A paper cup has the shape of a cone with height 10 cm and radius 3 cm (at the top). If water is poured into the cup at a rate of 2 cm3/s, how fast is the water level rising when the water is 5 cm deep?
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. F(x) = [" VI + sec t dt Hint:"V1 + sec t dt = - V1 + sec t dt
> For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve p = 20 + 1/10x, and the supply curve p
> If a supply curve is modeled by the equation p = 200 + 0.2x3/2, find the producer surplus when the selling price is $400.
> The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for |f (4)(x) | by hand. But computer algebra systems have no problem computing and graphing it, so we can easily find a v
> The supply function px (x) for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so ps is an increasing function of x
> A demand curve is given by p = 450/ (x + 8). Find the consumer surplus when the selling price is $10.
> Let f (x) = xe-x if x > 0 and f (x) = 0 for if x < 0. (a). Verify that is a probability density function. (b). Find P (1 < X < 2).
> Let f (x) = 3/64 x √16 – x2, for 0 < x < 4 and f (x) = 0 for all other values of x. (a). Verify that f is a probability density function. (b). Find P (X < 2).
> Let f (t) be the probability density function for the time it takes you to drive to school in the morning, where t is measured in minutes. Express the following probabilities as integrals. (a). The probability that you drive to school in less than 15 min
> Let f (x) be the probability density function for the lifetime of a manufacturer’s highest quality car tire, where is measured in miles. Explain the meaning of each integral. *40,000 (а) 30,000 f(x) dx (b) m f(x) dx 25,000
> A curve is defined by the parametric equations Find the length of the arc of the curve from the origin to the nearest point where there is a vertical tangent line. cos u - du sin u -du y = %3D
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(r) = V? + 4 dx
> A circular disk of radius is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a heig
> Evaluate limx→o = 1/x fx0(1 – tan 2t)1/t, dt.
> Find the point on the parabola y = 1 – x2 at which the tangent line cuts from the first quadrant the triangle with the smallest area.
> Estimate the area of the region enclosed by the loop of the curve x = t3 -12t, y = 3t2 + 2t + 5.
> If f is a differentiable function such that f (x) is never 0 and fx0 f (t) dt = [f (x)]2 for all x, find f.
> If f (x) = fx0x2 sin (t2) dt, find f'(x).
> If x sin πx = fx20f (t) dt, where f is a continuous function, find f (4).
> If f40e(x-2)4 dx = k, find the value of f40xe(x-2)4 dx.
> Show that |sin x – cos x | < √2 for all x.
> A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L. The man keeps the rope straight and taut. The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tange
> Evaluate the integral. f-23 (x2 – 3) dx
> Find d2/dx2 fx0(fsint1 √1 + u4, du) dt.
> One of the problems posed by the Marquis de l’Hospital in his calculus textbook Analyse des Infiniment Petits concerns a pulley that is attached to the ceiling of a room at a point C by a rope of length r. At another point B on the ceil
> Evaluate limx→∞ (1/√n √n + 1, + 1/√n √n + 2, + . . . + 1/√n √n + n.
> The figure shows a horizontal line y = c intersecting the curve y = 8x – 27x3. Find the number c such that the areas of the shaded regions are equal. y y = 8x – 27x y=c
> (a). Let An be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2π/n, show that (b). Show that limn→∞ An =
> Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. (a). If the target weight is 500 g, what is the probability that the machine produces a box with l
> Evaluate f10(3√1 – x7 - 7√1 – x3) dx.
> Suppose f is continuous, f (0) = 0, f (1) = 1, f'(x) > 0, and f10f (x) dx = 1/3. Find the value of the integral f10f-1 (y) dy.
> Evaluate f∞-1(x4/1 + x6) dx.
> Determine the values of the number for which the function f has no critical number: f(x) = (a? + a – 6) cos 2x + (a – 2)x + cos 1 %3D
> Evaluate the integral. f12 (2x – ex) dx
> Suppose the graph of a cubic polynomial intersects the parabola when y = x2, x = 0, x = a and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?
> A triangle with area 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
> Find the highest and lowest points on the curve x2 + xy + y2 = 12.
> For what value of a is the following equation true? + a = e lim x - a х —
> A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of is P (r, 0), and
> If f (x) = fg(x)01/√1 + t3, dt, where g (x) = fcosx0[1 + sin (t2)], dt, find f'(π/2).
> Find the approximations Ln, Rn, Tn and Mn for n = 5, 10 and 20. Then compute the corresponding errors EL, ER and ET. and EM (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.) What observations ca
> A solid is generated by rotating about the -axis the region under the curve y = f (x), where f is a positive function and x > 0. The volume generated by the part of the curve from x = 0 to x = b is b2 for all b > 0. Find the function f.
> In lies on ∆ABC, D, lies on AB, |CD| = 5cm, |AD| = 4 cm, |BD = 4cm, and CD ⊥ AB. Where should a point p be chosen on CD so that the sum |PA| + |PB| + |PC| is a minimum? What if |CD| = 2 cm?
> Find the volume of the largest circular cone that can be inscribed in a sphere of radius r.
> (a). Use an improper integral and information from Exercise 27 to find the work needed to propel a 1000-kg satellite out of the earth’s gravitational field. Exercise 27: (a). Newton’s Law of Gravitation states that t
> Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.
> Find the point on the hyperbola xy = 8 that is closest to the point (3, 0).
> A water skier skis over the ramp shown in the figure at a speed of 30 ft/s. How fast is she rising as she leaves the ramp? -15 ft
> A balloon is rising at a constant speed of 5 ft/s. A boy is cycling along a straight road at a speed of 15 ft/s. When he passes under the balloon, it is 45 ft above him. How fast is the distance between the boy and the balloon increasing 3 s later?
