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MATH23 MULTIVARIABLE CALCULUS

QUADRIC SURFACES. MATH23 MULTIVARIABLE CALCULUS. GENERAL OBJECTIVE. Determine functions of several variables. Identify the domain of several variables Discuss quadric surfaces, its equations and graphs. Differentiate various quadrics, based on equation, and graph.

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MATH23 MULTIVARIABLE CALCULUS

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  1. QUADRIC SURFACES MATH23 MULTIVARIABLE CALCULUS

  2. GENERAL OBJECTIVE • Determine functions of several variables. • Identify the domain of several variables • Discuss quadric surfaces, its equations and graphs. • Differentiate various quadrics, based on equation, and graph At the end of the lesson the students are expected to:

  3. Functions of Several Variables • A function f of two variables is a rule that assigns to each ordered pair of real numbers (x,y) in a set D a unique real number denoted by f(x,y). The set D is the domain of f and its range is the set of values that f takes on, that is, {f(x,y)|(x,y) an element of D}.

  4. Example 1. Find the domain of the following functions and evaluate f(3, 2). Domain of Functions • Determine the Domain of the following functions

  5. Quadric Surfaces • Is the graph traced by any quadratic or second degree equation in three variables x, y, z. • The general equation of a quadric surface is: Ax2 +Bx2+Cy2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0

  6. Cylinders • A cylinder is a surface that consists of all lines (called rulings) that are parallel to a given line and pass though a given plane curve. • A curve of two variables traced in three-dimensions • Examples of which are: • Circular Cylinder • Elliptical Cylinder • Parabolic Cylinder • Hyperbolic Cylinder

  7. z (-2, 0, 0) CYLINDERS and SPHERE: 1. x2 + y2 = 4 (0, 2, 0) y (0,-2, 0) (2, 0, 0) CIRCULAR CYLINDER x

  8. z (-1, 0, 0) 2. 4x2 + y2 = 4 (0, 2, 0) y (0,-2, 0) (1, 0, 0) ELLIPTICAL CYLINDER x

  9. z 3. x2 = y y x

  10. z 4. y2 = x y x

  11. z 5. z2 = y y x

  12. z 6. z2 = y – 1 y V(0, 1, 0) x

  13. SPHERE: z y (x – h)2 + (y – k)2 + (z – i)2 = r2 Ax2 + Ay2 + Az2 + Gx + Hy + Iz = J r = 0 (point) r = - (no locus) r = + (sphere) x

  14. EXAMPLE: Describe the locus of x2 + y2 + z2 + 2x – 4y – 8z + 5 = 0. Sketch the graph. SOLUTION: x2 + 2x + 1 + y2 – 4y + 4 + z2 – 8z + 16 = –5 + 1 + 4 + 16 (x + 1)2 + (y – 2)2 + (z – 4)2 = 16 C(–1, 2, 4) and r = 4

  15. z’ z (-1, 2, 8) (-5, 2, 4) (-1, 6, 4) (-1,- 2, 4) y y’ C(-1, 2, 4) (3, 2, 4) x (-1, 2, 0) x’

  16. QUADRIC SURFACES Common Types of Quadric Surfaces: Ellipsoid Hyperboloid of One Sheet Hyperboloid of Two Sheets Elliptic Paraboloid Hyperbolic Paraboloid Elliptic Cone

  17. Ellipsoid

  18. ELLIPSOID QUADRIC SURFACES y x

  19. Hyperboloid of One Sheet

  20. HYPERBOLOID OF ONE SHEET QUADRIC SURFACES

  21. Hyperboloid of Two Sheets

  22. HYPERBOLOID OF TWO SHEETS QUADRIC SURFACES

  23. Elliptic Paraboloid

  24. ELLIPTIC PARABOLOID QUADRIC SURFACES

  25. Hyperbolic Paraboloid

  26. HYPERBOLIC PARABOLOID z’ QUADRIC SURFACES x’ y’

  27. Elliptic Cone

  28. ELLIPTIC CONE QUADRIC SURFACES

  29. EXAMPLES: Sketch the quadric surface. 36x2+9y2+4z2=36 Solution:

  30. z (0,0,3) (-1,0,0) (0,-2,0) (0,2,0) (1,0,0) y (0,0,-3) x

  31. 2. 16x2+36y2-9z2=144

  32. (-4.2,0,4) z (0,-2.8,4) (0,2.8,4) y’ z=4 (-3,0,0) (4.2,0,4) (0,2,0) (0,-2,0) (-4.2,0,-4) x’ y (3,0,0) y” z=-4 (0,-2.8,-4) (0,2.8,-4) x (4.2,0,-4) x”

  33. 3. 4z2-4x2-y2=4

  34. z (-2.8,0,3) y’ (0,5.7,3) z=3 (0,-5.7,3) (0,0,1) (2.8,0,3) (-2.8,0,-3) x’ (0,0,-1) y z=-3 y” (0,-5.7,-3) (0,5.7,-3) (2.8,0,-3) x x”

  35. Example

  36. Example

  37. Example

  38. Example

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