Understanding the Slope-Intercept Form of the Tangent Line

When discussing the tangent line of a function at a specific point, it's important to understand how to derive its equation in the slope-intercept form. This tutorial will explore the tangent line of the function fx √x 9 at the point where x 0. We will go through the process step by step, from finding the function value to the calculation of the derivative and finally, the equation of the tangent line.

The Function and the Point of Tangency

Let's consider the function fx √x 9. We need to find the equation of the tangent line to this function at the point where x 0.

Step 1: Find the Function Value

First, we need to determine the value of the function at x 0. f(0) √0 9 9 Therefore, the point of tangency is (0, 9).

Step 2: Calculate the Derivative

To find the slope of the tangent line, we need to calculate the derivative of the function at x 0. The derivative of fx √x 9 is derived as follows:

fx' d/dx (√x 9) 1/(2√x)

Now, we evaluate the derivative at x 0: fx'(0) 1/(2√0)

However, this results in an undefined value because 1/0 is undefined. To determine the slope, we need to examine the limit as x approaches 0 from the right:

fx' 1/(2√x) as x → 0^ → ∞

Since the slope is approaching infinity, the tangent line is vertical at x 0.

Step 3: Equation of the Tangent Line

Since the slope is undefined and the tangent line is vertical, the equation of the tangent line can be written as:

x 0

This equation is the slope-intercept form of a vertical line, which does not exist in the standard form y mx b. Thus, the equation of the tangent line at the point (0, 9) is:

x 0

Conclusion

In conclusion, when the slope of the tangent line is undefined (as in the case of a vertical line), the standard slope-intercept form cannot be used. Therefore, the equation of the tangent line to the function fx √x 9 at the point where x 0 is:

x 0

This tutorial provides a clear methodology for finding the tangent line of a function at a specified point, even when the slope is undefined.

Key Takeaways

When calculating the derivative of a function to find the slope of the tangent line, be cautious of points where the derivative is undefined. A vertical line has an undefined slope and its equation can be written as x a, where a is the x-coordinate of the tangency point. The slope-intercept form y mx b is only applicable to non-vertical lines.

Frequently Asked Questions

What is the Slope-Intercept Form?

The slope-intercept form of a line's equation is y mx b, where m represents the slope of the line and b is the y-intercept.

How do you find the Derivative of a Function?

The derivative of a function is found using various differentiation rules. For a function fx √x 9, the derivative is fx' 1/(2√x).

What does it Mean if the Slope is Undefined?

If the slope of a line is undefined, it indicates a vertical line. The equation of such a line takes the form x a, where a is the x-coordinate of the point through which the line passes.