Sixth Term Examination Paper (STEP) Pure Mathematics: Calculus Syllabus

Describe the method of integration by substitution.

To find $\int f(g(x))g'(x)\,dx$, let $u=g(x)$ so $du=g'(x)\,dx$, giving $\int f(u)\,du$. For definite integrals also change the limits to values of $u$.

State the integration by parts formula.

$$\int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx,\qquad \int_a^b u\,dv=[uv]_a^b-\int_a^b v\,du.$$

What is a reduction formula and how is it typically derived?

A recurrence expressing an integral $I_{n}$ in terms of $I_{n-1}$ (or $I_{n-2}$), usually obtained by integration by parts; repeated application reduces to a base case such as $I_{0}$ or $I_{1}$.

How do you integrate a proper rational function using partial fractions?

Factorise the denominator, express the fraction as a sum of simpler fractions (e.g. $\frac{A}{x-a}+\frac{B}{x-b}$ or $\frac{Ax+B}{x^{2}+px+q}$), then integrate each term (giving logs and arctangents). If improper, divide first.

State the Fundamental Theorem of Calculus for definite integrals.

If $F'(x)=f(x)$ then $\int_a^b f(x)\,dx=F(b)-F(a)=[F(x)]_a^b$.

What does $\int_a^b f(x)\,dx$ represent geometrically, and how is signed area treated?

It is the signed area between the curve $y=f(x)$ and the $x$-axis from $x=a$ to $x=b$; regions below the axis count negatively.

State the symmetry results for $\int_{-a}^{a} f(x)\,dx$ when $f$ is even and when $f$ is odd.

If $f$ is even, $\int_{-a}^{a}f\,dx=2\int_0^a f\,dx$. If $f$ is odd, $\int_{-a}^{a}f\,dx=0$.

Define an improper integral with an infinite upper limit and state what convergence means.

$\int_a^{\infty}f(x)\,dx=\lim_{R\to\infty}\int_a^{R}f(x)\,dx$. It converges if this limit exists (is finite); otherwise it diverges. (Similarly handle infinite discontinuities of the integrand.)

For which values of $p$ does $\int_1^{\infty}\frac{dx}{x^{p}}$ converge, and to what?

It converges iff $p>1$, with value $\frac{1}{p-1}$. For $p\le 1$ it diverges.

How do you find the area between two curves $y=f(x)$ and $y=g(x)$ on $[a,b]$ with $f\ge g$?

$$A=\int_a^b\big(f(x)-g(x)\big)\,dx$$ (upper curve minus lower curve).

Give the formula for the area under a parametric curve $x=x(t)$, $y=y(t)$.

$$A=\int y\,dx=\int_{t_1}^{t_2} y(t)\,\frac{dx}{dt}\,dt$$ with limits taken at the parameter values corresponding to the endpoints.

State the volume of revolution about the $x$-axis for $y=f(x)$ between $x=a$ and $x=b$.

$$V=\pi\int_a^b y^{2}\,dx=\pi\int_a^b \big(f(x)\big)^{2}\,dx$$

State the volume of revolution about the $y$-axis for a region bounded by $x=g(y)$ between $y=c$ and $y=d$.

$$V=\pi\int_c^d x^{2}\,dy=\pi\int_c^d \big(g(y)\big)^{2}\,dy$$

Give the formula for the mean (average) value of $f$ over $[a,b]$.

$$\bar f=\frac{1}{b-a}\int_a^b f(x)\,dx$$

State the arc length formula for $y=f(x)$ from $x=a$ to $x=b$.

$$s=\int_a^b\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx$$

Give the arc length formula for a parametric curve $x=x(t)$, $y=y(t)$.

$$s=\int_{t_1}^{t_2}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}\,dt$$

State the surface area of revolution about the $x$-axis for $y=f(x)$ on $[a,b]$.

$$S=2\pi\int_a^b y\,\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx$$

State the trapezium rule for $\int_a^b y\,dx$ with $n$ strips of width $h=\frac{b-a}{n}$.

$$\int_a^b y\,dx\approx\frac{h}{2}\Big[y_0+y_n+2(y_1+y_2+\cdots+y_{n-1})\Big]$$

State Simpson's rule (with $n$ even strips, $h=\frac{b-a}{n}$).

$$\int_a^b y\,dx\approx\frac{h}{3}\Big[y_0+y_n+4(y_1+y_3+\cdots)+2(y_2+y_4+\cdots)\Big]$$ with 4× on odd-indexed and 2× on even-indexed interior ordinates.

