Find The Slope Of The Tangent Line At The Point Of Tangency (y, X) On The Inverse Of Y = F(x).x Є [a, B].1 X,x E [3, 8] At (5.2, 5) On The Inversey=22x--5x

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Introduction

In calculus, the slope of the tangent line to a curve at a given point is a fundamental concept used to study the behavior of functions. When dealing with the inverse of a function, finding the slope of the tangent line requires a different approach. In this article, we will explore how to find the slope of the tangent line at the point of tangency (y, x) on the inverse of y = f(x), where x belongs to the interval [a, b]. We will use the given function y = 2x^2 - 5x as an example and find the slope of the tangent line at the point (5, 2).

Understanding the Inverse of a Function

The inverse of a function y = f(x) is denoted as f^(-1)(x) and is defined as a function that satisfies the property f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. In other words, the inverse of a function undoes the action of the original function. To find the inverse of a function, we need to swap the roles of x and y and solve for y.

Finding the Inverse of the Given Function

To find the inverse of the given function y = 2x^2 - 5x, we need to swap the roles of x and y and solve for y.

y = 2x^2 - 5x

Swapping the roles of x and y:

x = 2y^2 - 5y

Solving for y:

2y^2 - 5y - x = 0

Using the quadratic formula:

y = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-x)}}{2(2)}

Simplifying:

y = \frac{5 \pm \sqrt{25 + 8x}}{4}

This is the inverse of the given function.

Finding the Slope of the Tangent Line

To find the slope of the tangent line at the point (5, 2), we need to find the derivative of the inverse function.

\frac{dy}{dx} = \frac{d}{dx} \left( \frac{5 \pm \sqrt{25 + 8x}}{4} \right)

Using the chain rule:

\frac{dy}{dx} = \frac{1}{4} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{25 + 8x}} \cdot 8

Simplifying:

\frac{dy}{dx} = \frac{1}{\sqrt{25 + 8x}}

Evaluating the derivative at the point (5, 2):

\frac{dy}{dx} = \frac{1}{\sqrt{25 + 8(5)}} = \frac{1}{\sqrt{65}}

This is the slope of the tangent line at the point (5, 2).

Conclusion

In this article, we have explored how to find the slope of the tangent line at the point of tangency (y, x) on the inverse of y = f(x), where x belongs to the interval [a, b]. We have used the given function y = 2x^2 - 5x as an example and found the slope of the tangent line at the point (5, 2). The process involves finding the inverse of the function, finding the derivative of the inverse function, and evaluating the derivative at the point of interest.

Example Use Cases

  1. Physics: In physics, the slope of the tangent line to a curve represents the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the position of an object as a function of time represents the velocity of the object.
  2. Engineering: In engineering, the slope of the tangent line to a curve represents the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the stress on a material as a function of strain represents the modulus of elasticity of the material.
  3. Economics: In economics, the slope of the tangent line to a curve represents the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the demand for a product as a function of price represents the elasticity of demand.

Code Implementation

Here is a Python code implementation of the steps involved in finding the slope of the tangent line:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = 2*x**2 - 5*x

# Find the inverse of the function
f_inv = sp.solve(f - x, x)[0]

# Find the derivative of the inverse function
f_inv_prime = sp.diff(f_inv, x)

# Evaluate the derivative at the point (5, 2)
slope = f_inv_prime.subs(x, 5)

print(slope)

Q: What is the inverse of a function?

A: The inverse of a function y = f(x) is denoted as f^(-1)(x) and is defined as a function that satisfies the property f(f^(-1)(x)) = x and f^(-1)(f(x)) = x. In other words, the inverse of a function undoes the action of the original function.

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the roles of x and y and solve for y. This can be done using algebraic manipulations, such as solving a quadratic equation or using the quadratic formula.

Q: What is the derivative of the inverse function?

A: The derivative of the inverse function is found using the chain rule and the fact that the derivative of the inverse function is the reciprocal of the derivative of the original function.

Q: How do I find the slope of the tangent line on the inverse of a function?

A: To find the slope of the tangent line on the inverse of a function, you need to find the derivative of the inverse function and evaluate it at the point of interest.

Q: What is the significance of the slope of the tangent line on the inverse of a function?

A: The slope of the tangent line on the inverse of a function represents the rate of change of the original function with respect to the input variable. This can be useful in a variety of applications, such as physics, engineering, and economics.

Q: Can you provide an example of finding the slope of the tangent line on the inverse of a function?

A: Let's consider the function y = 2x^2 - 5x. To find the slope of the tangent line on the inverse of this function, we need to find the inverse of the function, find the derivative of the inverse function, and evaluate it at the point (5, 2).

Q: How do I use the slope of the tangent line on the inverse of a function in real-world applications?

A: The slope of the tangent line on the inverse of a function can be used in a variety of real-world applications, such as:

  • Physics: The slope of the tangent line on the inverse of a function can represent the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the position of an object as a function of time represents the velocity of the object.
  • Engineering: The slope of the tangent line on the inverse of a function can represent the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the stress on a material as a function of strain represents the modulus of elasticity of the material.
  • Economics: The slope of the tangent line on the inverse of a function can represent the rate of change of a quantity with respect to another quantity. For example, the slope of the tangent line to a curve representing the demand for a product as a function of price represents the elasticity of demand.

Q: What are some common mistakes to avoid when finding the slope of the tangent line on the inverse of a function?

A: Some common mistakes to avoid when finding the slope of the tangent line on the inverse of a function include:

  • Not finding the inverse of the function correctly: Make sure to swap the roles of x and y and solve for y correctly.
  • Not finding the derivative of the inverse function correctly: Make sure to use the chain rule and the fact that the derivative of the inverse function is the reciprocal of the derivative of the original function.
  • Not evaluating the derivative at the correct point: Make sure to evaluate the derivative at the point of interest.

Q: Can you provide a Python code implementation of finding the slope of the tangent line on the inverse of a function?

A: Here is a Python code implementation of finding the slope of the tangent line on the inverse of a function:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
f = 2*x**2 - 5*x

# Find the inverse of the function
f_inv = sp.solve(f - x, x)[0]

# Find the derivative of the inverse function
f_inv_prime = sp.diff(f_inv, x)

# Evaluate the derivative at the point (5, 2)
slope = f_inv_prime.subs(x, 5)

print(slope)

This code uses the SymPy library to find the inverse of the function, find the derivative of the inverse function, and evaluate the derivative at the point (5, 2). The output of the code is the slope of the tangent line at the point (5, 2).