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.

Curves (x- 1) (y- 2) =5 and (x - 1)^2 +(y + 2)^2= r^2, intersect at four points A, B, C and D. If centroid of tri(ABC) lies on line y = 3x- 4, then find the locus of point D.

draw the line which start point (1,1,0) and end point (3,4,0)by using parametric equation.

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

xample 3: Sketch the graph of the curve y-fx) =x-1)

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?

1 EX-6 – Sketch the curve : y=_(x³- 6 x²+9x+6). 6. 3

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)?

Let the curve C be the mirror image of the parabola y" the line x+y + 4 = 0 If A and B are the points of intersection of C with the line y3D -5, then the distance between A and B is = 4z with respect to