Use the cross product to find the sin(theta) where theta is the angle between v and w where v = i + j and w = i + j + k.

View AnswerSuppose that you are climbing a hill whose shape is given by z = 483 - 0.03x2-0.03y2, and that you are at the point (50, 60, 300). In which direction (unit vector) should you proceed initially in orde

View AnswerParametrize the intersection of the surfaces using t = y as the parameter. (If there are multiple correct answers, give only one answer.) y^2 - z^2 = x - 2, y^2 + z^2 = 16.

View AnswerThe length of the day in Boulder (Latitude 40 degrees N) can be modeled approximately by L(t) = 3 cos (2 /365 (t + 10) )+ 12 where L is given in hours and t is the day of the year. a) Evaluate L(355);

View AnswerFind the general solution of the given second-order differential equation. 3y'' + 2y' + y = 0

View AnswerFind the volume V of the described solid S. A cap of a sphere with radius r and height h V = ___

View AnswerFind the radius of convergence, R, and the convergence interval for x of the series sum of (x - 8)^n / (n^7 + 1) from n = 0 to infinity.

View AnswerFind a parametrization, using cos(t) and sin(t), of the following curve: The intersection of the plane y=3 with the sphere x2+y2+z2=58

View AnswerLasha Talakhadze of the Republic of Georgia broke a world record in the snatch and grab and clean and jerk weight lifting competition in Rio with 105 , 258 kilograms respectively. Determine the amount

View AnswerFind the centroid of the region bounded by the given curves. Give your answers correct to two decimal places. y = x^3, x + y = 2, y = 0.

View AnswerFind a vector parametrization of the curve x = -5Z^2 in the xz-plane. Use t as the parameter in your answer.

View AnswerFor the following sequences evaluate the ratio \frac{a_n + 1}{a_n}. Simplify your answer. (a) a_n = \frac{2^n}{(2n-1)!} (b) a_n = \frac{(2n)!}{3^{2n + 1}}

View AnswerIntegrate f(x,y,z)=16xz over the region in the first octant (x,y,z%3E=0) above the parabolic cylinder z=y2 and below the paraboloid z=8-2x2-y2

View AnswerFind the limit for the following functions if they exist. lim(x,y)-%3E(0,0) (x+14y)^2/(x^2 + 196y^2), and lim(x,y)-%3E(0,0) (5x^3 + 7y^3)/(x^2 + y^2)

View AnswerA ladder 25 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 2 feet per second. How fast is the top of the ladder sliding down the wal

View AnswerGiven a_{n}=\frac{\sqrt{n^2-1}}{2n+1}, find \lim_{n \to \infty }a_{n}. Does \sum_{n=1}^{\infty}a_{n} converge or diverge? Explain why. Suppose b_{n} are such that \sum_{i=1}^{n}b_{i}=a_{n}. Does the s

View AnswerA spacecraft moves along a path described by the parametric equations; x = 10*(sqrt(1+t^4)-1) and y = 40t^(3/2), for the first 100 seconds after launch. Here, x and y are measured in meters, and t is

View AnswerA steel pipe is being carried down a hallway 9 ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 6 ft wide. What is the length of the longest pipe that can be carrie

View AnswerBismuth-210 has a half-life of 5.0 days. A sample originally has a mass of 800 mg. Find a formula y(t) for the mass remaining after (t) days. Find the mass remaining after 30 days. When is the mass re

View AnswerA poster is to have an area of 180 in^2 with 1 inch margins at the bottom and sides and a 2 inch margin at the top. Find the exact dimensions that will give the largest printed area.

View AnswerFind the radius of convergence, R, of the series: \sum_{n=2}^{\infty} (-1)^n \frac{xn}{8^nln\;n}

View AnswerUse Green's theorem to compute the area inside the ellipse( x^2/13^2)+(y^2/6^2)=1. Use the fact that the area can be written as dub intD dxdy=(1/2) int partialD (-y dx+x dy) . Hint: x(t)=13cos(t).

View Answer200 ounces of a drink that contains 60% fruit juice is mixed with x ounces of filtered water. If 100 ounces of filtered water is used, what percent of juice does that resulting juice contain?

View AnswerDetermine whether the following sequence converges or diverges. If it converges, find the limit. a_n = ((-1)^n)/(2*sqrt(n)).

View AnswerFind the cross product axb, and verify that it is orthogonal to both a and b. a = j + 7k b = 2i - j +4k

View AnswerFind the arc length parametrization of the circle in the plane z=9 with a radius of 4 and center located at (1,4,9).

