Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3...
Part B Please 2. Consider a point (xo, y0, 2o) in space and a vector (a, b, c). We can use this vector to define a plane as the set of all points (x,y, z) such that the vector (x - xo, y - yo, z - zo) connecting (z, y,z) with (xo, Yo, 20) is orthogonal to (a, b, c) (the normal vector to the plane). (a) Use a dot product to write the equation of a plane through...
Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r)) = K for some constant K is orthogonal to the tangent vector T() of each curve C described by the vector function on the surface passing through Po (xo,yo, zo). Hint, remember that the tangent vector T(o) R'(), so prove that Vfo R'O) 0 Extra Credit Prove that the V fo at each point Po (xo. yo, zo) on the surface f(x(t),y(t),z(r))...
Find the plane determined by the intersecting lines. L1 x= -1 +41 y=2+t z= 1 - 4 L2 x = 1 - 4s y= 1 + 25 z=2-2s The equation of the plane is (Type an equation.)
a) Show that the equation 23 : 1 f(z, Y, z) := +y+ defines a smooth surface S. b) Show that for any (r, y, z) E S, the gradient vector (fz(x, y, z), fy(, y, z), f:(x, y, z)) of f is a normal vector to S. (Hint: let a = x(t), y = y(t), z = z(t) be a curve in the surface passing through a point (o, Yo, 2o) in S, where ro = r(0), yo: y(0),...
Determine if lines L1 and L2 are parallel, oblique or cut. If they intersect, determine the point of intersection. L1 = x / 1 = y-1 / -1 = z-2/3 L2 = x-2/2 = y-3 / -2 = x / 7
x =-y+2 = -z+2 The symmetric equations for 2 lines in 3-D space are given as: 1. L,: x-2 = -y+1 = z+1 a) Show that lines L1 and L2 are skew lines. b) Find the distance between these 2 lines x =1-t y=-3+2t passes through the plane x+ y+z-4=0 2. The line Determine the position of the penetration point. a. Find the angle that the line forms with the plane normal vector n. This angle is also known as...
Consider the real valued function of two variablesz-f(x,y), which satisfies Ar2 +4y2+ 224 (a) Describe all points (ro. yo. f(ro. yo)), such that the tangent plane to f(x, y) at (o.yof(ro. yo)) the point (2,0,0). Include scription. (b) Describe all points (roo,f(o, yo)), such that the tangent plane to f(x, y) at (zo. yo. f(o. yo) passes through the point (2,0,27), where >0 is an arbitrary constant. With the aid of a computer or by hand, provide a sketch of...
Example A.3 Surface normal vector. Let S be a surface that is represented by f(x, y, z) -c, where f is defined and differentiable in a space. Then, let C be a curve on S through a point P-Go, yo,Zo) on S, where C is represented by rt)[x(t), y(t), z(t)] with r(to) -[xo. Vo, zol. Since C lies on S, r(t) must satisfy f(x, y. z)-c, or f(x(t), y(t), z(t))-c. Show that vf is orthogonal to any tangent vector r'(t)...
(1 point) Estimate f(-3.01,-1.97) given that f(-3,-2) = 5, fx(-3,-2) = -2 and fy(-3, -2) = 3. f(-3.01,-1.97) ~ (1 point) Find the linearization of the function z = x,y at the point (-5, 49). L(x, y) = (1 point) Find the equation of the tangent plane to z = ex + y + y4 + 6 at the point (0,3,91). z =
Consider the function f(x, y) = x^3 − 2xy + y^2 + 5. (a) Find the equation for the tangent plane to the graph of z = f(x, y) at the point (2, 3, f(2, 3)). (b) Calculate an estimate for the value f(2.1, 2.9) using the standard linear approximation of f at (2, 3). (c) Find the normal line to the zero level surface of F(x, y, z) = f(x, y) − z at the point (2, 3, f(2,...