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ScotlandMaths

Relationships and Calculus

6 dot points across 6 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do you find the gradient of a curve at any point, and how does the derivative reveal stationary points and the shape of a graph?

Differentiation of polynomial, root and reciprocal functions and of sine and cosine, the gradient of a curve and the equation of a tangent, increasing and decreasing functions, and stationary points and their nature.
A focused answer to the SQA Higher Mathematics differentiation content, covering differentiating polynomial, root, reciprocal and trigonometric functions, the gradient of a curve and the tangent equation, increasing and decreasing functions, and stationary points and their nature.
10 min answer →

How do you reverse differentiation to find a function from its gradient, and how does the definite integral measure area under a curve?

Integration as the reverse of differentiation, the indefinite integral of polynomial and trigonometric functions with the constant of integration, the definite integral, and the use of integration to find the area under a curve and the area between two curves.
A focused answer to the SQA Higher Mathematics integration content, covering integration as the reverse of differentiation, the indefinite integral of polynomial and trigonometric functions, the constant of integration, the definite integral, and finding the area under a curve and between two curves.
9 min answer →

How do you complete the square, read the discriminant, and use the factor theorem to factorise and solve polynomial equations?

Completing the square and the properties of the quadratic, the discriminant and the nature of the roots, the condition for a quadratic to be always positive or always negative, the factor and remainder theorems, and solving and sketching polynomials.
A focused answer to the SQA Higher Mathematics polynomials and quadratics content, covering completing the square, the discriminant and the nature of the roots, the always positive or always negative condition, the factor and remainder theorems, and solving and sketching polynomials.
9 min answer →

How do the compound angle formulae let you expand and simplify trigonometric expressions, and how does the wave function combine a sine and a cosine into one?

The addition (compound angle) formulae for sine and cosine, the double angle formulae, their use in proving identities and solving equations, and the wave function that expresses a sin x plus b cos x in the form k sin of x plus a.
A focused answer to the SQA Higher Mathematics addition formulae content, covering the compound and double angle formulae for sine and cosine, their use in proving identities and solving equations, and the wave function that writes a sin x plus b cos x as a single sine wave.
9 min answer →

How does the chain rule let you differentiate and integrate a function nested inside another, such as a bracket raised to a power or a sine of a linear expression?

Differentiating composite functions with the chain rule, including expressions of the form a function of a linear expression and sine and cosine of a linear expression, and reversing the process to integrate functions of the form (ax + b) to the n, sin(ax + b) and cos(ax + b).
A focused answer to the SQA Higher Mathematics chain rule content, covering how to differentiate composite functions including powers of brackets and trigonometric functions of a linear expression, and how to reverse the chain rule to integrate (ax + b) to the n, sin(ax + b) and cos(ax + b).
9 min answer →

How do you solve trigonometric equations across a full interval, and how do the identities help you reduce them to a solvable form?

Solving trigonometric equations in degrees and radians over a given interval, using the CAST diagram and the symmetry of the graphs, the trigonometric identities, and equations that reduce to a quadratic in a single trigonometric ratio.
A focused answer to the SQA Higher Mathematics trigonometric equations content, covering solving equations in degrees and radians over a given interval, the CAST diagram and graph symmetry, the trigonometric identities, and equations that reduce to a quadratic in one ratio.
9 min answer →