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4. Prove that d - (csc (x)) = -csc (x) cot (x). dx

4. Prove that d - (csc (x)) = -csc (x) cot (x). dx

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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4. Prove that d dx - (csc (x)) = −csc (x) cot (x). 5. Fill in the blank with "all", "no", or "some" to make the following statements true. Note that "some" means one or more instances, but not all. • If your answer is "all", then give a brief explanation as to why. • If your answer is "no", then give an example and a brief explanation as to why. • If your answer is "some", then give two specific examples that illustrate why your answer it not "all" or "no". Be sure to explain your two examples. An example must include either a graph or a specific function. f(x) g(x) (a) For functions f and 9, if either f(x) or g(x) is not differentiable at x = 2. (b) For is defined but not differentiable at x = 2, then functions f and g, if f and g are two functions whose second derivatives are defined, then (f g)" = f ·g" + f" ·g. (c) For functions f and g, (f(x) · g(x))' = f'(x) · g'(x). In mathematics, we consider a statement to be false if we can find any examples where the statement is not true. We refer to these examples as counterexamples. Note that a counterexample is an example for which the "if" part of the statement is true, but the "then" part of the statement is false.
Transcribed Image Text:4. Prove that d dx - (csc (x)) = −csc (x) cot (x). 5. Fill in the blank with "all", "no", or "some" to make the following statements true. Note that "some" means one or more instances, but not all. • If your answer is "all", then give a brief explanation as to why. • If your answer is "no", then give an example and a brief explanation as to why. • If your answer is "some", then give two specific examples that illustrate why your answer it not "all" or "no". Be sure to explain your two examples. An example must include either a graph or a specific function. f(x) g(x) (a) For functions f and 9, if either f(x) or g(x) is not differentiable at x = 2. (b) For is defined but not differentiable at x = 2, then functions f and g, if f and g are two functions whose second derivatives are defined, then (f g)" = f ·g" + f" ·g. (c) For functions f and g, (f(x) · g(x))' = f'(x) · g'(x). In mathematics, we consider a statement to be false if we can find any examples where the statement is not true. We refer to these examples as counterexamples. Note that a counterexample is an example for which the "if" part of the statement is true, but the "then" part of the statement is false.
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