Mathematics R1: free practice, theory and problems

1. Logarithms and exponential equations

What is it about?

The logarithm answers the question "which exponent do I need?". It is used to solve equations where the unknown is in the exponent, for example in growth and decay.

Concepts and formulas

• $\lg a = x \iff 10^x = a$, and $\ln a = x \iff e^x = a$.
• $\lg(ab) = \lg a + \lg b$, $\lg\dfrac{a}{b} = \lg a - \lg b$, $\lg a^n = n \lg a$ (similarly for $\ln$).
• $e^{kx} = a$ gives $x = \dfrac{\ln a}{k}$.
• Exponential growth: $f(x) = a \cdot b^x$. A growth factor $b > 1$ gives growth, $0 < b < 1$ gives decline.

Example

$3 \cdot 2^x = 48$ gives $2^x = 16$, so $x = \dfrac{\lg 16}{\lg 2} = 4$.

Practise logarithms and exponential equations in the app →

2. Differentiation and curve analysis

What is it about?

In R1 you differentiate more types of functions and use the derivative to analyse the graph: where it increases and decreases, maxima and minima, and inflection points.

Concepts and formulas

• $(e^x)' = e^x$, $(\ln x)' = \dfrac{1}{x}$.
• Product rule: $(uv)' = u'v + uv'$.
• Quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$.
• Chain rule: $\big(g(u(x))\big)' = g'(u) \cdot u'(x)$.
• Maximum/minimum where $f'(x) = 0$ and $f'$ changes sign. Inflection point where $f''(x) = 0$ and $f''$ changes sign.

Example

$f(x) = (2x + 1)^3$ gives $f'(x) = 3(2x + 1)^2 \cdot 2 = 6(2x+1)^2$.

Practise differentiation and curve analysis in the app →

3. Vectors in the plane

What is it about?

A vector has both length and direction. In the plane it is written with coordinates $[x, y]$.

Concepts and formulas

• Vector from $A(x_1, y_1)$ to $B(x_2, y_2)$: $\overrightarrow{AB} = [x_2 - x_1,\ y_2 - y_1]$.
• Length: $|[x, y]| = \sqrt{x^2 + y^2}$.
• Dot product: $\vec u \cdot \vec v = x_1x_2 + y_1y_2 = |\vec u||\vec v|\cos v$.
• $\vec u \perp \vec v \iff \vec u \cdot \vec v = 0$. Parallel vectors: $\vec u = t\vec v$.

Practise vectors in the plane in the app →

4. Probability and combinatorics

What is it about?

Combinatorics counts how many possibilities there are. Probability is favourable outcomes divided by possible outcomes, when all are equally likely.

Concepts and formulas

• Ordered without replacement: $n \cdot (n-1) \cdots (n-k+1)$. Unordered: $\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$.
• Conditional probability: $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$. Independent events: $P(A \cap B) = P(A)P(B)$.
• Bayes' theorem: $P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}$.
• Binomial distribution: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$.

Practise probability and combinatorics in the app →

5. Limits and continuity

What is it about?

A limit describes what $f(x)$ approaches when $x$ approaches a number (or tends to infinity), even if the function is not defined exactly there. Limits are the foundation of differentiation.

Concepts and formulas

• $\lim\limits_{x \to a} f(x) = L$: $f(x)$ gets as close to $L$ as we like when $x$ is close enough to $a$.
• "$\tfrac{0}{0}$" does not mean the answer is 0 or 1. Factorise and cancel first: $\lim\limits_{x \to 3}\dfrac{x^2 - 9}{x - 3} = \lim\limits_{x \to 3}(x + 3) = 6$.
• When $x \to \infty$ in a fraction of polynomials: divide by the highest power in the denominator. $\lim\limits_{x \to \infty}\dfrac{2x + 1}{x - 4} = 2$.
• $f$ is continuous at $a$ when $f(a)$ exists and $\lim\limits_{x \to a} f(x) = f(a)$: the graph can be drawn without lifting the pencil.
• Piecewise function: the two parts must give the same value at the joint for $f$ to be continuous.
• The definition of the derivative: $f'(x) = \lim\limits_{h \to 0}\dfrac{f(x + h) - f(x)}{h}$.

Example

$f(x) = \begin{cases} x^2 + k, & x < 1 \\ 2x + 3, & x \ge 1 \end{cases}$. Continuity at $x = 1$ requires $1 + k = 5$, so $k = 4$.

Practise limits and continuity in the app →

6. Differentiation rules in practice

What is it about?

Here you practise the rules you need to differentiate everything in R1: powers, $e^x$, $\ln x$, and combinations with the product, quotient and chain rules.

