University:
University of California, Berkeley
Course:
MATH 51 | Calculus I
Academic year:

2022
Views:

295

Pages:

1
Author:

CometConductor

A Function of Variables Definition: A f unction f o f t wo v ariables i s a r ule t hat a ssigns t o e ach o rdered p air o f 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) ∈ D }. xy−5 . Sketch the domain D of f . (Swok Sec 16.1 Ex 1) Exercise 1. Let f (x, y) = √ 2 2 y−x Exercise 2. Find the domain of the function g(x, y) = p 4 − x2 − y 2 . (Briggs Sec 12.2 Ex 1) Class Exercise 1. Find and sketch (#14-20 even) p the domain of the function. p √ √ (a) f (x, y) = xy (b) f (x, y) = x2 − y 2 (c) f (x, y) = y + 25 − x2 − y 2 (d) f (x, y) = arcsin(x2 + y 2 − 2) Definition: If f is a function of two variables with domain D, then the graph of f is the set of all points (x, y, z) in R2 such that z = f (x, y) and (x, y) is in D. Exercise 3. Find the domain and range of the following functions. Then sketch a graph. (Briggs Sec 12.2 Ex 2) p (a) f (x, y) = 2x + 3y − 12 (b) g(x, y) = x2 + y 2 (c) h(x, y) = 1 + x2 + y 2 Class Exercise 2. Sketch the graph of the function. (#24-30 even) (a) f (x, y) = 2 − x (b) f (x, y) = e−y p (c) f (x, y) = 1 + 2x2 + 2y 2 (d) f (x, y) = 4x2 + y 2 Definition: The level curves of a function f of two variables are the curves with equations f (x, y) = k, where k is a constant (in the range of f ). Exercise 4. Graph f (x, y) = 100 − x2 − y 2 and plot the level curves f (x, y) = 0, f (x, y) = 51, f (x, y) = 75 in the domain of f in the plane. (Hass Sec 14.1 Ex 5) Exercise 5. If f (x, y) = y 2 − x2 , sketch some level curves of f . (Swok Sec 16.1 Ex 4) Exercise 6. Find and sketch the level curves of the following surfaces: (a) f (x, y) = y − x2 − 1 2 2 and (b) f (x, y) = e−x −y . (Briggs Sec 12.2 Ex 3) Class Exercise 3. Draw a contour map of the function showing several level curves. (#44-50 even) (a) f (x, y) = x3 − y (b) f (x, y) = ln(x2 + 4y 2 ) y (c) f (x, y) = y sec x (d) f (x, y) = x2 +y 2 Definition: A f u nction o f t h ree v a riables, f , i s a r u le t h at a s signs t o e a ch o r dered triple (x, y, z) in a domain D ⊂ R3 a unique real number denoted by f (x, y, z). Class Exercise 4. Find and sketch the domain of the function: f (x, y, z) = ln(16−4x2 −4y2 −z2)