Linear Approximation
Approximate f near a using its tangent line.
Key idea
Approximate f near a using its tangent line.
In depth
Near x = a, f(x) ≈ f(a) + f′(a)(x − a). The tangent line is a good local estimate when the function is smooth. The error grows as you move away.
Worked example
√4.1 ≈ 2 + (1/4)(0.1) = 2.025. The true value is 2.02485.
Formula
Real-world uses
- Estimating square roots and small changes in measurement error without a calculator
Common mistake
Using it far from the base point.
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 Linear Approximation?
Approximate f near a using its tangent line. It is a topic at the expert level in Calculus.
What is the formula for Linear Approximation?
The key formula is f(x) ≈ f(a) + f′(a)(x−a).
Where is Linear Approximation used in real life?
Real-world examples: Estimating square roots and small changes in measurement error without a calculator.
What should I learn before Linear Approximation?
Make sure you are comfortable with Related Rates, the previous topic in Calculus.
What is a common mistake with Linear Approximation?
Using it far from the base point.
How can I practise Linear Approximation?
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 Linear Approximation?
Move on to Differentials next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Differentials – Use dy = f′(x)dx for small changes.
- Second Derivative Test – f″ > 0 means concave up; f″ < 0 concave down.