Concepts and formulas

• $(x^n)' = nx^{n-1}$, $(e^x)' = e^x$, $(\ln x)' = \dfrac{1}{x}$.
• The chain rule: $\big(e^{kx}\big)' = ke^{kx}$, $\big(\ln u\big)' = \dfrac{u'}{u}$, $\big((ax + b)^n\big)' = na(ax + b)^{n-1}$.
• The product rule: $(uv)' = u'v + uv'$. Example: $(x e^x)' = e^x + xe^x = (1 + x)e^x$.
• The quotient rule: $\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$. Example: $\left(\dfrac{x}{x + 1}\right)' = \dfrac{1}{(x + 1)^2}$.

Method

1. Look at the structure: is it a product, a quotient or a function of a function?
2. Choose the rule and write down $u$, $v$ (or the inner function $u$) and their derivatives.
3. Put it together and simplify. Insert numbers at the very end.

Practise differentiation rules in practice in the app →

7. Optimisation

What is it about?

Optimisation is finding the largest or smallest value of something: largest area, least material, largest profit. The derivative is zero at maxima and minima, so that is where you look.

Method

1. Draw a figure and name the quantities.
2. Write the quantity you want to optimise as a function of one variable. Use the constraint (e.g. a fixed perimeter) to remove the others.
3. Find the domain (lengths cannot be negative).
4. Solve $f'(x) = 0$ and check with a sign chart or $f''(x)$ that it is a maximum or minimum.
5. Check the endpoints and answer the question with units.

Example: a fence against a wall

You have 40 m of fence and want to enclose a rectangle against a wall (the wall is one long side). With width $x$ the length is $40 - 2x$, and the area is $A(x) = x(40 - 2x) = 40x - 2x^2$. $A'(x) = 40 - 4x = 0$ gives $x = 10$ m. Then the length is 20 m and the area $A(10) = 200$ m².

Example: a box from a sheet

From a square sheet with side 30 cm you cut away squares with side $x$ in the corners and fold up. The volume is $V(x) = x(30 - 2x)^2$, where $0 < x < 15$. $V'(x) = 0$ gives $x = 5$ cm and $V = 2000$ cm³.

Practise optimisation in the app →

8. Parametric curves

What is it about?

A parametric representation describes a curve by letting both $x$ and $y$ be functions of a parameter $t$ (often time). Think of a ball or a drone: the position is $\vec r(t) = [x(t), y(t)]$.

Concepts and formulas

• Straight line through the point $P(x_0, y_0)$ with direction vector $[a, b]$: $x = x_0 + at$, $y = y_0 + bt$.
• The velocity vector is the derivative: $\vec v(t) = \vec{r}^{\,\prime}(t) = [x'(t), y'(t)]$. It points along the tangent.
• The speed is its length: $|\vec v(t)| = \sqrt{x'(t)^2 + y'(t)^2}$.
• The acceleration: $\vec a(t) = \vec{v}^{\,\prime}(t)$.
• Circle centred at the origin with radius $r$: $[r\cos t, r\sin t]$.
• The curve crosses the $y$-axis when $x(t) = 0$ and the $x$-axis when $y(t) = 0$.

Example

$\vec r(t) = [2t, t^2]$. Then $\vec v(t) = [2, 2t]$, and at $t = 2$ the speed is $|[2, 4]| = \sqrt{20} \approx 4.47$.

Practise parametric curves in the app →

9. Logic and proof

What is it about?

In mathematics something is true only once it is proved. You must be able to use the symbols for implication and equivalence correctly, and write simple proofs.

Concepts and formulas

• $A \Rightarrow B$ (implication): if $A$ is true, then $B$ is true. Example: $x = 3 \Rightarrow x^2 = 9$.
• $A \Leftarrow B$: the arrow goes the other way. $x^2 = 9 \Leftarrow x = 3$.
• $A \Leftrightarrow B$ (equivalence): both ways. $2x + 1 = 7 \Leftrightarrow x = 3$.
• $x^2 = 9 \Rightarrow x = 3$ is false, because $x = -3$ is a counterexample.
• Direct proof: start from what you know and work your way to the claim.
• Contrapositive proof: $A \Rightarrow B$ is the same as $\text{not } B \Rightarrow \text{not } A$.
• Proof by contradiction: assume the claim is false and show that this leads to something impossible.
• Counterexample: one example is enough to show that a claim is false.

Example: direct proof

Claim: the sum of three consecutive integers is divisible by 3. Proof: $n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)$, which is divisible by 3. $\blacksquare$

Example: even numbers

An even number can be written $2n$ and an odd number $2n + 1$. The sum of two odd numbers: $(2m + 1) + (2n + 1) = 2(m + n + 1)$, i.e. an even number.

Practise logic and proof in the app →