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Calculus III Hughs-Hallett

Calculus III Hughs-Hallett. Math 232 A,B Br. Joel Baumeyer. Multivariable Calculus. A function in Three Dimensional Space (3-D), z = f(x,y):

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Calculus III Hughs-Hallett

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  1. Calculus III Hughs-Hallett Math 232 A,B Br. Joel Baumeyer

  2. Multivariable Calculus • A function in Three Dimensional Space (3-D), z = f(x,y): • is a function with two independent variables; it is still a rule that assigns for each of the two independent variables, x and y, one and only one value to the dependent variable z. • e.g. h(x,t) = 5 + cos(0.5x - t) pg 4

  3. 3-D Graphing Basics • x coordinate measures the distance from the yz-plane whose name is: x = 0 • y coordinate measures the distance from the xz-plane whose name is: y = 0 • z coordinate measures the distance from the xy-plane whose name is: z = 0 • distance:

  4. Graphs of Functions in 2 Variables • The graph of a function of two variables, f, is the set of all points (x,y,z) such that z = f(x,y). • The domain of such a function is is a subset of points in the real Euclidean plane. In general, the graph of a function of two variables is a surface in 3-space. • In particular, the graph of a linear functionin 3-space is a plane: ax + by + cz = d.

  5. Section of a Graph of a Function in 3-D • For a function f(x,y), the function we get by holding x fixed and letting y vary is called a section of f with x fixed. The graph of the section of f(x,y) with x = c is the curve, or cross-section, we get by intersecting the graph of f with the plane x = c. We define a section of f with y fixed similarly.

  6. Contour Lines (or Level Curves) • Contour lines, or level curves, are obtained from a surface by slicing it by horizontal planes. • If z = f(x,y) then for some value c, where z = c, c = f(x,y) is a level curve for the function.

  7. Equation of a Plane in 3-D • If a plane has slope m in the x direction, slope n in the y direction, and passes through the point: then its equation is: • if we write: then: f(x,y) = c + mx + ny

  8. Level Surfaces • A function of three variable f(x,y,z), is represented by a family of surfaces of the form: . f(x,y,z) = c, each of which is called a level surface.

  9. 3-D « 4-D • A single surface representing a two-variable function: z = f(x,y) can always be thought of a one member of the family of level surfaces representing a three variable function: G(x,y,z) = f(x,y) - z. • The graph of z = f(x,y) is the level surface of G = 0.

  10. Limits and Continuity • The function f as a limit at the point (a,b), written: if the difference |f(x,y) - L| is as small as we wish whenever the distance from the pint (x,y) to the point (a,b) is sufficiently small, but not zero. • A function f is continuous at a point if • A function is continuous if it is continuous at each point of its domain

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