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mathematics
calculus

Questions and Answers of Calculus

A point P on the rim of a wheel of radius a is initially at the origin. As the wheel rolls to the right along the x-axis, P traces out a curve called a cycloid (see Figure 18). Derive parametric
Find the length of one arch of the cycloid of problem 18. First show that
Suppose that the wheel of Problem 18 turns at a constant rate w = dÎ¸/dt, where t is time. Then Î¸ = wt.(a) Show that the speed ds/dt of P along the cycloid is(b) When is the
Find the length of each curve.(a)(b) x = t - sint, y=1 - cost, 0 ‰¤ t ‰¤ 4Ï€
Find the length of each curve.(a)(b) x = accost + at sint, y = asint - at cost, -1 ‰¤ t ‰¤ 1
In Problems, find the area of the surface generated by revolving the given curve about the x-axis. a. y = 6x, 0 ≤ x ≤ 1 b. y = √25 - x2, -2 ≤ x ≤ 3 c. y = x3/3, 1 ≤ x ≤ √7
If the surface of a cone of slant height C and base radius r is cut along a lateral edge and laid flat, it becomes the sector of a circle of radius C and central angle Î¸ (see Figure
Show that the area of the part of the surface of a sphere of radius a between two parallel planes h units apart (h < 2a) is 2irah. Thus, show that if a right circular cylinder is circum-scribed about
Figure 20 shows one arch of a cycloid. Its parametric equations (see Problem 18) are given byx = a(t - sin t), y = a(1 - cos t), 0 s t s 2Ï€(a) Show that the area of the surface generated
The circle x = a cos t, y = a sin t, 0 ≤ t ≤ 2π, is revolved about the line x = b, 0 < a < b, thus generating a torus (doughnut). Find its surface area.
Sketch the graphs of each of the following parametric equations. (a) x = 3 cos t, y = 3 sin t, 0 ≤ t ≤ 2π (b) x = 3 cos t, y = sin t, 0 ≤ t ≤ 2π (c) x = t cos t, y = t sin t, 0 ≤ t ≤
Find the lengths of each of the curves in Problem 35. You will first have to set up the appropriate integral and then use a computer to evaluate it. In problem 35 (a) x = 3 cos t, y = 3 sin t, 0 ≤
Using the same axes, draw the graphs of y = xn on [0, 1] for n = 1, 2, 4, 10, and 100. Find the length of each of these curves. Guess at the length when n = 10,000
In Problems, sketch the graph of the given parametric equation and find its length. a. x= t3/3, y = t2/2; 0 ≤ t ≤ 1 b. x = 3t2 + 2, y = 2t3 - 1/2; 1 ≤ t ≤ 4
A force of 6 pounds is required to keep a spring stretched foot beyond its normal length. Find the value of the spring constant and the work done in stretching the spring 1/2 foot beyond its natural
Find the work done in pumping all the oil (density S = 50 pounds per cubic foot) over the edge of a cylindrical tank that stands on one of its bases. Assume that the radius of the base is 4 feet, the
Do Problem 13 assuming that the tank has circular cross sections of radius 4 + x feet at height x feet above the base. In problem 13 Find the work done in pumping all the oil (density S = 50 pounds
A volume v of gas is confined in a cylinder, one end of which is closed by a movable piston. If A is the area in square inches of the face of the piston and x is the distance in inches from the
A cylinder and piston, whose cross-sectional area is 1 square inch, contain 16 cubic inches of gas under a pressure of 40 pounds per square inch. If the pressure and the volume of the gas are related
Find the work done by the piston in Problem 16 if the area of the face of the piston is 2 square inches. In problem 16 A cylinder and piston, whose cross-sectional area is 1 square inch, contain 16
One cubic foot of gas under a pressure of 80 pounds per square inch expands adiabatically to 4 cubic feet according to the law pv1.4 = c. Find the work done by the gas.
A cable weighing 2 pounds per foot is used to haul a 200-pound load to the top of a shaft that is 500 feet deep. How much work is done?
