Equation of tangent line example 1 | Derivative applications | Differential Calculus | Khan Academy

TL;DR

The equation of the tangent line to a given curve at the point x=1 is a horizontal line with y=e/3.

Transcript

We have the curve y is equal to e to the x over 2 plus x to the third power. And what we want to do is find the equation of the tangent line to this curve at the point x equals 1. And when x is equal to 1, y is going to be equal to e over 3. It's going to be e over 3. So let's try to figure out the equation of the tangent line to this curve at this... Read More

Key Insights

• 🫥 The slope of the tangent line is equal to the derivative of the curve at the point of tangency.
• 📏 The derivative of the curve is found using the product rule.
• 😑 Substituting the x-coordinate of the point of tangency into the derivative expression gives the slope of the tangent line.
• 🫥 A slope of 0 indicates a horizontal tangent line.

Questions & Answers

Q: How is the slope of the tangent line at a specific point related to the derivative of the curve at that point?

The slope of the tangent line at a point is equal to the value of the derivative of the curve at that point. This relationship allows us to find the slope of the tangent line using calculus.

Q: What is the equation for the derivative of the given curve?

The derivative of the curve y=e^(x/2)+x^3 is found using the product rule. The derivative is e^x(2+x^3)^-1 + e^x(3x^2)(2+x^3)^-2.

Q: How is the slope of the tangent line at x=1 calculated?

To find the slope of the tangent line at x=1, substitute x=1 into the derivative expression. Simplifying the expression gives a slope of 0.

Q: What is the equation of the tangent line to the curve at x=1?

Since the slope of the tangent line is 0, the equation of the tangent line is a horizontal line. The y-coordinate of the point of tangency, y=e/3, becomes the equation of the tangent line: y=e/3.

Summary & Key Takeaways

• The goal is to find the equation of the tangent line to the curve y=e^(x/2)+x^3 at the point x=1, where y=e/3.

• The slope of the tangent line is equal to the derivative of the curve at that point.

• The derivative of the curve is found using the product rule, and when evaluated at x=1, the slope is 0.

• Since the slope is 0, the equation of the tangent line is a horizontal line with y=e/3.