NZ Scholarship Calculus cheat sheet

For f∘g, solve x∈dom(g) and g(x)∈dom(f); for f^{-1}, restrict f to an interval where it is one-to-one and swap domain with range.

Rational function asymptotes: vertical at zeros of the denominator that do not cancel; if numerator degree < denominator degree, horizontal y=0; if equal, y=leading coefficient ratio; if numerator degree is one larger, divide to find the oblique asymptote.

Remainder and factor theorems: P(x) divided by (x-a) leaves remainder P(a); (x-a) is a factor exactly when P(a)=0; repeated factor (x-a)^m makes P(a)=P'(a)=...=P^{(m-1)}(a)=0.

Real polynomial conjugate roots: non-real roots occur in conjugate pairs with equal multiplicity; an odd-degree real polynomial has at least one real root.

Log and exponential: ln(e^x)=x for all real x, e^{ln x}=x for x>0; a^x=e^{x ln a}; log(xy)=log x+log y only when x,y>0.

Inequality solution rule: move all terms to one side and use a sign table at zeros and undefined points; do not multiply by an expression of unknown sign unless split into cases.

Absolute value: |x|<a (a>0) iff -a<x<a; |x|>a iff x<-a or x>a; |x-a|<r is the interval (a-r,a+r).

Partial fractions: divide first if degree numerator ≥ denominator; for distinct linear factors use A/(x-a); repeated (x-a)^m uses A_1/(x-a)+...+A_m/(x-a)^m; irreducible quadratic gets (Bx+C)/(ax²+bx+c).

Transformations: y=af(b(x-h))+k gives horizontal scale 1/|b|, reflection if b<0, vertical scale |a|, reflection if a<0, then shift right h and up k; apply horizontal changes before translations.

Derivative definition: f'(x)=lim_{h→0}[f(x+h)-f(x)]/h; at a point this is the slope of the tangent and instantaneous rate of change.

Rules: (uv)'=u'v+uv'; (u/v)'=(u'v-uv')/v²; d/dx f(g(x))=f'(g(x))g'(x); d/dx [f(x)]^{g(x)} via logarithmic differentiation.

Standard derivatives: d/dx x^n=nx^{n-1}; d/dx e^{kx}=ke^{kx}; d/dx ln x=1/x; d/dx sin x=cos x; d/dx cos x=-sin x; d/dx tan x=sec²x; d/dx arctan x=1/(1+x²).

Parametric curve: dy/dx=(dy/dt)/(dx/dt) when dx/dt≠0; d²y/dx²=[d/dt(dy/dx)]/(dx/dt); vertical tangent when dx/dt=0 and dy/dt≠0.

Implicit differentiation: differentiate F(x,y)=0 term by term, using d/dx y = y' and d/dx y²=2yy', then solve algebraically for y'.

Extrema on [a,b]: candidates are endpoints and points where f'=0 or f' is undefined; compare f values; second derivative test f''(c)>0 gives local minimum, f''(c)<0 local maximum.

Monotonicity and concavity: f'>0 on an interval means increasing, f'<0 decreasing; f''>0 concave up, f''<0 concave down; inflection requires a change in sign of f''.

Sketching checklist: domain, intercepts, symmetry, asymptotes, critical points, intervals of increase/decrease, concavity, and behaviour as x→±∞.

Mean value theorem: if f is continuous on [a,b] and differentiable on (a,b), some c in (a,b) satisfies f'(c)=[f(b)-f(a)]/(b-a); Rolle's theorem is the case f(a)=f(b).

FTC: if f is continuous on [a,b] and F'=f, then ∫_a^b f(x)dx=F(b)-F(a); d/dx∫_a^x f(t)dt=f(x).

Standard antiderivatives: ∫x^n dx=x^{n+1}/(n+1)+C for n≠-1; ∫1/x dx=ln|x|+C; ∫e^{kx}dx=e^{kx}/k+C; ∫cos x dx=sin x+C; ∫sin x dx=-cos x+C.

Substitution: ∫f(g(x))g'(x)dx=∫f(u)du with u=g(x); for definite integrals change limits to g(a),g(b) or substitute back before evaluating.

Integration by parts: ∫u dv=uv-∫v du; choose u by LIATE preference (log, inverse trig, algebraic, trig, exponential) and dv as the remaining factor.

Trig integrals: for ∫sin^m x cos^n x dx, if m or n is odd reserve one factor for substitution; if both even use power-reduction identities; ∫sec²x dx=tan x+C.

Trig substitutions: √(a²-x²) use x=a sinθ; √(a²+x²) use x=a tanθ; √(x²-a²) use x=a secθ, then simplify with identities and convert back.

Partial fractions: decompose proper rational functions before integrating; distinct linear factors give logarithms, repeated factors give powers and logs, irreducible quadratics give arctangents after completing the square.

Area and volume: area between f≥g on [a,b] is ∫_a^b(f-g)dx; rotating y=f(x) about the x-axis gives V=π∫_a^b f(x)²dx; about y=k use radius |f(x)-k|.

Improper integrals: replace an infinite limit or singular point by a limit variable; convergence means the limit is finite; ∫_1^∞ 1/x^p dx converges iff p>1.

Pythagorean and reciprocal identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=csc²θ; secθ=1/cosθ, cscθ=1/sinθ, cotθ=cosθ/sinθ.

Sum and difference: sin(A±B)=sinA cosB±cosA sinB; cos(A±B)=cosA cosB∓sinA sinB; tan(A±B)=(tanA±tanB)/(1∓tanA tanB).

Double-angle: sin2θ=2sinθcosθ; cos2θ=cos²θ-sin²θ=2cos²θ-1=1-2sin²θ; tan2θ=2tanθ/(1-tan²θ).

General solutions: sinθ=s gives θ=nπ+(-1)^n arcsin s; cosθ=c gives θ=2nπ±arccos c; tanθ=t gives θ=nπ+arctan t, for integer n and valid s,c.

Inverse trig ranges: arcsin x has range [-π/2,π/2]; arccos x has range [0,π]; arctan x has range (-π/2,π/2); use these to resolve multiple angles.

Polar form: z=r(cosθ+i sinθ)=re^{iθ}, r=|z|≥0; multiplication multiplies moduli and adds arguments; division divides moduli and subtracts arguments modulo 2π.

De Moivre and roots: z^n=r^n(cos nθ+i sin nθ); nth roots are r^{1/n}[cos((θ+2kπ)/n)+i sin((θ+2kπ)/n)] for k=0,...,n-1.

Complex conjugates and quadratics: conjugate of a+bi is a-bi; z times its conjugate is |z|²; for real coefficients, non-real roots occur in conjugate pairs; quadratic formula gives roots directly.

Argand loci: |z-a|=r is a circle centre a radius r; arg(z-a)=θ is a ray from a at angle θ excluding a; |z-a|=|z-b| is the perpendicular bisector of segment ab.