Problem 2. For each of the following formulas for f (x), find the largest possible set A ⊆ Rso that f : A → R is a function, and prove that f is differentiable at every point a ∈ A.(a) f (x) = sec x = 1cos x .(b) f (x) = csc x = 1sin x .(c) f (x) = tan x = sin xcos x .(d) f (x) = cot x = cos xsin x
Question
Problem 2. For each of the following formulas for f (x), find the largest possible set A ⊆ Rso that f : A → R is a function, and prove that f is differentiable at every point a ∈ A.(a) f (x) = sec x = 1cos x .(b) f (x) = csc x = 1sin x .(c) f (x) = tan x = sin xcos x .(d) f (x) = cot x = cos xsin x
Solution
(a) The function f(x) = sec x = 1/cos x is defined for all real numbers except for those where cos x = 0. This occurs at x = (2n+1)π/2, where n is an integer. Therefore, the largest possible set A for which f is a function is A = R - {(2n+1)π/2 : n ∈ Z}. The function f is differentiable at every point a ∈ A because the secant function is the derivative of the arccosine function, which is differentiable for all real numbers.
(b) The function f(x) = csc x = 1/sin x is defined for all real numbers except for those where sin x = 0. This occurs at x = nπ, where n is an integer. Therefore, the largest possible set A for which f is a function is A = R - {nπ : n ∈ Z}. The function f is differentiable at every point a ∈ A because the cosecant function is the derivative of the arcsine function, which is differentiable for all real numbers.
(c) The function f(x) = tan x = sin x/cos x is defined for all real numbers except for those where cos x = 0. This occurs at x = (2n+1)π/2, where n is an integer. Therefore, the largest possible set A for which f is a function is A = R - {(2n+1)π/2 : n ∈ Z}. The function f is differentiable at every point a ∈ A because the tangent function is the derivative of the arctangent function, which is differentiable for all real numbers.
(d) The function f(x) = cot x = cos x/sin x is defined for all real numbers except for those where sin x = 0. This occurs at x = nπ, where n is an integer. Therefore, the largest possible set A for which f is a function is A = R - {nπ : n ∈ Z}. The function f is differentiable at every point a ∈ A because the cotangent function is the derivative of the arccotangent function, which is differentiable for all real numbers.
Similar Questions
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