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Unformatted text preview: MATH237: EXAMAID SOS
NIALL W. MACGILLIVRAY (EDITOR); VINCENT CHAN (WRITER) 1. October 31st, 2010 ScalarFunctions(1.1 − 1.2) DEFINITION 1.1. Suppose A and B are sets. A function f is a rule that determines how a subset of A is associated with a subset of B , such that each element in A for which f is deﬁned is sent to exactly one element in B . This subset of A for which f is deﬁned is called the domain of f and denoted by D(f ), while the subset of B which is attained by f is called the range of f and denoted by R(f ). We typically denote f by f : A → B . DEFINITION 1.2. f : R → R is a scalar function, i.e. when the domain is a subset of R and the range is a subset of R. EXAMPLE 1. Deﬁne f : R2 → R by f (x, y ) = x2 + 2y 2 − 2x. Find the domain and range of f. SOLUTION. Clearly D(f ) = R2 . For the range, notice x2 +y 2 −2x ≥ x2 − 2x, which has a minimum at x = 1 (using what we know from 1dimension). Then f (x, y ) ≥ f (−1, 0) = −1. √ For any value z ∈ [0, ∞[, we can take x = 1 + 1 + z and y = 0. Indeed, this is a possible value for x we can solve for using z = x2 − 2x and the quadratic equation. Thus, R(f ) = [−1, ∞[. EXAMPLE 2. Deﬁne f : R2 → R by f (x, y ) =
x + y  x−y  . Find the domain and range of f . SOLUTION. We only need x = y  for f to be deﬁned. The domain is then R2 \ {(x, y ) : x = y }. As for the range, notice we can obtain 1 and 1 by setting y = 0 and x = 0 respectively. Now, suppose for z ∈ R, we have f (x, y ) = z . Then x + y  = z x− z y , so that: (1 − z )x = −(1 + z )y  We have already dealt with the case z = ±1. Otherwise, y  = − 1−z x. Since x, y  ≥ 0, 1+z 1−z we have − 1+z ≥ 0. Then either both 1 − z and −1 − z are positive, or both are negative. That is, we have z < −1 or z > 1. 1 In either case, we can set . and WATERLOO SOS E XAM AID: MATH237 to obtain , so the range is (including the case ) E XERCISE 1. Find the domain and range of the following functions. (a) (b) (c) It is difﬁcult to sketch scalar functions from their deﬁnition alone. To help us, we look at level curves and crosssections. D EFINITION 1.3. For , we deﬁne the level curves of to be the curves given by . . where is a constant (in the range of ). Sketching level curves is an exercise in solving for equations in two variables. The course notes cover three excellent examples, which are fairly standard. E XAMPLE 3. Deﬁne a function the surface . S OLUTION. From example 1, . Recall that the equation by . Sketch the level curves and use them to sketch and . For values of , we examine determines an ellipse with major axis length , minor axis length , shifted units right and units up. This formula is a more general form of what we have: since there are no terms, we will have and since there is a term, we will want (to get ). Then if , Thus, describes an ellipse with major axis length , minor axis length , shifted 1 unit right. If , we get just the single point , and this is the exceptional level curve. We can thus generate the image of , and we get a surface called an elliptical paraboloid. 2 WATERLOO SOS E XAM AID: MATH237 E XERCISE 2. Sketch the level curves and use them to sketch the surface . D EFINITION 1.4. For , we deﬁne the crosssections of or to be the curves given by where are constants. They correspond to the intersection of the surface with the vertical planes respectively. , E XAMPLE 4. Deﬁne a function by yield , . Sketch the crosssections. while the crosssections yield S OLUTION. The crosssections . Sketching these for E...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Scalar, Sets

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