n + 4n + 3
> Determine whether the series is convergent or divergent by expressing Sn as a telescoping sum (as in Ex am ple 6). If it is convergent, find its sum. 2 Σ n² – 1 ,2
> Determine whether the sequence converges or diverges. If it converges, find the limit. sin 2n 1 + + Jn
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1 Σ п(n + 1) A-1 +
> Find the radius of convergence and interval of convergence of the series. (4x + 1)" n? 00 n-1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. E [(0.8)*-1 – (0.3)*]
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ 2 arctan n
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. -1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 3" 24
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 24 Σ 34
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 Σ cos A-1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k? Σ k? – 1 00 - 1 k-2
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k(k + 2) Σ -i (k + 3)2 k-1
> Explain what it means to say that ∑∞n-1 an = 5.
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. n - 1 Σ A-1 3n Зп — 1
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? E (-1)*-'ne- (1 error|< 0.01)
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. Σ n-0
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 Σ A-o 3*+1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10" Σ (-9)*-1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 E 6(0.9)“-! A-1 n-1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 1 + 0.4 + 0.16 + 0.064 +
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10 – 2 + 0.4 – 0.08 + ·
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 4 + 3 ++ + ...
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3 – 4 +4 - + ... 16
> (a). Explain the difference between (b). Explain the difference between aj and 内I and E aj
> (a). What is the difference between a sequence and a series? (b). What is a convergent series? What is a divergent series?
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)*+1 (lerror|< 0.00005) A-1
> (a). What is an alternating series? (b). Under what conditions does an alternating series converge? (c). If these conditions are satisfied, what can you say about the remainder after n terms?
> On what interval is the curve y = fx0 t2/t2 + t + 2, dt concave downward?
> If f (x) = fsinxo√1 + t2 dt and g (y) = fy3 f(x) dx, find g"(π/6).
> If f (x) = fx0 (1 – t2) et2 dt, on what interval is f increasing?
> Evaluate the integral. f01 10x dx
> Evaluate the integral. f12 x-2 dx
> Evaluate the integral. f1/2√3/2 6/√1 – t2 dt
> find f. f'(x) = √x (6 + 5x) f (1) = 10
> find f. f'(x) = 8x3 + 12x + 3, f (1) = 6
> Find f. f"(x) = 1 - 6x, f (0) = 8
> Find f. f"(x) = 6x + 12x2
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. *cos X y = * (1 + v?)1º dv Jsin x
> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) — 4 — 3(1 + х?)-1, F(1) = 0
> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 5x* – 2x', F(0)=4
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 27 + 6 cos x COS %3D
> Find the most general antiderivative of the function. (Check your answer by differentiation.) g(8) = cos e – 5 sin e
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 3e* + 7 sec?x
> Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Expl
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. 2 3 lim +
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim 4
> The area labeled B is three times the area labeled A. Express a in terms of b. y. y= e" y= e" B A a
> Suppose h is a function such that h (1) = -2, h'(1) = 2, h"(1) = 3, h (2) = 6, h'(2) = 5, h"(2) = 13, and h" is continuous everywhere. Evaluate f21 h"(u) du.
> A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has e
> (a). If f is continuous, prove that fπ/20 f (cos x) dx = fπ/20 f (sin x) dx (b). Use part (a) to evaluate fπ/20 cos2 x dx and x fπ/20 sin 2x dx.
> If a and b are positive numbers, show that r(1 – x)* dx = [ x*(1 – x)* dx
> If f is continuous on R, prove that fbaf (x + c) dx = fb+ca+c f (x) dx For the case where f (x) > 0, draw a diagram to interpret this equation geometrically as an equality of areas.
> If f is continuous on R, prove that fbaf (-x) dx = f-1-bf (x) dx For the case where f (x) > 0 and 0 < a < b, draw a diagram to interpret this equation geometrically as an equality of areas.
> If f is continuous and f90 f (x) dx = 4, find f30 xf (x2) dx.
> If f is continuous and f40f (x) dx = 10, find f20f (2x) dx.
> The velocity graph of a car accelerating from rest to a speed of 120 km/hover a period of 30 seconds is shown. Estimate the distance traveled during this period. (km/h) 80 40 10 20 30 (seconds)
> Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower b
> Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f (t) = ½ sin (2πt/5) has o
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. C 3r + 1) dx x² + 1
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 3r u? - 1 du J2x u? + g(x) = Hint: "f(u) du = f(u) du + f* f(u) du /2x
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. 1 C x² + 4 dx X + 2
> Show that f∞0e-x2 dx = f10 √-ln y, dy by interpreting the integrals as areas.
> Show that fπ0x2e-x2 dx = 1/2 f∞0e-x2 dx.
> Estimate the numerical value of f∞0e-x2 dx by writing it as the sum of f40e-x2 dx and f∞4e-x2 dx. Approximate the first integral by using Simpson’s Rule with n = 8 and show that the second integral is smaller than f∞4e-4x dx, which is less than 0.0000001
> Determine how large the number a has to be so that 1 dx < 0.001 x2 + 1
> Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of
> As we will see in Section 7.4, a radioactive substance decays exponentially: The mass at time t is m (t) = m (0) ekt, where m (0) is the initial mass and is a negative constant. The mean life M of an atom in the substance is For the radioactive carbon
> The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.46 108 km.) There are five locations L1
> The average speed of molecules in an ideal gas is Where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and is the molecular speed. Show that 3/2 "p'e-Mu/(2RT) du 4 M 2RT 8RT V TM
> Evaluate the definite integral. Fe4e dx/x√ ln x,
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = sin't dt - :
> A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a). If its height is 4 ft and the radius at the top is 4 ft, find the work required t
> If f∞-∞ f (x) dx is convergent and a and b are real numbers, show that L f(x) dx + [* f(x) dx = [°_f(x) dx + [° f(x) dx
> (a). Show that f∞-∞ x dx is divergent. (b). Show that This shows that we can’t define lim ,xd x dx = 0 Lf(x) dx = lim , f(x) dx
> Evaluate the definite integral. fa0 x√a2 – x2, dx
> A hole of radius is bored through the middle of a cylinder of radius R > r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. : p= 2 quarter-circle
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. р3 10 y4 (4, 3)
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. у%3 1/х, у— 0, х—D 1, х%3D2
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = e*, y= 0, x= 0, x= 1
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. Зх + 2у 3 6, у%3D0, х%3D0
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tan x y = Jo
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = 4 - x², y = 0
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. %3D %3D P.(1, - 2), Р:(3, 4), Р:(-3, —7), Р.(6, — 1)
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. т — 6, т — 5, ms — 10; P.(1,5), Р.(3, —2), Р.{(-2, —1)
> Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m,= 25 ++ -2 m3= 10 ++++ 7 m2 = 20 ++ 3
> A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson’s Rule to estimate the force of the water against the plate. Depth (m) 2.0 2.5 3.0 3.
