Math Problem Statement
For the functions below, find the tangent line to the inverse of the function at the given x value. You may assume the functions indeed have inverses on the interval [0, ∞). (a) f (x) = x3 + 2x + 5, x = 5
GPT 3.5
Solution
To find the tangent line to the inverse of the function at a given value, we can follow these steps:
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Find such that : This will help us determine the point on the inverse function.
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Calculate : This will help us find the slope of the tangent line to the inverse function.
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Use the slope of the tangent line: The slope of the tangent line to the inverse function at a point is the reciprocal of the derivative of the original function at that point.
Step 1: Solve for such that
We need to solve: Simplifying this gives: Factoring out : This gives as a solution. Since has no real roots, we have: Thus, .
Step 2: Calculate
We compute the derivative:
Step 3: Evaluate at
Now, we find:
Step 4: Find the slope of the tangent line to the inverse
The slope of the tangent line to the inverse function at is:
Step 5: Equation of the tangent line
The equation of a line in point-slope form is: Where and . Substituting in: This simplifies to:
Conclusion
The equation of the tangent line to the inverse of the function at is:
If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:
- What is the value of for other values of ?
- How can we confirm that the function is one-to-one?
- What would happen if we changed the function to ?
- How do we graph the function and its inverse?
- Can we find the tangent line for another point on the inverse function?
Tip: When working with inverse functions, always ensure that the original function is one-to-one over the interval of interest.
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Math Problem Analysis
Mathematical Concepts
Calculus
Inverse Functions
Differentiation
Formulas
f'(x) = 3x^2 + 2
y - y_1 = m(x - x_1)
Theorems
Inverse Function Theorem
Point-Slope Form of a Line
Suitable Grade Level
Grades 11-12