33. Figure 14.13 shows the saddle-shaped surface z = S(x, y). (a) What is the sign of f,(0, 5)? (b) What is the sign of f,(0, 5)? (0, 5, 3)

The curve in the plane defined by the equation (*) (x^2 + xy + y^2 = 7) is a rotated ellipse, shown in the figure. (c) The curve has two horizontal tangent lines. Find their equations.(d) Why can we see from the figure that the curve is not the graph of a function y of x?Show directly from (∗) that y is not a function of x.

Describe the traces of the curve x2 + 9z2 = 9 in the vertical planes (those parallel to the xz-plane).

Point P(x,y) moves with a distance from (0,1) one-half of its distance from line y = 4. What is the equation of the locus?

If you will draw the graph of y=x^2 (x-1), how will the graph behave at the x-axis?   A) The graph crosses both (0,0) and (1,0). B) The graph crosses (0,0) and is tangent to the x-axis at (1,0). C) The graph crosses (1,0) and is tangent to the x-axis at (0,0). D) The graph is tangent to the x-axis at both (0,0) and (1,0).

In two-dimensional Euclidean space, consider a curve defined by the equation x^2 + y^2 - 6x + 4y - 12 = 0. Determine the type of curve and its center.

V (sec-"(-3z?))³ If y= Vsec-1 (-3x²), then dy - dz 21-3z지·v9z¢_1 True False

A circular disk of radius 5 meters centered at the origin has been unevenly heated. The temperature at any point on the disk is given by T(x, y) = 50e-2x²-y² degrees Celsius, where x and y are measured in meters. (a) Where on the disk is the temperature the greatest? What is the temperature at that point? (b) Find the direction of greatest temperature increase at the point (1, 1). (c) What is the rate of temperature change if you move away from the point (1,1) in the direction of the point (2,0)?

Calculate y" and y" y = x² (8 x + 3)

I really need help figuring out how to do this problem, using the cheat sheet formulas needed for the equation