Points where there are Horizontal and Vertical Tangent Lines using Implicit Differentiation

TL;DR

Using implicit differentiation, we can find the points on the graph where the equation has horizontal and vertical tangent lines.

Transcript

find the points at which the graph of the equation below has horizontal and vertical tangent lines we're going to use implicit differentiation to do this so the graph of this equation will have horizontal tangent lines whenever dy/dx is zero and it'll have vertical tangent lines whenever dy/dx is undefined so the first thing we'll do in this proble... Read More

Key Insights

• ☺️ Implicit differentiation helps find the derivative of equations involving both x and y.
• 🫥 The slope of the tangent line is given by dy/dx.
• 🫥 Horizontal tangent lines occur when dy/dx is zero.
• 🫥 Vertical tangent lines occur when dy/dx is undefined.
• ❣️ Solving the equations obtained from setting dy/dx equal to zero or undefined gives the x and y values for the tangent lines' points.

Questions & Answers

Q: How do we use implicit differentiation to find the derivative of the equation?

Implicit differentiation is done by differentiating both sides of the equation with respect to x. The chain rule is applied to differentiate terms involving y, resulting in dy/dx.

Q: What conditions determine the presence of horizontal and vertical tangent lines?

Horizontal tangent lines occur when dy/dx is zero, and vertical tangent lines occur when dy/dx is undefined.

Q: How do we find the x-values for horizontal tangent lines?

To find the x-values for horizontal tangent lines, we solve the equation -50x - 200 = 0, which results in x = -4.

Q: How do we find the y-values for vertical tangent lines?

To find the y-values for vertical tangent lines, we solve the equation 32y - 160 = 0, which results in y = 5.

Summary & Key Takeaways

• Implicit differentiation is used to find the derivative of the equation and find dy/dx.

• Horizontal tangent lines occur when dy/dx is zero, and vertical tangent lines occur when dy/dx is undefined.

• The equation is solved for horizontal and vertical tangent lines by setting the derivative equal to zero and finding the corresponding x and y values.