4.2 Critical Points and Extreme Values
... – Note that the domain of f is the set of all real numbers except -3. – The first derivative of f is given by f '(x) = [ 2x (x + 3) - (x 2 + 7 )(1) ] / (x + 3) 2 – Simplify to obtain f '(x) = [ x 2 + 6 x - 7 ] / (x + 3) 2 – Solving f '(x) = 0 ...
... – Note that the domain of f is the set of all real numbers except -3. – The first derivative of f is given by f '(x) = [ 2x (x + 3) - (x 2 + 7 )(1) ] / (x + 3) 2 – Simplify to obtain f '(x) = [ x 2 + 6 x - 7 ] / (x + 3) 2 – Solving f '(x) = 0 ...
is the input, which is a list. Then, you can test your curried function
... by which you separate the even numbers from the input list. c. Using the logic you have used in part b, write another curried function to create a tuple of two lists, such that the first list contains all the items from the input list satisfying a given predicate ...
... by which you separate the even numbers from the input list. c. Using the logic you have used in part b, write another curried function to create a tuple of two lists, such that the first list contains all the items from the input list satisfying a given predicate ...
Math 535 - General Topology Fall 2012 Homework 2 Solutions
... number N such that Un = R for all n ≥ N . Consider a sequence y with yn = 0 for all n ≥ N and yn ∈ Un for 1 ≤ n < N . Then we have y ∈ U ∩ R∞ . Because “large boxes” form a basis of the product topology, every open neighborhood of x intersects R∞ . Therefore R∞ is not closed. Remark. In fact, the ar ...
... number N such that Un = R for all n ≥ N . Consider a sequence y with yn = 0 for all n ≥ N and yn ∈ Un for 1 ≤ n < N . Then we have y ∈ U ∩ R∞ . Because “large boxes” form a basis of the product topology, every open neighborhood of x intersects R∞ . Therefore R∞ is not closed. Remark. In fact, the ar ...
Problem Set 3 – Special Functions
... a. Describe the domains and ranges of f and g. The domain of both is the set of all real numbers. The range of f is ...
... a. Describe the domains and ranges of f and g. The domain of both is the set of all real numbers. The range of f is ...
V.3 Quotient Space
... Remark If we assign the indiscrete topology on Y , any function p : X → Y would be continuous. But such a topology is too trivial to be useful and the most interesting one would be the finest topology. Definition 1 (1st definition) Given p : X → Y , a function from a topological space X onto a set Y ...
... Remark If we assign the indiscrete topology on Y , any function p : X → Y would be continuous. But such a topology is too trivial to be useful and the most interesting one would be the finest topology. Definition 1 (1st definition) Given p : X → Y , a function from a topological space X onto a set Y ...
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... Theorem 3.20. If f : X → Y is a.λ.c. and K is λ-compact relative to X, then f (K) is N -closed relative to Y. Proof. Let {Gα : α ∈ ∇} be any cover of f (K) by regular open sets of Y . Then {f −1 (Gα ) : α ∈ ∇} is a cover of K by λ-open sets of X. Hence there exists a finite subset ∇0 of ∇ such that ...
... Theorem 3.20. If f : X → Y is a.λ.c. and K is λ-compact relative to X, then f (K) is N -closed relative to Y. Proof. Let {Gα : α ∈ ∇} be any cover of f (K) by regular open sets of Y . Then {f −1 (Gα ) : α ∈ ∇} is a cover of K by λ-open sets of X. Hence there exists a finite subset ∇0 of ∇ such that ...