Unit 4 Differentiation Tools for Approximation and Limits (AP Calculus BC)

Local linearity

The idea that a differentiable function looks nearly like a straight line when viewed very close to a point; near x=a, f(x) is well-approximated by its tangent line.

Differentiable (at x=a)

A condition guaranteeing the function has a well-defined derivative at x=a, which implies the graph has a tangent line there and exhibits local linearity.

Tangent line (at x=a)

The line that touches y=f(x) at (a,f(a)) with slope f′(a), representing the best linear approximation near x=a.

Linearization

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

Point of linearization (a)

The “center” x-value where f(a) and f′(a) are known and where the approximation L(x) is anchored.

Tangent line approximation

Using L(x)=f(a)+f′(a)(x−a) to estimate function values: f(x)≈L(x) for x near a.

Derivative (limit definition)

f′(a)=lim(x→a) [f(x)−f(a)]/(x−a); the limiting slope of secant lines approaching x=a.

Linear approximation statement

Near x=a, the change in f is approximated by a linear change: f(x)−f(a)≈f′(a)(x−a), so f(x)≈f(a)+f′(a)(x−a).

Actual change in input (Δx)

The real change in x from the base point: Δx=x−a.

Actual change in output (Δy)

The real change in the function value: Δy=f(x)−f(a).

Differential (dx)

A small (infinitesimal-style) change in x used in linear approximation; at x=a, dx is taken as a small input change.

Differential (dy)

The change predicted by the tangent line: dy=f′(a)dx (approximately equals Δy when changes are small).

Differentials relationship

Near x=a, Δy≈f′(a)Δx; in differential form, dy=f′(x)dx and at x=a, dy≈f′(a)dx.

Concave up

A shape where f′′(x)>0; the graph bends upward and the tangent line tends to lie below the curve near the point.

Concave down

A shape where f′′(x)<0; the graph bends downward and the tangent line tends to lie above the curve near the point.

Underestimate (by linearization)

When the tangent-line approximation L(x) gives a value less than the true f(x) near a; typically occurs when f is concave up (f′′>0) near a.

Overestimate (by linearization)

When the tangent-line approximation L(x) gives a value greater than the true f(x) near a; typically occurs when f is concave down (f′′<0) near a.

Indeterminate form

A limit form that does not by itself determine the limit’s value (e.g., 0/0, ∞/∞, 0·∞, ∞−∞).

Indeterminate quotient forms (for L’Hôpital)

The two main quotient forms where L’Hôpital’s Rule applies directly: 0/0 and ∞/∞.

L’Hôpital’s Rule

If lim(x→a) f(x)/g(x) is 0/0 or ∞/∞ and conditions hold, then lim f/g = lim f′/g′ (if the derivative limit exists or is infinite).

Conditions for L’Hôpital’s Rule

f and g must be differentiable near the limit point, g′(x)≠0 near that point, and the original limit must produce 0/0 or ∞/∞ before applying the rule.

Rechecking the form (L’Hôpital habit)

After each application of L’Hôpital’s Rule, substitute again to see whether the new expression is still indeterminate before applying the rule again.

Rewriting 0·∞ (into a quotient)

Convert an indeterminate product like 0·∞ into a fraction (e.g., x ln x = (ln x)/(1/x)) so it becomes an indeterminate quotient form (∞/∞ or 0/0) where L’Hôpital may apply.

Conjugate method (for ∞−∞)

A technique to rewrite expressions like √(x^2+x)−x by multiplying by the conjugate to eliminate subtraction and produce a quotient that can be simplified or evaluated.

Common L’Hôpital mistake: quotient rule misuse

An error where students differentiate the entire fraction using the quotient rule; L’Hôpital’s Rule requires differentiating numerator and denominator separately (f′/g′), not the derivative of f/g.