View AnswerShow that the line integral is independent of path by finding a function f such that f = F. C2xe ydx + (2y x2e y)dy, C is any path from (1, 0) to (5, 1)

View AnswerFind the interval of convergence of the following series. The sum of (-1)^n * ((x + 2)^n)/(n^2) from n = 0 to infinity.

View AnswerUse Green's Theorem to calculate the circulation of F = (y)I + (2xy)j around the unit circle, oriented counterclockwise.

View AnswerA) Find the area between x-axis and under the curve of f(x) = x^2 +x+3 over the interval ~[1,3] B) Evaluate \int^4_{0}f(x)dx \left\{\begin{matrix} 12x, x%3C 1 \\ 8 x\geq 1 \end{matrix}\right.

View AnswerFind the average value of the function f(x) = 7 sin2x cos3xon the interval [-pi, pi]. (Round your answer to two decimal places.)

View AnswerA cube is located such that its top four corners have the coordinates (-1, -2, 2), (-1,3,2), (4, -2, 2) and (4,3,2). Give the coordinates of the center of the cube.

View AnswerFind the absolute maximum and absolute minimum of the function f(x,y)=xy-2y-4x+8 on the region on or above y=x^2 and on or below y=7.

View AnswerUse the arc length formula to find the length of the curve y = sqrt(2 - x^2), 0 less than or equal to x less than or equal to 1. Check your answer by noting that the curve is part of a circle.

View AnswerFind the arc length of the curve below on the given interval by integrating with respect to x. y = (x^3)/3 + 1/(4x); [1, 3]. (Give an exact answer, using radicals as needed.)

View AnswerA bee with velocity vector r'(t) starts out at (-4,3,1) at t=0 and flies around for 7 seconds. Where is the bee located at time 7 if integral from t 0 to 7 r'(u)du=0?

View AnswerAt what points does the helix r(t) = sin t, cos t, t intersect the sphere x^2 + y^2 + z^2 = 8^2?

View AnswerCompute the flux of the vector field F=8x^2y^2zk through the surface S which is the cone sqrt(x^2+y^2)=z, with 0=%3Cz=%3CR, oriented downward.(a) Parameterize the cone using cylindrical coordinates.

View AnswerConsider the solid shaped like an ice cream cone that is bounded by the functions z = x2 + y2 and z = 18 - x2 - y2. Set up an integral in polar coordinates to find the volume of this ice cream cone.

View AnswerSuppose the solid W is one-quarter of a circular cylinder of height 8 and radius 4 centered about the z-axis in the first octant. Find the limits of integration for an interated integral of the form W

View AnswerFind the directional derivative of the function f(x,y) = tan^(-1)(xy) at the point (3,2) in the direction of the vector v = 5i + 4j.

View AnswerUse the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves y = sqrt(x - 1), y = 0, and x = 5 about the line y = 3.

View AnswerA population is modeled by the differential equation given below. dP/dt = 1.3P(1 - P/4200) (a) For what values of P is the population increasing? (b) For what values of P is the population decreasing?

View AnswerLet C be the curve of intersection of the parabolic cylinder x^2 = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point ( 5 , 25 / 2 , 125 / 6 ).

View AnswerFind the centroid of the region bounded by the given curves. y = 6 sin 2x, y = 6 cos 2x, x = 0, x = pi/ 8

View AnswerUse Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x, y) = 10x + 10y; x^2 + y^2 = 50.

View AnswerFind the derivative of the function g(x) = integral of 1/sqrt(3 + t^4) dt from tan x to 4x^2.

View AnswerA sphere of radius 3 inches is sliced with two parallel planes: one passes through the equator and the other is H inches above the first plane. The resulting portion of the sphere between the two plan

View AnswerFind the dimensions of the box with volume 8000 cm^3 that has minimal surface area. (Let x, y, and z be the dimensions of the box.)

View AnswerFind the centroid of the region bounded by the curves y = 2x^3 - 2x and y = 2x^2 - 2. Sketch the region and plot the centroid to see if your answer is reasonable.

View AnswerFind the derivative of the following functions: A) f(x) = e^(x^2 + x) - sqrt(ln x). B) f(x) = x^3 e^x. C) f(x) = (e^x)/(1 + x^2).

View AnswerFind the volume of the wedge-shaped region contained in the cylinder x^2+y^2=49 and bounded above by the plane z=x and below by the xy-plane.

