5.2 Calculus - AHL content

Continuity and differentiability are local conditions.

Continuity requires the function value and both one-sided limits to agree; differentiability additionally requires matching finite slopes.

Example

|x| is continuous at 0 but not differentiable there because its left slope is −1 and right slope is 1.

Check the function, limit and derivative separately at the suspected point.

Differentiability implies continuity, but continuity alone does not imply differentiability.

L’Hôpital’s rule compares leading limiting behaviour.

For suitable 0/0 or ∞/∞ forms, the ratio limit can equal the ratio of derivatives; the indeterminate form must be identified first.

Example

limₓ→0 sin x/x=lim cos x/1=1.

Verify the form and conditions before differentiating, then simplify and re-check the limit.

The rule is not a general quotient shortcut and does not apply directly to every finite ratio.

Implicit differentiation keeps both variables changing.

Differentiate each term with respect to x, treating y as y(x), so every y term contributes a factor dy/dx.

Example

From x²+y²=25, 2x+2y y′=0, hence y′=−x/y where y≠0.

Collect dy/dx terms, then substitute the point only after differentiating.

A vertical tangent can make the solved slope undefined even though the curve is smooth.

Further calculus combines rules with the original question.

Higher derivatives, substitutions and mixed integrals are tools; choose them from the structure and required quantity rather than from a memorised sequence.

Example

For y=e^(x²), y′=2xe^(x²) and y″=(2+4x²)e^(x²), showing why the chain and product rules recur.

Annotate the outer and inner functions, then check whether the result is a rate, area or approximation.

More algebra is not automatically more accurate; domain and constants still govern the answer.

Advanced integration techniques expose hidden structure.

Substitution reverses a chain rule, integration by parts reverses a product rule, and partial fractions split rational terms into integrable pieces.

Example

∫2x cos(x²)dx uses u=x² and becomes sin(x²)+C.

Choose the technique that simplifies the derivative pattern, then back-check by differentiating.

A substitution must transform the differential as well as the expression.

Volumes of revolution depend on slices and axis choice.

Rotating a region creates disks, washers or shells; the radius and thickness must match the axis and variable of integration.

Example

Rotating y=x from 0 to 1 about the x-axis gives V=π∫₀¹x²dx=π/3.

Sketch the region, identify outer/inner radius or shell height, and state the limits.

Using a radius measured from the wrong axis can produce a plausible but incorrect volume.

A differential equation becomes a model after a condition selects a solution.

The equation states a rate relationship; solving it gives a family, and an initial or boundary condition chooses the member that fits the system.

Example

dy/dx=2y with y(0)=3 gives y=3e^(2x), not the whole family Ce^(2x).

Separate variables or use the appropriate method, then substitute the condition and check the derivative.

A mathematical solution can still be physically invalid if it violates domain or sign constraints.

A Maclaurin series approximates a function near zero.

The series uses derivatives at 0: f(x)=f(0)+f′(0)x+f″(0)x²/2!+…; truncation creates an approximation error.

Example

e^x≈1+x+x²/2 for small x, so e^.1≈1.105 using three terms.

State the expansion point and order, then judge whether the input lies in a range where the truncation is useful.

A convergent series can still be inaccurate when too few terms are used far from the expansion point.