SQA Higher Mathematics Area 3 Applications: recurrence relations, the circle and applying calculus

What Area 3 actually demands

Applications takes the algebra and calculus of the first two areas and uses them on richer, often contextual problems. The examiners test whether you can model a situation, choose the right method, and interpret a result in context. This guide walks through all four topics of the area, then sets out the patterns the SQA repeats. Each topic has a matching dot-point page with practice questions; this overview ties them together.

Sequences and recurrence relations

The area opens with recurrence relations of the form $u_{n+1} = au_n + b$. You generate terms, state the condition for a limit ($-1 < a < 1$), and find it from $L = aL + b$, giving $L = \dfrac{b}{1 - a}$. In context, $a$ is the proportion kept each step and $b$ is the fixed amount added, so the limit is the long-term steady value.

The circle

The circle with centre $(a, b)$ and radius $r$ has equation $(x - a)^2 + (y - b)^2 = r^2$, and the general form $x^2 + y^2 + 2gx + 2fy + c = 0$ has centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$. To find where a line meets a circle, substitute and solve the quadratic; its discriminant decides whether the line cuts, touches as a tangent, or misses. A tangent is perpendicular to the radius at the point of contact.

Applying differential calculus

Applying differential calculus covers optimisation: write the quantity as a function of one variable, differentiate, solve $f'(x) = 0$, and check the nature. On a closed interval, the greatest and least values occur at a stationary point or an endpoint. The derivative is a rate of change, so in motion velocity is $\dfrac{ds}{dt}$ and acceleration is $\dfrac{dv}{dt}$.

Applying integral calculus

Applying integral calculus finds the area between curves as $\displaystyle\int_a^b \big(f(x) - g(x)\big)\,dx$ between the intersections, treating regions below the x-axis separately so areas do not cancel. Integrating a rate accumulates a quantity, so integrating velocity recovers displacement and integrating a flow rate gives a total amount.

How Area 3 is examined

A typical SQA profile for Applications:

• Modelling. Setting up a recurrence relation or an optimisation function from a description.
• Coordinate geometry. The circle, its intersection with lines, and tangents.
• Optimisation and rates. Maximum and minimum problems and motion.
• Area and accumulation. Definite integrals for area between curves and total change.

Check your knowledge

A mix of recall and method questions covering Area 3. Attempt them, then check against the solutions.

1. State the condition for $u_{n+1} = au_n + b$ to have a limit. (1 mark)
2. Find the limit of $u_{n+1} = 0.6u_n + 8$. (2 marks)
3. State the centre and radius of $(x - 2)^2 + (y + 3)^2 = 16$. (2 marks)
4. Find the $x$-value that maximises $y = 8x - x^2$. (2 marks)
5. Find $\displaystyle\int_0^2 6x\,dx$. (2 marks)