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.

A limit converges when nearby function values approach one finite value; it diverges when no finite common value exists. For a polynomial, first principles gives $f'(x)=\lim_{h\to0}[f(x+h)-f(x)]/h$; for $f(x)=x^2$, simplifying produces $2x+h\to2x$. Higher derivatives may be written $d^ny/dx^n$ or $f^{(n)}(x)$. At a point, differentiability implies continuity, but a corner, cusp, vertical tangent or discontinuity prevents a finite two-sided derivative.

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.

At a finite point or at infinity, first substitute and confirm an indeterminate $0/0$ or $\infty/\infty$ form. Then $\lim f/g=\lim f'/g'$ when the rule's conditions and the derivative-ratio limit hold; if the new ratio remains indeterminate, repeat. Alternatively use leading Maclaurin terms, for example $\sin x=x-x^3/6+\cdots$, so $\sin x/x\to1$. Products, differences and powers must first be algebraically converted to an eligible quotient or series form.

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.

Related-rate workflow: write one equation connecting all changing quantities, differentiate with respect to time and only then substitute the instant's values. For a circle, $A=\pi r^2$ gives $dA/dt=2\pi r\,dr/dt$; at $r=3$ and $dr/dt=2$, $dA/dt=12\pi$ square units per unit time. For optimization, include feasible endpoints as candidates when the optimum can occur on the boundary.

Further derivative pairs unlock advanced integrals.

Key derivatives are $(\tan x)'=\sec^2x$, $(\sec x)'=\sec x\tan x$, $(\cosec x)'=-\cosec x\cot x$, $(\cot x)'=-\cosec^2x$, $(a^x)'=a^x\ln a$, $(\log_a x)'=1/(x\ln a)$, $(\arcsin x)'=1/\sqrt{1-x^2}$, $(\arccos x)'=-1/\sqrt{1-x^2}$ and $(\arctan x)'=1/(1+x^2)$.

Example

Because $1/[(x+1)(x+2)]=1/(x+1)-1/(x+2)$, partial fractions give $\int dx/[(x+1)(x+2)]=\ln|x+1|-\ln|x+2|+C$. For $\int dx/[1+(2x+1)^2]$, the linear inner derivative gives $\tfrac12\arctan(2x+1)+C$.

Match an integrand to a derivative pair, include the reciprocal inner-gradient factor for a linear composite, and use partial fractions before integrating a rational expression when required.

Inverse-trig derivative domains and logarithmic absolute values matter; a memorised form without its domain or inner-gradient factor is incomplete.

Advanced integration techniques expose hidden structure.

Substitution reverses a chain rule. Integration by parts reverses a product rule: $\int u\,dv=uv-\int v\,du$; repeat it when the remaining integral still contains a product suited to the same process.

Example

For $\int xe^x\,dx$, choose $u=x$ and $dv=e^x dx$. Then $du=dx$, $v=e^x$, so the integral is $xe^x-\int e^x dx=e^x(x-1)+C$.

Use substitution when one factor is the derivative of an inner function; use parts when differentiating one factor simplifies it. Transform limits as well as the differential in a definite substitution.

Partial fractions belong to AHL 5.15. In this objective, do not mix them into the choice between substitution and integration by parts.

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.

Area between $x=f(y)$ and the $y$-axis from $y=c$ to $y=d$ is $A=\int_c^d|f(y)|\,dy$, splitting where the curve crosses the axis. Rotation about the $x$-axis gives $V=\pi\int_a^b(R(x)^2-r(x)^2)\,dx$; about the $y$-axis use radii expressed in $y$, $V=\pi\int_c^d(R(y)^2-r(y)^2)\,dy$. Sketch the region so the radius is measured perpendicular to the chosen axis.

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.

Method map: separable equations rearrange to $g(y)dy=f(x)dx$; homogeneous $dy/dx=F(y/x)$ uses $y=vx$; linear $y'+P(x)y=Q(x)$ uses integrating factor $\mu=e^{\int P(x)dx}$ so $(\mu y)'=\mu Q$. Euler's method for $y'=f(x,y)$ uses $x_{n+1}=x_n+h$ and $y_{n+1}=y_n+h f(x_n,y_n)$. Apply the initial condition after integration and check the solution in the original equation.

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.

Standard series: $e^x=1+x+x^2/2!+\cdots$, $\sin x=x-x^3/3!+x^5/5!-\cdots$, $\cos x=1-x^2/2!+x^4/4!-\cdots$, $\ln(1+x)=x-x^2/2+x^3/3-\cdots$, and $(1+x)^p=1+px+p(p-1)x^2/2!+\cdots$. Obtain related series by substituting, multiplying, differentiating or integrating within the valid interval; a differential equation can determine successive coefficients after substituting $y=\sum a_nx^n$.