A high-speed bullet train accelerates and decelerates at the rate of 4ft/s2. Its maximum cruising speed is 90 mi/h. (a). What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at that speed for 15 minutes? (b). Suppose that the train starts from rest and must come to a complete stop in 15 minutes. What is the maximum distance it can travel under these conditions? (c). Find the minimum time that the train takes to travel between two consecutive stations that are 45 miles apart. (d). The trip from one station to the next takes 37.5 minutes. How far apart are the stations?
> Use Newton’s method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. 3 sin(x?) = 2x
> When gas expands in a cylinder with radius r, the pressure at any given time is a function of the volume: P = P (V). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: F = πr2p. Show that t
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. х — у— 1, у— х? — 4х + 3; about y— 3
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. x = 2y – y', x = 0; about the y-axis
> The region enclosed by the given curves is rotated about the specified line. Find the volume of the resulting solid. у 3 1/х, х 3D 1, х— 2, у %3D 0; about the x-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x, y = Vx; about x = 2
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. у 3D1+ sec x, yу 3 3; about y —1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = e", y = 1, x = 2; about y = 2
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x=1- y, x= y² – 1
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у %3 х? — 2х, у%3х+4
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = x², y? = x
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у 3 In x, ху — 4, х%3D1, х— 3
> A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.
> Let g (x) = fx0f (t) dt, where f is the function whose graph is shown. (a). Evaluate g (0) and g (6). (b). Estimate g (x) for x = 1, 2, 3, 4, and 5. (c). On what interval g is increasing? (d). Where does have a maximum value? (e). Sketch a rough graph of
> Find the values of p for which the integral converges and evaluate the integral for those values of p. dx
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = e", y = x? – 1, x=-1, x = 1 x = -1,
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. sin?x - dx V- x,
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. sec'x -dx lo xVx
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. 2 + e -
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. x' + 1 .3
> Show that 1/3 Tn + 3.2Mn = S2n.
> Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.
> Sketch the region and find its area (if the area is finite). s = {(x, y) | –2 < r< 0, 0 < y< 1//I + 2 }
> A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surfa
> Find the area of the shaded region. YA x= y? - 4y (-3, 3) x= 2y - y?
> Let g (x) = fx0 f (t) dt, where f is the function whose graph is shown. (a). Evaluate g (x) for x = 0, 1, 2, 3, 4, 5 and 6. (b). Estimate g (7). (c). Where does b have a maximum value? Where does it have a minimum value? (d). Sketch a rough graph of g.
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < x < n/2, 0 < y< sec?r}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | x > 0, 0 < y< xe*}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < y < 2/(x² + 9)}
> Find the area enclosed by the x-axis and the curve x = 1 + et, y = t – t2.
> Find the area enclosed by the curve x = t2 – 2t, y = √t and the y-axis.
> Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 < θ < 2π, to find the area that it encloses.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. csc x dx J2/2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx e - 1
> Graph f (x) = sin (ex) and use the graph to estimate the value of t such that ft+1tf (x) dx is a maximum. Then find the exact value of t that maximizes this integral.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dy 4y – 1 Je
> Find the area of the shaded region. yA x= y?- 2 y=1 x= e y=-
> A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a (t) = 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (d
> If the birth rate of a population is b (t) = 2200e0.024t people per year and the death rate is d (t) = 1460e0.018t people per year, find the area between these curves for 0 < t < 10. What does this area represent?
> A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 29.0, 27.6, 26.3, 28.8, 20.5, 15.1, 8.7 and 2.8. Use Simpson’s Rule to estima
> The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool. 5.6 5.0 6.8 4.8 4.8 7.2 6.2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dx 3— х
> Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use Simpson’s Rule to estimate how much farther Kell
> Sketch the curves y = cos x and y = 1 – cos x, 0 < x < π, and observe that the region between them consists of two separate parts. Find the area of this region.
> Sketch the region that lies between the curves y = cos x and y = sin 2x and between x = 0 and x = π/2. Notice that the region consists of two separate parts. Find the area of this region.