A 10-pound monkey hangs at the end of a 20-foot chain that weighs 2 pound per foot. How much work does it do in climbing the chain to the top? Assume that the end of the chain is attached to the
A space capsule weighing 5000 pounds is propelled to an altitude of 200 miles above the surface of the earth. How much work is done against the force of gravity? Assume that the earth is a sphere of
According to Coulomb's Law, two like electrical charges repel each other with a force that is inversely proportional to the square of the distance between them. If the force of repulsion is 10 dynes
A bucket weighing 100 pounds is filled with sand weighing 500 pounds. A crane lifts the bucket from the ground to a point 80 feet in the air at a rate of 2 feet per second, but sand simultaneously
In Problems, assume that the shaded region is part of a vertical side of a tank with water (Î´ = 62.4 pounds per cubic foot) at the level shown. Find the total force exerted by the water
A force of 0.6 Newton is required to keep a spring with a natural length of 0.08 meter compressed to a length of 0.07 meter. Find the work done in compressing the spring from its natural length to a
Find the total force exerted by the water on all sides of a cube of edge length 2 feet if its top is horizontal and 100 feet below the surface of a lake.
Find the total force exerted by the water against the bottom of the swimming pool shown in Figure 19, assuming it is full of water.
Find the total force exerted by the fluid against the lateral surface of a right circular cylinder of height 6 feet, which stands on its circular base of radius 5 feet, if it is filled with oil (δ =
A conical buoy weighs m pounds and floats with its vertex V down and h feet below the surface of the water (Figure 20). A boat crane lifts the buoy to the deck so that V is 15 feet above the water
Initially, the bottom tank in Figure 21 was full of water and the top tank was empty. Find the work done in pumping all the water into the top tank. The dimensions are in feet.
It requires 0.05 joule (Newton-meter) of work to stretch a spring from a length of 8 centimeters to 9 centimeters and another 0.10 joule to stretch it from 9 centimeters to 10 centimeters. Determine
For a certain type of nonlinear spring, the force required to keep the spring stretched a distance s is given by the formula F = ks4/3. If the force required to keep it stretched 8 inches is 2
Two similar springs S1 and S2, each 3 feet long, are such that the force required to keep either of them stretched a distance of s feet is F = 6s pounds. One end of one spring is fastened to an end
In each of Problems, a vertical cross section of a tank is shown. Assume that the tank is 10 feet long and full of water, and that the water is to be pumped to a height 5 feet above the top of the
For the homogeneous lamina shown in Figure 17, find m, My, Mx, x, and y.
John and Mary weighing 180 and 110 pounds, respectively, sit at opposite ends of a 12-foot teeter board with the fulcrum in the middle. Where should their 80-pound son Toni sit in order for the board
Use Pappus's Theorem to find the volume of the solid obtained when the region bounded by y = x3, y = 0, and x = I is revolved about the y-axis. Do the same problem by the method of cylindrical shells
Use Pappus's Theorem to find the volume of the torus obtained when the region inside the circle x2 + y2 = a2 is revolved about the line x = 2a.
Use Pappus's Theorem together with the known volume of a sphere to find the centroid of a semicircular region of radius a.
Prove Pappus's Theorem by assuming that the region of area A in Figure 20 is to be revolved about the y-axis,And
The region of Figure 20 is revolved about the line y = K, generating a solid.(a) Use cylindrical shells to write a formula for the volume in terms of w(y).(b) Show that Pappus's formula, when
Consider the triangle T of Figure 21.(a) Show that y = h/3 (and thus that the centroid of a triangle is at the intersection of the medians).(b) Find the volume of the solid obtained when T is
A regular polygon P of 2n sides is inscribed in a circle of radius r. (a) Find the volume of the solid obtained when P is revolved about one of its sides. (b) Check your answer by letting n → ∞.
Let f be a nonnegative continuous function on [0,1].(a) Show that(b) Use part (a) to evaluate
Let 0 ≤ f(x) ≤ g(x) for all x in [0, 1], and let R and S be the regions under the graphs of f and g, respectively. Prove or disprove those yR ≤ yS.
