Stoc Lecture M/W... 4. (6 points) Recall that while the graph of y = z crosses...
0 intersect only at (0,0) g(r)at z arctan(3z) Show that the graph y f(x) and its tangent line y po Consider the ftunction f(x) Intermediate steps: 1) The lIne tangent to y f(x)atz -0isy g(x) where g(r) 9(a)- 2Let H(x) f(x) - 9(x) The derivative ot H (x)s H'(z) = which is zero only when x = Rolle's theorem to H (x) on the interval [ri, 0]. Get a contradiction. 4) Now assume that we have zp O where f(2)-9(T2)...
e 09, 201 (6) 2 points An equation for the level curve of f(z, y) = In(z+y) that passes through the point (0, e2) is A. z + y = e2 B. I+y e C. z+y 3. D. None of the above (7) 2 points The gradient of f(z,y, z) = ep at the point (-1,-1,2) is A. (2e2,e2,2e2). B. (-e,-e,2e2). C. (-2e2,-2e2, e) D. (-2e2,-e,-e) (8) 2 points Let f be a function defined and continuous, with continuous first...
Algebraic Geometry Robin Hartshorne: Blowing Up Curve Singularities Part (d): 5.6. Blowing Up Cure Singularities. (a) Let Y be the cusp or node of (Ex. 5.1). Show that the curve 1. obtained b;y blowing up Y at O = (0.0) is nonsingular (cf. (4.9.1) and (Ex. 4.10) . (b) We define a node (also called ordinary douhle poinı) to be a double point (i.e., a point of multiplicity 2) of a plane curve with distinct tangent directions (Ex. 5.3). If...
Algebraic Geometry Robin Hartshorne: Blowing Up Curve Singularities Part (b): 5.6. Blowing Up Curve Singularities. (a) Let Y be the cusp or node of (Ex. 5.1). Show that the curve 1. obtained b;y blowing up Yat O (0.0) is nonsingular (cf. (4.9.1) and (Ex. 4.10)) (b) We define a node (also called ordinary double poini) to be a double point (i.e., a point of multiplicity 2) of a plane curve with distinct tangent directions (Ex. 5.3. If P s a...
2. Use Definition to find the equation of the tangent line to the graph of the equation y- 1/2 at -2 3. Find the points on the graph of y2-/2 at which the tangent line is parallel to the line y - 3. 4. Sketch the graph of a continuous function f that satisfies all of the stated conditions. f(0) 2, f(-2)- (2)-0, f(-2) f(O)-f'(2)-0 f"(z) > o if-2<zco, f,(z) < 0 if <-2 or x > 0; 2. Use...
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...
19. [-13 Points] DETAILS HARMATHAP12 9.3.008. MY NOTES In Example 6 in this section, we were given f(x) = 3x2 + 2x + 11 and found f'(x) = 6x + 2. (a) Find the instantaneous rate of change of f(x) at x = 6. (b) Find the slope of the tangent to the graph of y = f(x) at x = 6. (c) Find the point on the graph of y = f(x) at x = 6. (x, y) =...
UU. LIUC JUULIULIS. 1) Find the equation of the tangent plane to the graph z = 2x2 + 2xy + y2 + 1 at the point P(-1, -3, 18). 2) Find all critical value(s) and classify as maxima/minima/saddle points/none. F(x,y) = 2x + 4y - x2 - y2 - 3 3) Find the directional derivative of z = xy +x in the direction of v= <3,-4> at the point Q(1,4). Also find the direction of maximum increase at this point....
#10 and #12 8. Find all points (.y) where fCx.y) -3x2 + 7xy -4y2 + x + y has possible relative maximum or minimum values 9. Find all points (x,y, z) where f(x,y,z) 5+ 8x 4y+x2+y2 z2has possible relativema imun or minimum value 10. Both first partial derivatives of f(x.y)-x-4xyy are zero at the points (0 11. Find all points (x,y) where f(e.y) 2x2+3xy + 5y has possible relative maximum or minimum values. Then, use the 12. Use the second...
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,...