Mathematics R2: free practice, theory and problems

1. Integration

What is it about?

Integration is the opposite of differentiation. The definite integral gives the area between the graph and the $x$-axis (with sign).

Concepts and formulas

• $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ ($n \ne -1$), $\int \dfrac{1}{x}\,dx = \ln|x| + C$, $\int e^{kx}\,dx = \dfrac{1}{k}e^{kx} + C$.
• $\int \sin x\,dx = -\cos x + C$, $\int \cos x\,dx = \sin x + C$.
• Definite integral: $\int_a^b f(x)\,dx = F(b) - F(a)$.
• Integration by parts: $\int u'v\,dx = uv - \int uv'\,dx$. Substitution: let $u = g(x)$.
• Volume of revolution about the $x$-axis: $V = \pi\int_a^b f(x)^2\,dx$.

Practise integration in the app →

2. Sequences and series

What is it about?

A sequence is a list of numbers. A series is the sum of the terms of a sequence.

Concepts and formulas

• Arithmetic sequence (constant difference $d$): $a_n = a_1 + (n-1)d$, sum $S_n = \dfrac{n(a_1 + a_n)}{2}$.
• Geometric sequence (constant ratio $k$): $a_n = a_1 k^{n-1}$, sum $S_n = a_1\dfrac{k^n - 1}{k - 1}$.
• An infinite geometric series converges when $|k| < 1$, with sum $S = \dfrac{a_1}{1 - k}$.

Practise sequences and series in the app →

3. Differential equations

What is it about?

A differential equation is an equation where the unknown is a function, and the equation contains the derivative. They describe growth, cooling, falling with air resistance and much more.

Concepts and formulas

• $y' = ky$ has the solution $y = Ce^{kx}$. With $y(0) = y_0$, $C = y_0$.
• Separable equations $y' = f(x)g(y)$: separate the variables and integrate both sides: $\int \dfrac{dy}{g(y)} = \int f(x)\,dx$.
• First-order linear $y' + ay = b$: solution $y = \dfrac{b}{a} + Ce^{-ax}$.
• Logistic growth: $y' = ky(M - y)$, where $M$ is the carrying capacity.

Practise differential equations in the app →

4. Trigonometric functions

What is it about?

Sine and cosine functions describe everything that oscillates: sound, tides, alternating current and day length.

Concepts and formulas

• $f(x) = A\sin(cx + \varphi) + d$: amplitude $|A|$, period $p = \dfrac{2\pi}{c}$, equilibrium line $y = d$.
• $(\sin x)' = \cos x$, $(\cos x)' = -\sin x$, $(\tan x)' = \dfrac{1}{\cos^2 x}$.
• Angles in radians: $180^\circ = \pi$.
• $\sin x = a$ has the solutions $x = \sin^{-1}a + n\cdot 2\pi$ and $x = \pi - \sin^{-1}a + n\cdot 2\pi$.

Practise trigonometric functions in the app →

5. Vectors in space

What is it about?

In space, vectors have three coordinates $[x, y, z]$. With the dot product and the cross product you find angles, areas, volumes and equations of planes.

Concepts and formulas

• Length: $|[x, y, z]| = \sqrt{x^2 + y^2 + z^2}$.
• Dot product: $\vec u \cdot \vec v = x_1x_2 + y_1y_2 + z_1z_2$.
• The cross product $\vec u \times \vec v$ is perpendicular to both, and its length is the area of the parallelogram they span.
• Plane through $P(x_0, y_0, z_0)$ with normal vector $[a, b, c]$: $a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$.
• Sphere with centre $(x_0, y_0, z_0)$ and radius $r$: $(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = r^2$.

Practise vectors in space in the app →

6. Integration techniques

What is it about?

Many integrals cannot be read directly from the table. Then you need a technique: substitution (the chain rule backwards), integration by parts (the product rule backwards) or partial fractions.

Concepts and formulas

• Substitution: when you see a function and its derivative, set $u = g(x)$ and $du = g'(x)\,dx$.

$\int 2x(x^2 + 1)^3\,dx$: with $u = x^2 + 1$ this becomes $\int u^3\,du = \dfrac{u^4}{4} + C = \dfrac{(x^2 + 1)^4}{4} + C$.

• Integration by parts: $\int u'v\,dx = uv - \int uv'\,dx$. Choose $v$ as the factor that gets simpler when differentiated (often $x$ or $\ln x$).