Approximate the centroid of the lamina shown in Figure 22. All measurements are in centimeters, and the horizontal measurements occur 5 centimeters apart.
The geographic center of a region (county, state, country) is defined to be the centroid of that region. Use the map in Figure 24 to approximate the geographic center of Illinois. All distances are
Do Problem 3 if 8(x) = 1 + x3. In problem 3 A straight wire 7 units long has density δ(x) = √x at a point x units from one end. Find the distance from this end to the center of mass.
The masses and coordinates of a system of particles in the coordinate plane are given by the following: 2, (1, 1); 3, (7. 1); 4, (-2, -5); 6, (-1, 0); 2, (4, 6). Find the moments of this system with
The masses and coordinates of a system of particles are given by the following: 5, (-3, 2); 6, (-2, -2); 2, (3, 5); 7, (4, 3); 1, (7, -1) . Find the moments of this system with respect to the
Verify the expressions for AMy, AMx, My and Mx in the box in Figure 10.
In Problems find the centroid of the region bounded by the given curves. Make a sketch and use symmetry where possible. a. y = 2 - x, y = 0, x = 0 b. y = 2 - x2, y = 0
In Problems, a discrete probability distribution for a random variable X is given. Use the given distribution to find(a) P(X ‰¥ 2)(b) E(X).1.2. 3.
Prove the three properties of the CDF in Theorem A.
A continuous random variable X is said to have a uniform distribution on the interval [a, b] if the PDF has the form(a) Find the probability that the value of A' is closer to a than it is to b. (b)
The median of a continuous random variable X is a value x0 such that P(X ≤ x0) = 0.5. Find the median of a uniform random variable on the interval [a, b].
Without doing any integration, find the median of the random variable that has PDF f(x) = 15/512x2(4 - x)2, 0 ≤ x ≤ 4.
Find the value of k that makes f(x) = kx (5 - x), 0 ≤ x ≤ 5, a valid PDF.
Find the value of k that makes f(x) = kx2 (5 - x)2, 0 ≤ x ≤ 5, a valid PDF.
The time in minutes that it takes a worker to complete a task is a random variable with PDF f(x) = k(2 - |x - 2|), 0 ≤ x ≤ 4. (a) Find the value of k that makes this a valid PDF. (b) What is the
The daily summer air quality index (AQI) in St. Louis is a random variable whose PDF is f(x) = kx2 (180 - x), 0 ≤ x ≤ 180. (a) Find the value of k that makes this a valid PDF. (b) A day is an
Suppose that X is a continuous random variable. Explain why P(X = x) = 0. Which of the following probabilities are the same? Explain. P(a < X < b), P(a ≤ X ≤ b). P(a < X a ≤ b), P(a ≤ X < b)
Show that if A' is the complement of A. that is, the set of all outcomes in the sample space S that are not in A, then P(Ac) = 1 - P(A).
If X is a discrete random variable then the CDF is a step function. By considering values of x less than zero, between 0 and 1, etc., find and graph the CDF for the random variable X in Problem 1.
Suppose a random variable Y has CDFFind each of the following (a) P(Y (b) P(0.5 (c) The PDF of Y (d) Use the Parabolic Rule with n = 8 to approximate E(Y).
Suppose a random variable Z has CDFFind each of the following (a) P(Z > 1) (b) P(1 (c) The PDF of Z (d) E(Z)
In Problems, a PDF for a continuous random variable X is given. Use the PDF to find(a) P(X ‰¥ 2),(b) E(X),(c) The CDE1.2.
