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Verify that f has an inverse function. Then use the function f and the given real number a to find (f 1)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ 4, a = 121

Verify that f has an inverse function. Then use the function f and the given real number a to find (f 1)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ 4, a = 121

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.CR: Chapter 4 Review
Problem 47CR
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Verify that f has an inverse function. Then use the function f and the given real number a to find (f−¹)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ – 4, a = 121 Step 1 The given function is f(x) = x³ – 4. Differentiate f(x) with respect to x. f'(x) Submit Skip (you cannot come back)
Transcribed Image Text:Verify that f has an inverse function. Then use the function f and the given real number a to find (f−¹)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ – 4, a = 121 Step 1 The given function is f(x) = x³ – 4. Differentiate f(x) with respect to x. f'(x) Submit Skip (you cannot come back)
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Verify that f has an inverse function. Then use the function f and the given real number a to find (f−¹)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ - 4, a = 121 Step 1 The given function is f(x) = x³ – 4. Differentiate f(x) with respect to x. Step 2 f'(x) = The function f'(x) Therefore, f(x) X = = 3 x 3x² is always non-negative Step 3 To find f-¹(x), let y = f(x) and solve for x, y = x³ - 4 1/3 1/3 is monotonic on the domain of x and the inverse of the function exists. (y + 4) Interchange x and y in the expression for x. f−¹(x) = (x + 4)' y = (x + 4) Find f-1(x) by replacing y with f-¹(x). f−¹(121) = 2 1/3 1/3 f'(f−¹(121)) : = 1/3 || = ³(1 Step 4 To find (f−¹)'(x) use the theorem that states that if f is differentiable on an interval I and f has an inverse function g, that is g = f−¹, then g is differentiable at any x for which f'(g(x)) ‡ 0 and g'(x) = = Find the values of f-1(121) and then f'(f-¹(121)), f−¹(x) = (x + 4)¹/3 1/3 non-negative on the domain of f(x). 2 1 f'(g(x))
Transcribed Image Text:Verify that f has an inverse function. Then use the function f and the given real number a to find (f−¹)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) f(x) = x³ - 4, a = 121 Step 1 The given function is f(x) = x³ – 4. Differentiate f(x) with respect to x. Step 2 f'(x) = The function f'(x) Therefore, f(x) X = = 3 x 3x² is always non-negative Step 3 To find f-¹(x), let y = f(x) and solve for x, y = x³ - 4 1/3 1/3 is monotonic on the domain of x and the inverse of the function exists. (y + 4) Interchange x and y in the expression for x. f−¹(x) = (x + 4)' y = (x + 4) Find f-1(x) by replacing y with f-¹(x). f−¹(121) = 2 1/3 1/3 f'(f−¹(121)) : = 1/3 || = ³(1 Step 4 To find (f−¹)'(x) use the theorem that states that if f is differentiable on an interval I and f has an inverse function g, that is g = f−¹, then g is differentiable at any x for which f'(g(x)) ‡ 0 and g'(x) = = Find the values of f-1(121) and then f'(f-¹(121)), f−¹(x) = (x + 4)¹/3 1/3 non-negative on the domain of f(x). 2 1 f'(g(x))
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