Question

Math 32-_ Multivariable Calculus HW 3 (1) Consider the two straight lines L1 : (2-t, 3 + 2t,-t) and L2 : <t,-2 + t, 7-20 a) Verify that L1 and L2 intersect, and find their point of intersection. (b) Find the equation of the plane containing L1 and L2 (2) Consider the set of all points (a, y, z) satisfying the equation 2-y2+220. Find their intersection 0 and 2-0. Use that information to sketch a with the planes y =-3,-2,-1,0, 1, 2, 3 and x graph of these points. use them to sketch the surface of each of the following functions. (a) f(x, y)y2 (3) Graph appropriate level curves (that is, horizontal cross sections) and vertical cross sections, and (c) f(x,y) = eVz〒 (d) f(e, y) sin(Vy) (b) f(x, y) = ln(Vgr+P) (4) In class, we will encounter the notion of a partial derivative of a nice function f(, y) with respect to either x or y. The partial derivative of f with respect to x is denoted either fz or af/ax Algebraically, to find fr, we pretend y is constant and differentiate the usual way; similarly, to find fy, we pretend is constant. Geometrically, fr (o, yo) equals the slope of the tangent line to the curve that we get by intersecting the graphf(, y) with the plane yyo at the point (xo, yo, f(xo, yo)) (said differently, it's the slope in the r-axis direction); and fy(zo, yo) can be interpreted in a similar way, as the slope in the y-axis direction. Let us apply this notion to tangent planes. Consider the graph of a nice function f(x, y). Using the slopes f (o, yo) and fy(xo, yo), find two vectors that point in a direction tangent to the graph z f(x, y) at the point (zo, yo, f (xo, yo)), with one parallel to the az-plane and the other parallel to the yz-plane. (Hint: What can you say about the components of the vector (a, b, c) if you know that vector is parallel to the xz-plane? What is that vector's slope?) Use this information to find the equation of the plane tangent to the graph of f(x,) at the point (o, yo), explaining your reasoning. (5) (a) Suppose f(z, y) e. Find J(, ) and J(x,y) (b) Does there exist a nice function f (z, y) for which fa(, y) 32 +2xy3 7 and fy(x,)-

Answer 1

Given L_1: (2 -t, 3 + 2t, -y), L_2 = (t, -2 + t, 7 - 2t) L_1: x - 2/ -1 = y - 3 /2 = z/-1 = -z L_2: x/1 = y + 2/ 1 = z - 7/ - 2 = t a general point on L_1 is (2 -k_1, 3 + 2k_1, -k_1) and on L_2 is (k_2, -2 + K_2, 7-2k_2) if L_1, L_2 are intersect then for some k_1, k_2 2 -k_1 = k_2 (1) 3 + 2k_1 = -2 +k_2 (2) -k_1 = 7 -2k_2 (3) let equation (1) & (2) 2 -k_1 = k_2 3 + 2k_1 = -2 +k_2 putting the value of k_2 from (1) i (2) 3 + 2k_1 = -2 + 2-k_1 rightarrow 3 + 2k_1 = -k_1 rightarrow 3 = -3k_1 rightarrow kl_1 = -1 putting the value of k_1 in (1) k_2 = 2 - (-1) = 3 putting the value of k_1, k_2 in equation (3)