AP Calculus AB Unit 4: Linearization (Local Linearity, Differentials) and L'Hôpital's Rule

Local Linearity

The idea that a differentiable function behaves approximately like its tangent line near a specific input value.

Tangent Line

The linear approximation of a function at a point, given by the slope equal to the derivative at that point.

Linearization Formula

L(x) = f(a) + f'(a)(x - a), used to estimate function values near x = a.

Point-Slope Form

An equation of the form y - y1 = m(x - x1) used to describe the equation of a line.

Instantaneous Rate of Change

The value of the derivative at a specific point, measuring how quickly a function changes at that point.

Small Change Approximation

For small changes Δx, the change in output can be approximated as Δy ≈ f'(a)Δx.

Differential Form

dy = f'(x)dx, representing the linear estimate of change in output with respect to a small change in input.

Concave Up Function

A function that curves upwards, indicating that its second derivative f''(x) > 0.

Concave Down Function

A function that curves downwards, indicating that its second derivative f''(x) < 0.

L'Hôpital's Rule

A method for evaluating limits that produce indeterminate forms such as 0/0 or ∞/∞.

Indeterminate Forms

Forms like 0/0 or ∞/∞ where further analysis is required to determine the limit.

Quotient Rule Trap

The common error of using the quotient rule instead of differentiating numerator and denominator separately in L'Hôpital's Rule.

Monotonic Behavior

The behavior of a function in which the function is either entirely increasing or entirely decreasing.

Type A Rewrite for L'Hôpital

Converting a product form like 0·∞ into a quotient to apply L'Hôpital's Rule.

Type B Rewrite for L'Hôpital

Rewriting expressions involving powers like 1^∞ using logarithms before applying L'Hôpital's Rule.

Limit Divergence

When a limit approaches either ∞ or -∞ after applying L'Hôpital's Rule, indicating the function grows without bound.

Function's Value at a Point

The output or value of a function f at a specific input a, noted as f(a).

Small Input Change

The difference Δx = x - a, representing a small deviation from a known point a.

Error Estimation

Using derivatives to estimate potential errors in real-world measurements, such as volume changes in response to radius changes.

Quotient Form

A fraction form used in limits where both the numerator and denominator approach 0 or infinity.

Regular Expressions for Limits

Expressions that yield either 0/0 or ∞/∞, requiring validation prior to applying L'Hôpital's Rule.

Tangent Line Approximation

Using the slope and point of tangency to estimate values of a function near the tangent point.

Differentiability

The property of a function that allows it to be differentiated, usually at every point in an interval.

Measure of Change

The graphical representation of how output changes in response to input variations, often visualized through tangents.

Friendly Input Choice

Selecting a value a that simplifies calculations, making it easier to compute function values and derivatives.

Rounding Errors

Small inaccuracies that can arise from limitations in measurement precision.

Contextual Application

Using mathematical concepts in real-world scenarios to understand their practical implications.