Unit 4 Differentiation Applications: Approximations and Indeterminate Limits

Local linearity

The idea that a function may be curved overall but looks approximately like a straight line when zoomed in near a point.

Derivative at a point

A measure of the slope of the line that best matches (is tangent to) the function right near that point.

Tangent line

The line that “best matches” a curve at a specific point; used for local approximation.

Tangent line equation at x = a

The line given by y = f(a) + f'(a)(x − a).

Linearization

The tangent-line function L(x) = f(a) + f'(a)(x − a) used to approximate f(x) near x = a.

Anchor point

The point (a, f(a)) that the linearization/tangent line passes through and is built around.

Slope in linearization

The value f'(a), which determines how steep the tangent line is at x = a.

Approximation via linearization

When x is close to a, f(x) ≈ L(x), meaning the function value is estimated by the tangent line value.

Concavity (role in error)

The curve’s bending; as you move farther from a, concavity causes the function to drift away from the tangent line, reducing accuracy.

Δx (delta x)

A small change in input; if starting at a, the new input is a + Δx.

Δy (delta y)

The actual change in output: Δy = f(a + Δx) − f(a).

Differentials

A notation system for estimating small changes using derivatives, often paired with linearization.

dx

A small change in the input used in differentials (often taken as dx = Δx when changes are small).

dy

The estimated change in the output from the tangent line: dy = f'(x) dx (so near a, dy ≈ f'(a)Δx).

Error propagation (measurement error)

Using derivatives/differentials to translate a small input uncertainty (like dr) into an estimated output uncertainty (like dV).

Common linearization mistake: missing (x − a)

Writing L(x) = f(a) + f'(a)x instead of f(a) + f'(a)(x − a); the incorrect form only works when a = 0.

Common linearization mistake: confusing Δy and dy

Treating dy (estimated change) as the same as Δy (actual change), or treating dy as exact.

Indeterminate form

A limit expression (like 0/0 or ∞/∞) that does not determine the limit value without further analysis.

L’Hôpital’s Rule

A method for evaluating limits of quotients in 0/0 or ∞/∞ form by replacing f/g with f'/g' (when conditions are met).

0/0 form

An indeterminate form where both numerator and denominator approach 0, allowing L’Hôpital’s Rule (if differentiability conditions hold).

∞/∞ form

An indeterminate form where both numerator and denominator grow without bound, allowing L’Hôpital’s Rule (if differentiability conditions hold).

Indeterminate-form verification (L’Hôpital requirement)

The step of checking that substitution gives 0/0 or ∞/∞ before applying L’Hôpital; skipping this is a common pitfall.

Repeated L’Hôpital application

Applying L’Hôpital’s Rule more than once if the differentiated limit is still in 0/0 or ∞/∞ form.

Converting 0·∞ to a quotient

Rewriting a product like x ln x as a fraction (e.g., ln x ÷ (1/x)) to create an ∞/∞ or 0/0 form for L’Hôpital.

Conjugate (for resolving ∞ − ∞)

An algebra tool used to rewrite expressions like √(x²+x) − x by multiplying by √(x²+x) + x to turn a difference into a quotient.