Derivatives, Tangent Lines & Linear Approximation

Vocabulary flashcards covering key terms from the lecture on derivatives, tangent lines, and linear approximations.

Derivative (f'(x))

The instantaneous rate of change of a function, defined by f'(x)=lim_{h→0}[f(x+h)-f(x)]/h.

Derivative at a Point

The slope of the tangent line to the curve at x=a, given by f'(a).

Limit Definition of the Derivative

Using the limit as h→0 of the difference quotient to compute f'(x).

Tangent Line

The straight line that just touches a curve at a point and has slope equal to the function’s derivative there.

Point-Slope Form

Equation of a line using a point (x₁,y₁) and slope m: y−y₁ = m(x−x₁).

Slope-Intercept Form

Equation of a line written as y = mx + b, where m is slope and b is y-intercept.

Slope (m)

The measure of steepness of a line; for a curve, m = f'(a) at x = a.

Linearization (L(x))

The tangent-line approximation of f at x=a: L(x)=f(a)+f'(a)(x−a).

Linear Approximation

Using L(x) to estimate f(x) for x close to a; often called the tangent-line approximation.

When to Use Linearization

Best when x is near a because the tangent line closely matches the curve locally.

Concave Up

Curve bends upward (f''(x)>0); tangent line lies below the graph, giving an underestimate.

Concave Down

Curve bends downward (f''(x)<0); tangent line lies above the graph, giving an overestimate.

Overestimate

An approximation that is larger than the true value; occurs with concave-down curves using tangent lines.

Underestimate

An approximation that is smaller than the true value; occurs with concave-up curves using tangent lines.

Choosing a for Linearization

Select a value of x where f(a) and f'(a) are known and that is close to the x-value you wish to approximate.

Increasing Function

A function with f'(x)>0, meaning its graph rises as x increases.

Decreasing Function

A function with f'(x)<0, meaning its graph falls as x increases.

Using Tangent Line for Estimation

Plug the desired x into y−y₁ = m(x−x₁) (or y=mx+b) to approximate f(x) when exact evaluation is hard.