Find the area of the parallelogram spanned by the vectors <1, 0, -1> and <-2, 2, 0>. I have troubles with dot products.
Questions by Sun K. from Los Angeles, CA
A constant force F=<-1, 2, -5> acts on a particle as it moves from A(2, 4, 1) to B(-2, 1, 3). Find the work done by the force. I have trouble for finding the dot products.
Each student in a cooking class of 50 students is assigned to create a dessert, an appetizer, or both. The total number of students creating an appetizer is seven more than the number of students...
Right triangle A has base b, height h, and area x. Rectangle B has length 2b and width 2h. What is the area of rectangle B in terms of x? a) 2x b) 4x c) 6x d) 7x e) 8x
Evaluate (d^100/(dx^95 dy^2 dx^3))(ye^x(x)+cos(x)).
Find a potential function for the vector field f(x, y)=2x/y i+(1-x^2)/y^2 j.
A spring gun at ground level fires a golf ball at an angle of 45 degrees. The ball lands 10 m away. a) What was the ball's initial speed? b) For the same initial speed, find the two firing...
Find an equation of the plane that passes through the point P(-1, 2, 1) and contains the line of intersection of the planes x+y-z=2 and 2x-y+3z=1.
A satellite circles planet X in an orbit having a radius of 10^8 m. If the radius of planet X is 10^7 m and the free fall acceleration on the surface of X is 20 m/s^2, what's the orbital period of...
Find the arc length of the curve defined by r(t)=(t, sqrt(6)/2*t^2, t^3), -1<=t<=1. r'(t)=<1, sqrt(6)t, 3t^2> sqrt(1+6t^2+9t^4) But how do I simplify this?
A particle starts at the origin with initial velocity i+j-k. Its acceleration is a(t)=ti+j+tk. Find its position at t=1.
Find the speed of the particle with position function r(t)=e^3t i+e^-3t j+te^3t k when t=0.
Find the normal vector to the plane 3x+2y+6z=6.
Find dz/dy at (1, ln 2, ln 3) if z(x, y) is defined by the equation xe^y+ye^z+2lnx-2-3ln2=0.
Suppose the temperature in degrees Celsius at a point (x, y) is described by a function T(x, y) satisfying Tx(2, 7)=4, Ty(2, 7)=2. The position of a crawling ant after t seconds is given by x(t)=sqrt(1+t),...
Find the slope of the tangent line at theta=pi/2 for the curve in the xy plane with the polar equation r=theta.
Find the perpendicular distance between the parallel planes z+1=x+2y and 3x+6y-3z=4.
Find the area of the triangle with vertices A(2, 1), B(5, 3), and C(6, 4).
Find the mass of the plate with density p(x, y)=ky, where k is a positive constant and R is the region bounded by y=x^2 and x=y^2.
Evaluate the double integral from 0 to 1 and x^2 to 1 of (x^3)(sin(y^3)) dy dx by reversing the order of integration. I know how to solve the integral, I just don't know how to set up the new...
Find the maximal value of f(x, y)=3y+4x on the circle x^2+y^2=1.
z=(e^x)(cos(y)). Find (d^2*z)/(dx^2)+(d^2*z)/(dy^2).
Let z=(e^x)(cos(xy)). Find the z-intercept of the equation of the tangent plane to the surface at (1, pi/2, 0).
Find a parametric representation of the surface z=3x+4y.
Let z=(x^2+y^2)^(3/2). What is (d^2*z)/(dx*dy) at (sqrt(2), sqrt(2))? dz/dy=3y(x^2+y^2)^(1/2) d/dx=3x(x^2+y^2)^(1/2) What should I do next?
Find lim as (x, y)--->(0, 0) for (8x^2*y^2)/(x^4+y^4). lim as (0, y)--->(0, 0) =0/y^4=0 lim as (x, 0)--->(0, 0) =0/x^4=0 The limit is 0, right? But doesn't exist is the answer for...
Let aT be the tangential component of the acceleration vector of r(t)=
Let a=<2, 3, 1> and b=<-2, 5, -3>. Find the absolute value of a+b.
Find the curvature of the curve of intersection of the cylinder x^2+y^2=16 and the plane x+z=5 at (4, 0, 1).
Find the Cartesian equation for the curve described by the polar equation r=1/(1-sin(theta)).
Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x).
Find the domain of r(t)=
Find the curvature of the curve r(t)=(t+1)i+2(t^2-1)j+(t-2)k. r'(t)=i+4tj+k sqrt(1^2+(4t)^2+1^2)=sqrt(2+16t^2) r''(t)=4j r'(t)xr''(t)=? Please show your work step by step from...
If f(x, y, z)=(z^2+x^2-y)/(2x^2+y), what's f(x, -x^2, x^2)? Answer: x^2+2 I have no idea to do this.
What's the arc length for the curve defined by r(t)=3sin(t)i+3cos(t)j+4tk for 0<=t<=10? (Answer: 50) r'(t)=<3cost, -3sint, 4>
Determine the minimum surface area of a closed rectangular box with volume 8 ft^3.
Find the shortest distance between the planes 2x+3y-z=2 and 2x+3y-z=4.
If f(x, y, z)=sin(3x-yz), where x=e^(t-1), y=t^3, z=t-2, what's df/dt(1)? df/dt=(df/dx)(dx/dt)+(df/dy)(dy/dt)+(df/dz)(dz/dt) (3cos(3x-yz))(e^(t-1))+(cos(3x-z))(3t^2) +(cos(3x-y))(1) What...
If f(x, y, z)=sin(3x-yz), where x=e^(t-1), y=t^3, z=t-2, what's df/dt(1)?
What's the length of the curve r(t)=<2 cos t, 2 sin t, sqrt(5)> from 0<=t<=2pi?
True or False? The integral from 0 to 2pi, from 0 to 4, from r to 4 dz dr dtheta represents the volume enclosed by the cone z=sqrt(x^2+y^2) and the plane z=4.
Use Stokes' Theorem to evaluate the surface integral where F=
Use Stokes' Theorem to evaluate the line integral where F=<(x^2)(e^x)-y, sqrt(y^2+1), z^3> and where C is the boundary of the portion of z=4-x^2-y^2 above the xy-plane. The curl is <0, 0, 1> but...
Show that lim (x, y)-->(0, 0) (3x^2(y))/(x^4+y^2) does not exist.
Find the directional derivative of f(x, y)=x^2*y+4y^2 at (2, 1) for u=<1/2, sqrt(3)/2>. Answer: 2+6sqrt(3) <2xy, 8y> at (2, 1)=<4, 8> <4, 8>*<1/2, sqrt(3)/2>=2+4sqrt(3) This doesn't...
Find the directional derivative of f(x, y)=x^2*y+4y^2 at (2, 1) for u=<1/2, sqrt(3)/2>. Answer: 2+6sqrt(3) The gradient is <2xy, 8y> at (2, 1)=<4, 8> And <4, 8>*<1/2, sqrt(3)/2>=2+4sqrt(3) This...
Find the gradient of f(x, y)=2e^4x/y-2x at (2, -1). Answer: <-8e^-8-2, -16e^-8> Step by step, I really don't know this completely.
Find the gradient of f(x, y)=xe^xy^2+cosy^2.
Use implicit differentiation to find dz/dx and dz/dy for 3x^2*z+2z^3-3yz=0. Answer: dz/dx=-6xz/(3x^2+6z^2-3y) dz/dy=3z/(3x^2+6z^2-3y) This is Calculus 3 topic. Show your work through steps...