View AnswerCompute the length of the curve C with parametrization r(t) = (e^t, sqrt(2)t, e^(-t)), -1 less than or equal to t less than or equal to 1.

View AnswerDetermine whether the geometric series is convergent or divergent. 10 - 4 + 1.6 - 0.64 + ... If it is convergent, find its sum.

View AnswerFind the absolute maximum and absolute minimum values of the function, if they exist, over the indicated interval. f(x) = 1/3x^3 - 5x; [-8, 8].

View AnswerFind the exact area of the surface obtained by rotating the given curve about the x-axis. Using calculus with Parameter curves.x = 6t - 2t3, y = 6t2, 0 t 1

View Answer1. Determine whether the geometric series is convergent or divergent and find its sum. 9 + 8 + 64//9 + 512/81 + ... 2. Determine whether the geometric series is

View AnswerSolve the given initial-value problem. The DE is a Bernoulli equation. x2? dy/dx ? 2xy = 6y4, y(1) = 1/4

View AnswerFind the linear approximation of the function f(x, y, z) =x2 + y2 + z2 at (2, 6, 9) and use it to approximate the number 2.022 + 5.992 + 8.972. (Round your answer to five decimal places.)

View AnswerA particle starts at the point (-3, 0), moves along the x-axis to (3, 0), and then along the semicircle y = sqrt(9 - x^2) to the starting point. Use Green's Theorem to find the work done on this particle by the force field F(x, y) = %3C3x, x^3 + 3xy^2%3E

View AnswerUse Stokes' Theorem to evaluate S curl F dS. F(x, y, z) = 4y cos z i + ex sin z j + xey k, S is the hemisphere x2 + y2 + z2 = 49, z %3E= 0, oriented upward.

View AnswerFind a cubic function f(x) = ax3 + bx2 + cx + d that has a local maximum value of 4 at x = -4 and a local minimum value of 0 at x = 2.

View AnswerUse polar coordinates to find the volume of the solid bounded by the paraboloid z = 7 - 6x^2 - 6y^2 and the plane given by z = 1. Show steps needed to compute this answer by hand calculation.

View AnswerShow that the vector field F(x,y,z)=(4ycos(-2x),-2xsin(4y),0) is not a gradient vector field by computing its curl. How does this show what you intended?

View AnswerSet up the iterated integrals for both orders of integration. Then evaluate the double integral using the easier order. Double integral y dA, D is bounded by y = x - 56; x = y^2

View AnswerFind the surface area of the solid obtained by rotating the parametric curve c(t) = (cos^3 t, sin^3 t) about the x-axis for 0 less than equal t less than equal pi/2.

View AnswerFind Values of m so that the function y = e^mx is a solution of the given differential equation: y'' - 5y' + 6y = 0

View AnswerSolve the differential equation dR/dx=a(R2+16). Assume a is a non-zero constant, and use C for any constant of integration that you may have in your answer.

View AnswerThere is a rectangular box whose length x, width y and height z are changing at the following rates: the length x is decreasing at a rate of 2 ft/hour, the width y is increasing at a rate of 1 ft/hour

View AnswerFind all critical points of f(x) = -3x^4 + 16x^3 - 24x^2 + 36 and classify each as a relative minimum, relative maximum, or neither one.

View AnswerFind fx(1, 0) and fy(1, 0) and interpret these numbers as slopes for the following equation. f(x, y) = sqrt(4 - x^2 - 3y^2) fx(1, 0) = fy(1, 0) =

View AnswerA particle moves along the curve below. y = 8 + x3 As it reaches the point (1, 3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the x-coordinate of the point changing at that instan

View AnswerSolve the given differential equation by using an appropriate substitution. The DE is homogeneous. (x - y)dx + x dy = 0

View AnswerFor the surface with parametric equations r(s, t) = (st, s + t, s - t), find the equation of the

View AnswerFind the solution r(t) of the differential equation with the given initial condition: r'(t) = %3Csin2t, sin2t, 7t%3E, r(0) = %3C8, 7, 6%3E

View AnswerUsing disk or washers, find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 and y = 4 about the line y = 4.

View AnswerFind an equation of the tangent plane to the given surface at the specified point. z = 6(x - 1)^2 + 4(y + 3)^2 + 4, (2, -2, 14).

View AnswerIf a vector has direction angles alpha = pi/4 and beta = pi/3, find the third direction angle gamma.

