Tangent Planes
Linear approximation of surfaces.
Key idea
Linear approximation of surfaces.
In depth
For z = f(x, y) the tangent plane at (a, b) is z = f(a, b) + f_x(a, b)(x − a) + f_y(a, b)(y − b). It is the linear approximation in two variables.
Worked example
f = x² + y² at (1, 1): z = 2 + 2(x − 1) + 2(y − 1).
Real-world uses
- Approximating the height of a hilly surface near a point
- Error estimates in measurement
Common mistake
Using partials not evaluated at the 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 Tangent Planes?
Linear approximation of surfaces. It is a topic at the expert level in Calculus.
Where is Tangent Planes used in real life?
Real-world examples: Approximating the height of a hilly surface near a point; Error estimates in measurement.
What should I learn before Tangent Planes?
Make sure you are comfortable with Multivariable Chain Rule, the previous topic in Calculus.
What is a common mistake with Tangent Planes?
Using partials not evaluated at the point.
How can I practise Tangent Planes?
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 Planes?
Move on to Lagrange Multipliers 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.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.