How to find vertical tangents?

Understand the Problem

The question is asking how to identify vertical tangents of a curve or function on a graph. This typically involves finding the points where the derivative of the function is undefined or where the slope approaches infinity.

Answer

First, find the derivative $f'(x). Then, solve for the points where $f'(x)$ is undefined or approaches infinity. Verify the points on the original function $f(x).
Answer for screen readers

First, find the derivative $f'(x)$. Then, solve for the points where $f'(x)$ is undefined or approaches infinity. Verify the points on the original function $f(x)$.

Steps to Solve

  1. Find the derivative of the function

To identify vertical tangents, you need the first derivative of the function, which gives the slope of the tangent line at any point on the curve. Let's call the function $f(x)$. The derivative is $f'(x)$.

  1. Set the derivative equal to infinity or check where it is undefined

Vertical tangents occur where the derivative $f'(x)$ approaches infinity or where it is undefined. These points are where the slope of the tangent line is very steep.

  1. Solve for the points

Solve the equation $f'(x) = ext{undefined}$. This often involves setting the denominator of $f'(x)$ to zero if it is a rational function, or looking for points where $f'(x)$ has a vertical asymptote.

  1. Verify the points on the original function

Once you've found the points where $f'(x)$ is undefined, substitute these $x$-values back into the original function $f(x)$ to get the corresponding $y$-values. Verify these are points on the curve.

Example: For $f(x) = \sqrt[3]{x}$:

Find its derivative $f'(x) = \frac{1}{3} x^{-2/3}$. Notice $f'(x)$ is undefined when $x = 0$, indicating a possible vertical tangent.

  1. Combine the results

Combine the results from the previous steps to conclude where the vertical tangents are.

For $f(x) = \sqrt[3]{x}$, there's a vertical tangent at $(0,0)$.

First, find the derivative $f'(x)$. Then, solve for the points where $f'(x)$ is undefined or approaches infinity. Verify the points on the original function $f(x)$.

More Information

Vertical tangents represent locations on a graph where the slope is infinitely steep. They are essential in understanding the behavior of curves.

Tips

A common mistake is not verifying if the undefined points of $f'(x)$ lie on the original function $f(x)$. Always plug them back into the original function to confirm.

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