$\int xe^x\,dx = xe^x - \int e^x\,dx = (x - 1)e^x + C$.

• Partial fractions: $\dfrac{1}{x(x + 1)} = \dfrac{1}{x} - \dfrac{1}{x + 1}$, so $\int \dfrac{1}{x(x + 1)}\,dx = \ln|x| - \ln|x + 1| + C$.
• Definite integral with substitution: change the limits to $u$-values, or substitute back to $x$ before inserting.

Example

$\int_1^e \ln x\,dx$: by parts with $u' = 1$, $v = \ln x$ gives $[x\ln x - x]_1^e = (e - e) - (0 - 1) = 1$.

Practise integration techniques in the app →

7. Area and volume with integrals

What is it about?

The integral calculates the area under graphs, the area between graphs and the volume of solids formed when a graph is rotated about the $x$-axis.

Concepts and formulas

• Area between $f$ and $g$ where $f \ge g$ on $[a, b]$: $A = \int_a^b \big(f(x) - g(x)\big)\,dx$. The limits are often the intersection points, so solve $f(x) = g(x)$ first.
• If the graph lies below the $x$-axis, the integral gives a negative number. The area is then the absolute value, and you must split at the zeros.
• Volume of revolution about the $x$-axis: $V = \pi\int_a^b f(x)^2\,dx$ (the sum of thin circular discs with radius $f(x)$).
• Volume with known cross-section $A(x)$: $V = \int_a^b A(x)\,dx$.

Example: area between graphs

$y = x$ and $y = x^2$ intersect at $x = 0$ and $x = 1$. On $[0, 1]$, $x \ge x^2$, so $A = \int_0^1 (x - x^2)\,dx = \tfrac12 - \tfrac13 = \tfrac16$.

Example: cone

Rotate $y = \dfrac{r}{h}x$ from 0 to $h$: $V = \pi\int_0^h \dfrac{r^2}{h^2}x^2\,dx = \dfrac{\pi r^2 h}{3}$, the familiar cone formula.

Practise area and volume with integrals in the app →

8. Proof by induction

What is it about?

Induction is a proof technique for claims about all natural numbers $n$. Think of dominoes: if the first one falls, and each one knocks over the next, then all fall.

Method

1. Base case: show that the claim $P(1)$ is true.
2. Induction step: assume that $P(k)$ is true (the induction hypothesis), and show that then $P(k + 1)$ is also true.
3. Conclusion: the claim holds for all $n \ge 1$.

Known sums

• $1 + 2 + \dots + n = \dfrac{n(n + 1)}{2}$
• $1 + 3 + 5 + \dots + (2n - 1) = n^2$
• $1^2 + 2^2 + \dots + n^2 = \dfrac{n(n + 1)(2n + 1)}{6}$
• $1 + 2 + 4 + \dots + 2^{n - 1} = 2^n - 1$

Example

Claim: $1 + 3 + \dots + (2n - 1) = n^2$. Base case: $n = 1$ gives $1 = 1^2$. ✓ Step: assume $1 + 3 + \dots + (2k - 1) = k^2$. Add the next term $2(k + 1) - 1 = 2k + 1$: $k^2 + 2k + 1 = (k + 1)^2$. ✓ The claim then also holds for $k + 1$. $\blacksquare$

Practise proof by induction in the app →

9. Lines and planes in space

What is it about?

In space, lines are described with parametric equations and planes with an equation. With normal vectors and the dot product you find intersection points, angles and distances.

Concepts and formulas

• Line through $P(x_0, y_0, z_0)$ with direction $[a, b, c]$: $x = x_0 + at$, $y = y_0 + bt$, $z = z_0 + ct$.
• Plane: $ax + by + cz + d = 0$, where $\vec n = [a, b, c]$ is a normal vector (perpendicular to the plane).
• Distance from the point $Q(x_1, y_1, z_1)$ to the plane: $\dfrac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$.
• Line–plane intersection: insert the parametric form into the plane equation and solve for $t$.
• Angle between two planes = the angle between the normal vectors (or $180^\circ$ minus it).
• Volume of the parallelepiped spanned by $\vec a, \vec b, \vec c$: $|(\vec a \times \vec b)\cdot \vec c|$.

Example

The line $[1 + t, 2 - t, 3t]$ and the plane $x + y + z = 6$: $(1 + t) + (2 - t) + 3t = 6$ gives $t = 1$, i.e. the point $(2, 1, 3)$.

Practise lines and planes in space in the app →