Linear Approximation & Differentials Cheat Sheet

Key Facts

• The linear approximation of $f(x)$ near $x = a$ is $L(x) = f(a) + f'(a)(x - a)$.
• For a small change $\Delta x$, the change in the function is approximated by $\Delta y \approx f'(a)\Delta x$.
• The differential of $x$ is $dx$, and the differential of $y = f(x)$ is $dy = f'(x)dx$.
• Near $x = a$, the function value can be estimated by $f(a + \Delta x) \approx f(a) + f'(a)\Delta x$.
• The tangent line approximation is most accurate when $x$ is close to $a$ and $f$ is differentiable at $a$.
• Absolute error can be estimated by $|\Delta y - dy|$, where $\Delta y = f(a + \Delta x) - f(a)$.
• Relative error is $\frac{|\text{error}|}{|\text{actual value}|}$, and percent error is $\frac{|\text{error}|}{|\text{actual value}|} \cdot 100\%$.
• If $f''(x)$ is large in magnitude near $a$, the tangent-line approximation may become less accurate more quickly.

Vocabulary

Linear approximation

An estimate of a function near a point using the tangent line formula $L(x) = f(a) + f'(a)(x - a)$.

Tangent line

A line that touches a curve at a point and has slope equal to the derivative $f'(a)$ at that point.

Differential

A small estimated change in a dependent variable, written as $dy = f'(x)dx$.

Increment

A change in an input or output value, often written as $\Delta x$ or $\Delta y$.

Error estimate

A measure of how far an approximation is from the actual value, often written as $|\text{actual} - \text{approximation}|$.

Local linearity

The idea that a differentiable function looks nearly like a straight line when viewed very close to a point.