For a concave-up (convex) function, does the trapezium rule over- or under-estimate the integral?

It over-estimates: the chords lie above a convex curve, so the trapezia enclose more than the true area. (Concave-down curves are under-estimated.)

How is a differential equation formed from a physical statement like 'rate of growth is proportional to population'?

Translate the rate into a derivative and the proportionality into a constant: $\frac{dP}{dt}=kP$. More generally, express each described rate as a derivative and equate to the modelled relationship.

Describe the method for solving a first-order separable equation $\frac{dy}{dx}=f(x)g(y)$.

Separate variables: $\frac{1}{g(y)}\,dy=f(x)\,dx$, then integrate both sides: $\int\frac{dy}{g(y)}=\int f(x)\,dx+c$, and solve for $y$ if possible.

What is the integrating factor for the linear ODE $\frac{dy}{dx}+P(x)y=Q(x)$, and the resulting solution form?

Integrating factor $\mu=e^{\int P\,dx}$. Then $\frac{d}{dx}(\mu y)=\mu Q$, so $y=\frac{1}{\mu}\left(\int \mu Q\,dx+c\right)$.

For the second-order linear ODE $a y''+b y'+c y=0$, give the auxiliary equation and the form of the general solution when the roots are real and distinct.

Auxiliary equation $a m^{2}+bm+c=0$. If real distinct roots $m_1,m_2$: $y=Ae^{m_1 x}+Be^{m_2 x}$.

For $a y''+b y'+c y=0$, give the general solution forms for a repeated real root and for complex roots $m=\alpha\pm i\beta$.

Repeated root $m$: $y=(A+Bx)e^{m x}$. Complex roots $\alpha\pm i\beta$: $y=e^{\alpha x}\big(A\cos\beta x+B\sin\beta x\big)$.

What is the difference between initial conditions and boundary conditions for a differential equation?

Initial conditions specify the function (and derivatives) at a single value of the independent variable (e.g. $y(0)$ and $y'(0)$); boundary conditions specify values at two or more distinct points (e.g. $y(0)$ and $y(L)$). Both fix the arbitrary constants.

Write the differential equation for Newton's law of cooling and state its solution.

$\frac{d\theta}{dt}=-k(\theta-\theta_s)$ where $\theta_s$ is ambient temperature; solution $\theta=\theta_s+(\theta_0-\theta_s)e^{-kt}$.

Give the standard ODE for simple harmonic motion and its general solution, and name the model with a damping term.

SHM: $\ddot x+\omega^{2}x=0$ with $x=A\cos\omega t+B\sin\omega t$. Adding a velocity term $\ddot x+2\gamma\dot x+\omega^{2}x=0$ models damped oscillation.

State the Maclaurin series of a function $f(x)$.

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^{n}=f(0)+f'(0)x+\frac{f''(0)}{2!}x^{2}+\cdots$$

Write the Maclaurin series for $e^{x}$, $\sin x$ and $\cos x$.

$e^{x}=\sum_{n\ge0}\frac{x^{n}}{n!}$; $\sin x=\sum_{n\ge0}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!}$; $\cos x=\sum_{n\ge0}\frac{(-1)^{n}x^{2n}}{(2n)!}$.

State the Taylor series of $f$ about a general point $a$.

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^{n}=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^{2}+\cdots$$

Give the binomial series for $(1+x)^{n}$ valid for $|x|<1$ (non-integer $n$).

$$(1+x)^{n}=1+nx+\frac{n(n-1)}{2!}x^{2}+\frac{n(n-1)(n-2)}{3!}x^{3}+\cdots$$ and $\ln(1+x)=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\cdots$ for $-1<x\le1$.

How can a known series be used to evaluate a limit such as $\lim_{x\to0}\frac{\sin x - x}{x^{3}}$?

Expand: $\sin x=x-\frac{x^{3}}{6}+\cdots$, so $\sin x-x=-\frac{x^{3}}{6}+\cdots$, giving $\frac{\sin x-x}{x^{3}}\to-\frac{1}{6}$. Substituting leading terms of expansions resolves the indeterminate form.

How do you obtain a series for a product or composite, e.g. $e^{x}\sin x$ or $e^{\sin x}$, by combining known expansions?

Substitute or multiply the known Maclaurin series and collect terms up to the required power, discarding higher-order terms. E.g. $e^{x}\sin x=(1+x+\tfrac{x^2}{2}+\cdots)(x-\tfrac{x^3}{6}+\cdots)=x+x^{2}+\tfrac{x^{3}}{3}+\cdots$.