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 $a$, $f(x) \text{≈ } L(x)$, 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.

$\triangle y$ (delta y)

The actual change in output: $\triangle y = f(a + \triangle x) − f(a).$

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 $dx = \text{Δ}x$ when changes are small).

dy

The estimated change in the output from the tangent line: $dy = f'(x) \times dx$ (so near $a$, $dy \thickapprox f'(a) \triangle x$).

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 $\frac{0}{0}$ or $\frac{\text{∞}}{\text{∞}}$ form by replacing $\frac{f}{g}$ with $\frac{f'}{g'}$ (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).

$\text{∞/∞}$ 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 $\frac{0}{0}$ or $\frac{\text{∞}}{\text{∞}}$ 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 $\frac{0}{0}$ or $\frac{\text{∞}}{\text{∞}}$ form.

Converting $0 \times \infty$ to a quotient

Rewriting a product like $x \text{ln} x$ as a fraction (e.g., $\text{ln} x \big/ \big(1/x\big)$) to create an $\text{∞/∞}$ or $0/0$ form for L’Hôpital.

Conjugate (for resolving $\infty - \infty$)

An algebra tool used to rewrite expressions like $\sqrt{x²+x} - x$ by multiplying by $\sqrt{x²+x} + x$ to turn a difference into a quotient.