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SingaporeMaths

Calculus

9 dot points across 9 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do we locate and classify stationary points and solve optimisation problems?

Find and classify stationary points, determine increasing and decreasing intervals and concavity, and solve optimisation problems in context
A focused answer to the H2 Mathematics outcome on applications of differentiation. Finding stationary points, classifying them with the first and second derivative tests, concavity and points of inflexion, and optimisation.
10 min answer →

How does the definite integral measure area, and how do we handle areas below the axis or between curves?

Evaluate definite integrals, use them to find the area under a curve and between curves, and apply the fundamental theorem of calculus
A focused answer to the H2 Mathematics outcome on definite integrals and area. The fundamental theorem, evaluating definite integrals, signed area below the axis, and area between two curves.
10 min answer →

How do we solve first-order differential equations and interpret their solutions?

Solve first-order differential equations by direct integration and by separating variables, find particular solutions from conditions, and interpret solutions in context
A focused answer to the H2 Mathematics outcome on differential equations. Solving by direct integration and separation of variables, applying boundary conditions for particular solutions, and modelling growth and decay.
10 min answer →

How do the product, quotient and chain rules let us differentiate any combination of standard functions?

Differentiate standard functions and use the product, quotient and chain rules to differentiate products, quotients and composite functions
A focused answer to the H2 Mathematics outcome on differentiation techniques. The derivatives of standard functions, and the product, quotient and chain rules, with combined applications.
9 min answer →

How do we differentiate when y is defined implicitly or through a parameter?

Differentiate relations defined implicitly and curves defined parametrically, and find gradients, tangents and second derivatives in each case
A focused answer to the H2 Mathematics outcome on implicit and parametric differentiation. Differentiating implicit relations, finding dy/dx parametrically via the chain rule, and obtaining tangents and second derivatives.
10 min answer →

How do substitution, integration by parts and partial fractions extend what we can integrate?

Integrate standard functions and use substitution, integration by parts and partial fractions to evaluate a wide range of integrals
A focused answer to the H2 Mathematics outcome on integration techniques. Standard integrals, integration by substitution, integration by parts, and integrating rational functions via partial fractions.
10 min answer →

How does the Maclaurin series approximate a function as a power series, and how do we use the standard expansions?

Derive and use the Maclaurin series of a function, apply the standard series for common functions, and use series to obtain approximations
A focused answer to the H2 Mathematics outcome on the Maclaurin series. The general formula, deriving a series from repeated differentiation, the standard expansions, combining them, and approximating function values.
10 min answer →

How do derivatives give tangent and normal lines and connect related rates of change?

Find equations of tangents and normals to curves, and solve connected rates of change problems using the chain rule
A focused answer to the H2 Mathematics outcome on tangents, normals and related rates. Finding tangent and normal equations, the perpendicular gradient relation, and linking rates of change through the chain rule.
9 min answer →

How does integration give the volume of a solid formed by rotating a region about an axis?

Find volumes of revolution generated by rotating a region about the x-axis or y-axis, including the volume between two curves
A focused answer to the H2 Mathematics outcome on volumes of revolution. The disc formula for rotation about each axis, setting up the integral, and the volume of a region between two curves.
9 min answer →