View AnswerFind the work done by the force field F in moving an object from A to B. F(x,y)= 2y^{\frac{3}{2}}\mathbf{i} +3x\sqrt{y}\mathbf{j} A(1,1), B(2,9)

View AnswerFind the volume of the solid formed by revolving the region bounded by the graphs of y = x^4 + 2x^2 + 1, y = 1, and x = 1 about the line x = 2.

View AnswerA worker on a scaffolding 75 ft above the ground needs to lift a 500 lb bucket of cement from the ground to a point 30 ft above the ground by pulling on a rope weighing 0.5 lb/ft. How much work is req

View AnswerConsider the equation below. f(x) = 9 sin x + 9 cos x 0 x 2pi a) Find the intervals on which f is increasing. Find the intervals on which f is decreasing. b) Find the local minimum and maximum values

View AnswerConsider the vector field F(x,y,z)=xi+yj+zk. a) Find a function f such that F= f and f(0,0,0)=0. b) Use part a to compute the work done by F on a particle moving along the curve C given by r(t)=(1+3si

View AnswerConsider the solid that lies above the square (in the xy-plane), R = [0, 2] x [0, 2], and below the elliptic paraboloid z = 25 - x^2 - 2y^2. (A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower

View AnswerFind the centroid of the region in the first quadrant bounded by the given curves; y = x^3 and x = y^3.

View AnswerA spherical shell centered at the origin has an inner radius of 3 cm and an outer radius of 4 cm. The density of the material increases linearly with the distance from the center. At the inner surface

View AnswerWrite an equivalent series with the index of summation beginning at n = 1. sum n=0 infty (-1)n+1(n+1)xn

View AnswerUse implicit differentiation to find an equation of the tangent line to the curve at the given point. y sin 8x = x cos 2y, (?/2, ?/4)

View AnswerA Bernoulli differential equation is one of the form \frac{dy}{dx} + P(x)y = Q(x)y^n. Observe that, if n=0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u=y^{1-n} tr

View AnswerCalculate the derivative using implicit differentiation:{partial w / partial z}, {x^2w+w^8+wz^2+9yz=0}. Find dw/dz

View AnswerYou are given the following. F(x,y,z) =(x - 2z)i + (x +y + z)j + (x - 2y)k (a) Find the curl. i + j + k (b) Find the divergence of the vector field.

View AnswerSuppose that f(2)=-5, g(2)=4, f'(2)=-1, and g'(2)=3. a) h(x) = 4f(x) - 2g(x) Find h'(2). b) h(x) = f(x)g(x) Find h'(2). c) h(x) = f(x)/g(x) Find h'(2). d) h(x) = g(x)/(1+f(x)) Find h'(2).

View AnswerGive an equation of the plane whose trace in the yz-plane has equation y = 7z, and in the xy-plane has equation y =-4x

View AnswerFind the absolute maximum and minimum of the function f(x,y)=y(x)^(1/2)-y^2-x+3y on the domain 0 less than x less than 9, 0 less than y less than 3. Absolute minimum value: attained at Absolute maxim

View Answera) Find the point at which the given lines intersect. r = (3,3,0) + t(3,-3,3) r = (6,0,3) + s(-3,3,0) b) Find the equation of the plane that contains these lines.

View AnswerFind the exact length of the curve x = 6 + 6t^2, y = 3 + 4t^3, 0 less than or equal to t less than or equal to 5.

View AnswerUse a power series to approximate the definite integral, I, to six decimal places, i.e., so that the error is less than 0.0000005. Integral of 0 to 0.3 1/(1 + x^5) dx.

View AnswerEvaluate the following limit: Limit as x goes to 0 of ((1/1 + 2x^5) - 1 + 2x^5 - 4x^10)/(6x^15).

View AnswerWrite the equations in cylindrical coordinates. (a) 5z = 7x^2 + 7y^2 z = ___________ (b) 5x^2 + 5y^2 = 7^y r = ___________

View AnswerFind the solution of the differential equation that satisfies the given initial condition. du/dt = (2t + sec^2(t))/(2u), u(0) = -8

View AnswerSketch a contour diagram of each function. Then, decide whether its contours a predominantly lines, parabolas, ellipses or hyperbola. 1. z = -4x2 2. z = x2 - 4y2 3. z = y - 4x2 4. z = x2 + 3y2

View AnswerFind the equation of the tangent line to the curve f(theta) = theta*csc(theta) - (1/2)*cot(theta) at theta = pi/2.

