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)...
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...
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...
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...
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...
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...
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...
A vector A has a magnitude of 4 units and points in the -y-direction, while a vector has double the magnitude of A and points in the +x-direction. What are the magnitude and direction of the following vectors? Give the directions of each as an angle measured counterclockwise from the +x-direction. (a) AB magnitude units) direction * (counterclockwise from the +x-axis) (b) A magnitude direction units) (counterclockwise from the *x-axis) (c) B-A magnitude unit(s) (counterclockwise from the x-axis) direction