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ScotlandMaths

Applications of Algebra and Calculus

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you analyse a function's features to sketch its curve and apply rates of change?

Examine the properties of functions, including domain, asymptotes, symmetry and stationary points, to sketch the graph of a rational or other function, and apply differentiation to rates of change and optimisation problems.
A focused answer to the SQA Advanced Higher Mathematics functions and graph sketching content, covering domain and symmetry, vertical and non-vertical asymptotes of rational functions, stationary points and their nature, the systematic sketching of a curve, and applying differentiation to rates of change and optimisation.
12 min answer →

How do you build a power series for a function from its derivatives at zero?

Find the Maclaurin series expansion of a function using the standard formula, derive the standard expansions of exponential, logarithmic and trigonometric functions, and use known expansions to build the series of composite or product functions.
A focused answer to the SQA Advanced Higher Mathematics Maclaurin series content, covering the Maclaurin formula, deriving a series from successive derivatives at zero, the standard expansions of the exponential, logarithmic, sine and cosine functions, and combining known expansions for composite or product functions.
11 min answer →

How do you work with arithmetic and geometric sequences, their sums, and the convergence of a geometric series?

Work with arithmetic and geometric sequences and series, using the formulae for the nth term and the sum to n terms, the sum to infinity of a convergent geometric series, and the condition for convergence.
A focused answer to the SQA Advanced Higher Mathematics sequences and series content, covering arithmetic sequences and series, geometric sequences and series, the formulae for the nth term and the sum to n terms, the sum to infinity of a convergent geometric series, and the condition for convergence.
11 min answer →

How do you sum a series using standard formulae and prove a result for all positive integers by induction?

Apply the standard summation formulae for the sum of the first n natural numbers, their squares and their cubes, use sigma notation, and prove statements about series, divisibility and inequalities for all positive integers by mathematical induction.
A focused answer to the SQA Advanced Higher Mathematics summation and proof by induction content, covering sigma notation, the standard formulae for the sum of the first n natural numbers, squares and cubes, and the structure of a proof by mathematical induction applied to series, divisibility and inequalities.
12 min answer →

How do you expand a power of a binomial and pick out a specific term?

Use the binomial theorem to expand expressions of the form (a + b) to the power n for a positive integer n, using binomial coefficients, and find a general term or a specific term such as the constant term or the coefficient of a chosen power.
A focused answer to the SQA Advanced Higher Mathematics binomial theorem content, covering binomial coefficients and Pascal's triangle, the full expansion of (a + b) to the power n, the general term formula, and finding a specific term such as the constant term or the coefficient of a given power.
11 min answer →