Calculus 1

Content Covered:

- Rectangular Coordinates in 3-Space; Spheres; Cylindrical Surfaces

- Quadric Surfaces

- Partial Derivatives

- The Chain Rule

- Maxima and Minima of Functions of Two Variables

Directions:Kindly answer each of the following problems below. And also, please indicate the complete solution. I promise that I will give you a good rating after this. Thank you so much for helping me out.

1)

Find the center and radius of the sphere that has (1, 2, 8) and (3, 4, 24) as endpoints of a diameter. Hint: The midpoint of the line segment joining (3:1,y1,2:1) and (562,312, 2:2) 1 1 1 is (E (391 + 932): 5(311 + '92): 5031+ 232))- NOTE: Enter the erect answer. Radius: [ ] rm) = 24m + 4y Find the slope of the surface 3 = f (3:, y) in the ydirection at the point (16,4). NOTE: Enter the exact answer. fy(16a4) = Describe the set of all points in 3-space whose coordinates satisfy the inequality y~ + z~ + 8y - 20z > 5. Select all that apply. O All points inside the sphere x2 + () + 4) + (z - 10)2 = 121. O All points on the circular cylinder () + 4) + (z - 10)2 = 121. O All points inside the circular cylinder (y + 4) + (z - 10)2 = 121. O All points outside the circular cylinder () + 4) + (z - 10)2 = 121. O All points outside the sphere x2 + () + 4) + (z - 10) = 121.Describe the surface with equation x2+y2+z2+12x+6y+2235:0 The surface is a sphere with center ( n , n , n J and radius Identify the quadric surface z = 49 - x2 as an ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets, elliptic cone, elliptic paraboloid, or hyperbolic paraboloid. State the values of a, b, and c (if necessary; otherwise type 1 in the appropriate blank). with a = b C Ellipsoid Hyperboloid of one sheet Hyperbolic paraboloid Elliptic cone Elliptic paraboloid Hyperboloid of two sheets(c) Find the foci of the hyperbola in part (a). 0 (16,0, 25) O (i, 25] o (0, i5, 25) o (is, 0, 25) O (0, i 5'5, 25) (d) Describe the orientation of the focal axis of the hyperbola in part (a) relative to the coordinate axes. The focal axis is parallel tothe v xaxis y'axis ' z-axis View Policies Current Attempt in Progress This exercise refers to the hyperbolic paraboloid z = y2 - x2. (a) Find an equation of the hyperbolic trace in the plane z = -25. Oy = z - 1 O zz = - y+25 Ox + y = 25 O y 1 25 25 O x 2 25 25 (b) Find the vertices of the hyperbola in part (a). O (0, + 5, 25) O (0, + 5\\ 2, 25) O (+5\\2, 0, - 25) O (+5, 0, - 25) O (+5, 0, 25)Identify the surface by completing the squares. x2 + 16y2 - z2 - 6x + 32y + 8z = 0 O An ellipsoid. O An elliptic paraboloid. O An elliptic cone. O A hyperboloid of two sheets. O A hyperbolic paraboloid. O A hyperboloid of one sheet.az az Find and ay 2 = x In (1+xy Oz Ox Oz ayFind the partial derivative. 9p f (p, q) = e of aqFind fr(x, y) and fy(x, y). f(x, y) = (y' tan(5x)) fx(x, y) = fy ( x, y) =