Math Problem Statement
Show that f(x) = x to 3
dt / sqrt(1 + t^4) is one-to-one. f '(x) =
Correct: Your answer is correct. Since f '(x)
Correct: Your answer is correct. 0 for all x, f is increasing
Correct: Your answer is correct. on (−∞, ∞). Thus f is one-to-one. Find (f −1)'(0). (f −1)'(0) =
Incorrect: Your answer is incorrect.
Solution
Let's break this problem down carefully and provide the solution.
Step 1: Understanding
The function is defined as:
Step 2: Compute using the Fundamental Theorem of Calculus
By the Fundamental Theorem of Calculus, we have Note the negative sign because the variable of integration is the lower limit of the integral.
Since for all , the denominator is always positive. Thus, for all . This implies that the function is strictly decreasing on , and hence it is one-to-one.
Step 3: Use the formula for the derivative of the inverse function
We are tasked with finding . To do this, we need to find such that .
Step 4: Solve
This implies that because the integral of a positive function can only vanish when the upper and lower limits are the same.
Thus, . So .
Step 5: Compute
Final Answer:
Would you like further clarification or details on any steps?
Related Questions:
- How do you apply the Fundamental Theorem of Calculus in general problems?
- Why does the sign of determine whether a function is one-to-one?
- How is the formula for the derivative of the inverse function derived?
- What are conditions for the existence of an inverse for a function?
- How can you visualize the monotonicity of a function graphically?
Tip: Always verify whether a function is monotonic before concluding it has an inverse.
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Math Problem Analysis
Mathematical Concepts
Fundamental Theorem of Calculus
Derivative of an Inverse Function
Monotonicity
Integral Calculus
Formulas
f'(x) = -1 / sqrt(1 + x^4)
(f^{-1})'(y) = 1 / f'(f^{-1}(y))
f(x) = ∫(x to 3) dt / sqrt(1 + t^4)
Theorems
Fundamental Theorem of Calculus
Inverse Function Theorem
Suitable Grade Level
Undergraduate
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