View AnswerShow that the ellipsoid 3x^2 + 2y^2 + z^2 = 9 and the sphere x^2 + y^2 + z^2 - 8x - 6y -8z + 24 = 0 are tangent to each other at the point (1,1,2)

View AnswerDetermine the force due to hydrostatic pressure on the flat vertical side of a tank which has the shape in feet of the boundaries y=0, y=10lnx, y=10ln(-x) and the line y=10ln24. Note that water has density 62.4 lb/ft^3

View AnswerCompute Delta y and dy for the given values of x and dx = Delta x. (Round your answers to three decimal places.) y = sqrt x , x = 1 , Delta x = 1

View AnswerEvaluate the following limits: a) limit as x goes to two of (x^3 - 2x^2)/(10x-20) and b) limit as x goes to zero of (2x)ln(x^2).

View AnswerEvaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.) x3x2 + 64dx, x = 8 tan

View AnswerAn aquarium 4 m long, 3 m wide, and 2 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is 1000 kg/m^3, and use 9.8 m/s^2 for g.)

View AnswerSolve the initial value problem for r as a vector function of t. Differential equation: dr/dt = -7ti - 5tj - 5tk Initial condition : r(0) = 5i+j+2k

View AnswerFind the maximum rate of change of f at the given point and the direction in which it occurs. f(x, y, z) = (2x + y)/z, (3, 6, -1) Find direction of maximum rate of change (in unit vector)

View AnswerA lamina occupies the region inside the circle x^2 + y^2 = 10y but outside the circle x^2 + y^2 = 25. Find the center of mass if the density at any point is inversely proportional to its distance from

View AnswerFind the two x-intercepts of the function f(x) = -8x\sqrt{x+1} and find the optimum value. Also find the point/s at which the optimum exists.

View AnswerFind all points on the graph of the function f(x) = 2 cos x + cos2 xat which the tangent line is horizontal. (Use n as your arbitrary integer.)

View AnswerConsider the following equation. f(x,y) = y4/x, P(1,2). u=1/3(2i+swrt5j) a) Find the gradient of f. b) Evaluate the gradient at the point P. c) Find the rate of change of f at P in the direction of th

View AnswerA piece of wire 18 m long is cut into two pieces. One piece is bent into a square and the other is bent into an equilateral triangle. (a) How much wire should be used for the square in order to maximi

View AnswerTrevor bought 8 gallons of paint to paint his house. He used all but 1 quart. How many quarts of

View AnswerFind the volume of the given solid. Under the plane 7x + 2y - z = 0 and above the region enclosed by the parabolas y = x^2 and x = y^2.

View AnswerFind the second Taylor polynomial T2(x) for f(x)=ex2based atb=0. For values of x near 0,T2(x) = ex2. Approximate the value of 1ex20dxby computing 1T2(x)0dx.

View AnswerFind the solution r(t) of the differential equation with the given initial condition: r'(t) =

View AnswerFor the following function, determine the intervals on which the following functions are increasing and decreasing and classify each of the critical points as a relative minimum, a relative maximum, o

View AnswerUse the gradient to find the directional derivative of the function at P in the direction of Q. f(x,y) = 3x^2 - y^2 + 4, P(5, 3), Q(2, 2).

View AnswerA curve C is given by a vector function r(t), 4 t 6, with unit tangent T(t), unit normal N(t), and unit binormal B(t). Indicate whether the following line intergrals are positive, negative or zero.

View AnswerConsider the helix r(t)=(cos(-2t),sin(-2t),4t). Compute, at t=pi/6: the unit tangent vector T

View AnswerGravel is being dumped from a conveyor belt at a rate of 40 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is

View AnswerFind the exact length of the curve y = 2 + 2x^(3/2) for 0 less than or equal to x less than or equal to 1.

View AnswerA) Solve the given differential equation by undetermined coefficients y'' + 4y = 7 sin 2x B) Solve the given intial-value problem 5y'' + y' = -4x, y(0) = 0, y'(0) = -5y

View AnswerA warehouse selling cement has to decide how often and in what quantities to reorder. It is cheaper, on average, to place large orders, because this reduces the ordering cost per unit. On the other ha

View AnswerIf a ball is thrown vertically upward with a velocity of 80 ft/s, then its height after t seconds is S = 80t - 16t^2. What is the velocity of the ball when it is 96 ft above the ground on its way up?

View Answer