Find the following anti-derivative ∫1/x2 dx
In problems, find all values of x that satisfy the given relationship (a) sin x = 1/2 (b) cos x = -1
For the triangles shown in Problems, find all of the following in terms of x; sinÎ¸, cosÎ¸, tanÎ¸, cotÎ¸, tanÎ¸, cotÎ¸,
In problem solve the differential equation subject to the given condition. (a) y' = xy2, y = 1 when x = 0 (b) y' = cosx /y, y = 4 when x = 0
For Problems, letAnd find the following (a) F(1) (b) F' (x) (c) Dx F(x2)
In problems, evaluate the expression at the given values. (1) (1+h)1/h; h = 1, 1/5, 1/10, 1/50, 1/100 (2) (1+1/n)n; n = 1 , 10, 100, 1000
Use the approximations In 2 ≈ 0.693 and In 3 ≈ 1.099 together with the properties stated in Theorem A to calculate approximations to each of the following. For example, In 6 = ln(2 ∙ 3) = In 2
In problems, find the integrals(a) ˆ« 1/(2x+1) dx(b) ˆ« 1/(1-2x) dx(c)(d)
Use your calculator to make computation in problem 1 directly In problem 1 (a) In 6 (b) In 1.5 (c) In 81 (d) In √2 (e) In(1/36) (f) In 48
In problems, use Theorem A to write the expressions as the logarithm of a single quantity. (a) 2ln(x+1) - lnx (b) ½ ln (x-9) + ½ lnx (c) ln(x-2) - ln(x+2) + 2lnx
In Problems, find the indicated derivative. Assume in each case that x is restricted so that in is defined. (a) Dx In(x2 + 3x + π) (b) Dx In(3x3 + 2x) (c) Dx1n(x - 4)3
In problems, find dy / dx by logarithmic differentiation(a)(b) y = (x2 + 3x) 9x-2) (x2 +1) (c)
In problems, make use of the known graph of y = ln x to sketch the graphs of the equations. (a) y = ln |x| (b) y = ln √x (c) y = ln(1/x)
Sketch the graph of y = ln cos x + ln sec x on (-π/2, π/2) but think before you begin.
Find all local extreme values of f(x) = 2x2 ln x - x2 on its domain.
The rate of transmission in a telegraph cable is observed to be proportional to x2 In(1/x), where x is the ratio of the radius of the core to the thickness of the insulation (0 < x < 1). What value
Use the fact that ln x = -ln(1/x) and problem 43 to show that lim x→0 ln x = -∞
The region bounded by y = (x2 + 4)-1, y = 0, x = 1, and x = 4, is revolved about the y-axis, generating a solid, Find its volume.
Find the length of the curve y = x2/4 - ln √x. 1 ≤ x ≤ 2.
By appealing to the graph of y = 1/x, show that ½ + 1/3 + ... + 1/n < ln n < 1 + ½ + 1/3 + ... + 1/(n-1)
Prove Napier's inequality, which says that, for 0
Let f(x) = cos(In x).(a) Find the absolute extreme points on [0.1, 20].(b) Find the absolute extreme points on [0.01,20].(c) Evaluate
Draw the graphs of f(x) = x In(1/x) and g(x) = x2 In(1/x) on (0.1]. (a) Find the area of the region between these curves on (0, 1). (b) Find the absolute maximum value of |f(x) - g(x)| on (0, 1].
Follow the directions of Problem 59 for f(x) = x In x and g(x) = √x In x. In problem 59 (a) Find the area of the region between these curves on (0, 1). (b) Find the absolute maximum value of |f(x)
In problems, the graph of y = f(x) is shown. In each case, decide whether f has an inverse and, if so, estimate f-1(2)(a)(b) (c) (d) (e)
In problems, find a formula for f-1(x) and then verify that f-1(f(x)) = x and f(f-1(x)) = x (a) f(x) = x + 1 (b) f(x) = -x/3 + 1
Find the volume V of water in the conical tank of Figure 8 as a function of the height h. Then find the height ft as a function of volume V.
A ball is thrown vertically upward with velocity v0. Find the maximum height H of the ball as a function of v0. Then find the velocity v0 required to achieve a height of H.
In each of problems, the graph of y = f(x) is shown sketch the graph of y = f-1(x) and estimate (f-1)' (x).(a)(b) (c)
Suppose that both f and g have inverse and that h(x) = (f o g) (x) = f(g(x)). Show that h has an inverse given by h-1 = g-1 o f-1.

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