Tangent Lines
The line touching a curve with slope equal to the derivative.
Key idea
The line touching a curve with slope equal to the derivative.
In depth
The tangent line at x = a touches the curve with the same slope as the derivative. Its equation is y − f(a) = f′(a)(x − a). It is the best linear approximation to the curve near a.
Worked example
f(x) = x² at x = 3: f(3) = 9, f′(3) = 6, tangent y = 6x − 9.
Formula
Real-world uses
- Estimating the speed of a ball at one instant
- Drawing the direction of a curved road at one point
Common mistake
Using the curve’s height instead of the slope, or forgetting the point on the line.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Tangent Lines?
The line touching a curve with slope equal to the derivative. It is a topic at the advanced level in Calculus.
What is the formula for Tangent Lines?
The key formula is y − f(a) = f′(a)(x−a).
Where is Tangent Lines used in real life?
Real-world examples: Estimating the speed of a ball at one instant; Drawing the direction of a curved road at one point.
What should I learn before Tangent Lines?
Make sure you are comfortable with Definition of the Derivative, the previous topic in Calculus.
What is a common mistake with Tangent Lines?
Using the curve’s height instead of the slope, or forgetting the point on the line.
How can I practise Tangent Lines?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Tangent Lines?
Move on to Derivative as Velocity next.
Related advanced topics in Calculus
- Idea of a Limit – The value a function approaches as the input approaches a point.
- One-Sided Limits – Left and right limits must agree for the limit to exist.
- Limit Laws – Limits of sums, products and quotients are computed from the parts.
- Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
- Limits at Infinity – End behaviour of functions.
- Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.