> Show that f10(1 – x2) n dx = 22n(n!)2/ (2n + 1)! Hint: Start by showing that if denotes the integral, then 2k + 2 - Ik 2k + 3 Ik+1
> 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 cos x, y=x10
> 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 = x² In x, y = VI – I (х — 1
> 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 = y = x* – x, x 0 (x² + 1)*"
> Find the area of the shaded region. y. y = Vx+2 x= 2 1 х+1
> 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 sin(x²), y = x*
> Sketch the region enclosed by the given curves and find its area. y = 3x², y= 8r², 4x + y = 4, x> 0 %3D
> Sketch the region enclosed by the given curves and find its area. y = 1/x, y= x, y=}x, x>0
> Sketch the region enclosed by the given curves and find its area. у %3D сos x, у — 2 — cos x, 0x<2п
> Sketch the region enclosed by the given curves and find its area. y = e", y = xe", x= 0
> A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathema
> Sketch the region enclosed by the given curves and find its area. y = x', y = 4x – x?
> Sketch the region enclosed by the given curves and find its area. у — 12 — х', у-х*— 6 y= x² – 6
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у %3 sin x, у— 2х/т, х>0
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x= 2y', x= 4 + y? %3D
> To prove Theorem 1, let F and G be any two antiderivatives of f on I and let H = G - F. (a). If x1 and x2 are any two numbers in I with x1 < x2, apply the Mean Value Theorem on the interval [x1, x2] to show that H (x1) = H (x2). Why does this show that H
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. 4x + y? = 12, x= y
> Find the area of the shaded region. yA y = 5x – x? ((4, 4) y=x
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. -/2 e dy
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. J. Tr? + 2)²
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. -1 dw 2 –
> In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dx V1 +x 4.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 (x – 2)2
> Show that ½ (Tn + Mn) = T2n.
> (a). Graph the functions f (x) = 1/x1.1 and g (x) = 1/x0.9 in the viewing rectangles [0, 10] by [0, 1] and [0, 100] by [0, 1]. (b). Find the areas under the graphs of f and g from x = 1 to x = t and evaluate for t =10, 100, 104, 106, 1010, and 1020. (c).
> Sketch the region and find its area (if the area is finite). s = {(x, y) | x > -2, 0 < y<e?}
> If the units for are feet and the units for a (x) are pounds per foot, what are the units for da/dx? What units does f82 a(x) dx have?
> Sketch the region and find its area (if the area is finite). S = {(r, y) | x < 1, 0 < y< e*}
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. *2 (2' In z dz Jo
> Find the area under the curve y = 1/x3 from x = 1 to x = t and evaluate it for t = 10, 100, and 1000. Then find the total area under this curve for x > 1.
> Find the minimum value of the area of the region under the curve y = x + 1/x from x = a to x = a + 1.5, for all a > 0.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 33 " (x – 1)-1/5 dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. хр (х — 6)° J6 4.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. *14 dx -2 x + 2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 3 dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx. e2* + 3
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. Je x(In x)
> If x is measured in meters and f (x) is measured in newtons, what are the units for f0100 f (x) dx?
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx .3
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x? dx 9 + x*
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x'e dx
> The line y = mx + b intersects the parabola y = x2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB. y y =x B y = mx +b P
> (a). The base of a solid is a square with vertices located at (1, 0), (0, 1) and (0, -1). Each cross-section perpendicular to the x-axis is a semicircle. Find the volume of the solid. (b). Show that by cutting the solid of part (a), we can rearrange it t
> Which of the following integrals are improper? Why? (a) f dx. 2x - 1 (b) Jo 2x 1 dx :- 1 sin x dx -1+x* (d) * In(x – 1) dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 9. re dr
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. se-* ds -5s Jo
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. * cos at dt
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x +1 - dx x? + 2x
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx
> Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s and its downward acceleration is If the raindrop is initially 500 m above the
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. xedx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. L – 3y°) dy
