Questions from General Calculus

Q: Find the local maximum and minimum values and saddle point(s

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the impor...

Q: Find an equation of the plane that passes through the point (

Find an equation of the plane that passes through the point (1, 2, 3) and cuts off the smallest volume in the first octant.

Q: Show that the pyramids cut off from the first octant by any

Show that the pyramids cut off from the first octant by any tangent planes to the surface xyz − 1 at points in the first octant must all have the same volume.

Q: Find parametric equations for the tangent line to the curve of intersection

Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 4x2 + y2 + z2 = 9 at the point (-1, 1, 2).

Q: (a). The plane y + z = 3 intersects the

(a). The plane y + z = 3 intersects the cylinder x2 + y2 = 5 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1, 2, 1). (b). Graph the cylinder, the plane, a...

Q: (a). Two surfaces are called orthogonal at a point of

(a). Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F (x, y, z) = 0 and G (x, y, z) = 0 are orth...

Q: Suppose that the directional derivatives of f (x, y)

Suppose that the directional derivatives of f (x, y) are known at a given point in two nonparallel directions given by unit vectors u and v. Is it possible to find ∆f at this point? If so, how would y...

Q: Find the local maximum and minimum values and saddle point(s

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the impor...

Q: If f is homogeneous of degree n, show that fx

If f is homogeneous of degree n, show that fx (tx, ty) = tn-1 f (x, y)

Q: Equation 6 is a formula for the derivative dy/dx of

Equation 6 is a formula for the derivative dy/dx of a function defined implicitly by an equation F (x, y) = 0, provided that F is differentiable and Fy ± 0. Prove that if F has continuou...