> The angle of elevation of the sun is decreasing at a rate of 0.25 rad/h. How fast is the shadow cast by a 400-ft-tall building increasing when the angle of elevation of the sun is π/6?
> Evaluate the limit. lim x→(π/2)- (tan x) cos x
> The left, right, Trapezoidal, and Midpoint Rule approximations were used to estimate f20 f (x) dx, where f is the function whose graph is shown. The estimates were 0.7811, 0.8675, 0.8632, and 0.9540, and the same number of subintervals were used in each
> Evaluate the limit. lim x→1+ (x / 1 – x – 1/ ln x)
> Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines. (a). The x-axis (b). The y-axis (c). y = 2
> Evaluate the integral. f-55 e dx
> A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?
> Evaluate the limit. lim x→0+ x2 ln x
> Evaluate the limit. lim x→∞ x3 e-x
> Evaluate the limit. lim x→∞ e4x -1 – 4x/x2
> Find the local and absolute extreme values of the function on the given interval. Зх — 4 f(x) [-2, 2] x² + 1'
> Evaluate the limit. lim x→0 e4x – 1 – 4x /x2
> Evaluate the limit. lim x→0 1 – cos x / x2 + x
> Evaluate the limit. lim x→0 tan πx/ln (1 + x)
> For what values of the constants a and b is a point of inflection of the curve y = x3 + ax2 + bx + 1?
> If a diver of mass m stands at the end of a diving board with length L and linear density p, then the board takes on the shape of a curve y = f (x), where E and are positive constants that depend on the material of the board and g ( (a). Find an expre
> Evaluate the integral. f-10 (2x – ex) dx
> Find the local and absolute extreme values of the function on the given interval. f(x) = x/T- x, [-1, 1]
> Let R be the region in the first quadrant bounded by the curves y = x3 and y = 2x – x2. Calculate the following quantities. (a). The area of R (b). The volume obtained by rotating R about the x-axis (c). The volume obtained by rotating R about the y-axis
> Graph f (x) = e-1/x2 in a viewing rectangle that shows all the main aspects of this function. Estimate the inflection points. Then use calculus to find them exactly.
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> The figure shows two regions in the first quadrant: A (t) is the area under the curve y = sin (x2) from 0 to t, and B (t) is the area of the triangle with vertices O, P, and (t, 0). Find limt→0+A (t)/B (t). y y. P(t, sin(12)) P(t,
> If f' is continuous on [0, ∞] and limx→∞f (x) = 0, show that f∞0f' (x) dx = -f (0).
> If oil leaks from a tank at a rate r (t) of gallons per minute at time f020 r (t) dt, what does represent?
> Evaluate the integral. f18 3√x dx
> If n is a positive integer, prove that f10 (ln x) n dx = (-1) n n!
> If f' is continuous on [a, b], show that 2 fba f (x) f' (x) dx = [f (b)]2 – [f (a)]2
> Find a function f and a value of the constant a such that 2 f(t) dt = 2 sin x – 1
> Let be the region bounded by the curves y = tan (x2), x = 1 and y = 0. Use the Midpoint Rule with n = 4 to estimate the following quantities. (a). The area of (b). The volume obtained by rotating about the -axis
> If f is a continuous function such that for all x, find an explicit formula for f (x). , r) dt = xe² + [ e "f)di
> Find the local and absolute extreme values of the function on the given interval. f(x) = x' – 6x² + 9x + 1, [2, 4]
> Suppose that the temperature in a long, thin rod placed along the x-axis is initially C/(2a) if |x| a. It can be shown that if the heat diffusivity of the rod is k, then the temperature of the rod at the point x at time t is To find the temperature d
> A population of honeybees increased at a rate of r (t) bees per week, where the graph of is as shown. Use Simpson’s Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks. 12000 8000 400
> Let r (t) be the rate at which the world’s oil is consumed, where is measured in years starting at t = 0 on January 1, 2000, and r (t) is measured in barrels per year. What does f80r (t) dt represent?
> The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson’s Rule to estimate the distance traveled by the car. t (min) v (mi/h) t (min) v (mi/h) 40 6 56 1 42 7 57 2 45 8 57 3 49
> Evaluate the integral. f01 x4/5 dx
> (a). Evaluate the integral f∞0 xne-x dx for n = 0, 1, 2, and 3. (b). Guess the value of f∞0 xne-x dx when n is an arbitrary positive integer. (c). Prove your guess using mathematical induction.
> A particle moves along a line with velocity function v (t) = t2 - t, where is measured in meters per second. Find (a) the displacement and (b) the distance traveled by the particle during the time interval [0, 5].
> For what values of a is f∞0eax cos x dx convergent? Use the Table of Integrals to evaluate the integral for those values of a.
> Use the Comparison Theorem to determine whether the Integral is convergent or divergent. dx x' + 2
> Evaluate the integral or show that it is divergent. f62y/√y – 2, dy
> Find the volume of the solid obtained by rotating about the x-axis the region bounded by the curves y = e-2x, y = 1 + x and x = 1.
> Evaluate the integral or show that it is divergent. fe1dx/x √lnx
> Evaluate the integral or show that it is divergent. f101/2 – 3x, dx
> Evaluate the integral or show that it is divergent. f0-∞e-2x dx
> Evaluate the integral or show that it is divergent. f∞0ln x/x4, dx
> Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. y = sin x, 0 <I<T
> Evaluate the integral or show that it is divergent. f∞11/ (2x + 1)3, dx
