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Tangent Line

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Tangent Line

Study Guide

Key Definition

A tangent line is a line that touches a curve at a single point, where the slope of the line equals the slope of the curve at that point.

Important Notes

A tangent line only touches a curve at one point.
The point where the tangent line touches the curve is called the point of tangency.
Tangent lines to circles are perpendicular to the radius at the point of tangency.
The slope of the tangent line equals the slope of the curve at the point of tangency.
Tangents can be used to approximate the slope of a curve at a particular point.

Mathematical Notation

$\angle$ represents an angle

$r$ represents the radius

$\overline{AB}$ represents a line segment

$\frac{dy}{dx}$ represents the derivative

$\pi$ represents pi, approximately 3.14159

Remember to use proper notation when solving problems

Why It Works

Tangent lines provide a way to understand the behavior of curves at specific points using $derivatives$.

Remember

The tangent to a circle is always perpendicular to the radius at the point of tangency.

Quick Reference

Tangent Line:Touches the curve at exactly one point without crossing it.

Understanding Tangent Line

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Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A tangent line touches a curve at one point, sharing the same slope at that point.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following defines a tangent line?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you're designing a skate park and need to calculate the slope of a ramp tangent to a circular bowl to ensure safety. If the radius of the bowl is $10$ feet, what is the slope of the tangent line at the edge?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a circle centered at the origin with radius $r$. Determine the equation of the tangent line at any point $(x, y)$ on the circle.

Challenge Quiz

Single Choice Quiz

Advanced

Given a curve described by $y = x^2$, what is the equation of the tangent line at $x = 2$?

Recap

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Review key concepts and takeaways