Calculus: worked exam questions

Area Between a Curve and a Line

Find the area enclosed between a curve and a line by integrating the difference, a 0606 calculus question that starts with finding the intersection points.

Area Between a Curve and the x-axis

A 0606 definite-integral question where the region sits below the x-axis, worked to the mark, including why the integral comes out negative and how to report area.

Connected Rates of Change: Expanding Circle

A circle's radius grows at a known rate; find how fast the area grows using the chain rule dA/dt = dA/dr · dr/dt, a 0606 connected-rates question.

Connected Rates of Change: Expanding Sphere

A 0606 Paper 2 connected-rates question worked step by step using the chain rule, the topic students most often freeze on, made mechanical.

Connected Rates: Filling a Cylindrical Tank

Find how fast the water level rises in a cylindrical tank from the rate of volume increase, a 0606 connected-rates question with a clean chain rule.

Differentiation: Product, Quotient and Chain Rule

Three short 0606 differentiation parts, one each for the product, quotient and chain rule, worked with the factorising step examiners expect in the answer.

Finding the Equation of a Curve from Its Gradient

Given dy/dx and a point on the curve, integrate and find the constant to recover y, a 0606 question that tests integration plus the +C step students forget.

Finding Where a Tangent Is Parallel to a Line

Find the point where a curve's tangent is parallel to a given line by matching gradients, a 0606 calculus question linking differentiation to straight lines.

Greatest and Least Values on a Closed Interval

Find the greatest and least values of a cubic on a closed interval by comparing stationary points with the endpoints, a 0606 calculus question.

Integrating a Bracket Raised to a Power

Integrate a linear bracket raised to a power using the reverse chain rule, then evaluate a definite integral, a core 0606 integration technique.

Kinematics: Finding Velocity from Acceleration

Integrate acceleration to get velocity and displacement, using initial conditions to fix the constants, a core 0606 kinematics question.

Kinematics: Total Distance from a Velocity Function

A 0606 kinematics question: find when a particle is at rest, then the total distance travelled, the part where distance and displacement differ.

Maximum Volume of an Open Box

Maximise the volume of an open box made by cutting squares from a sheet, a classic 0606 optimisation question with a rejected root to watch for.

Optimisation: Minimum Surface Area of a Cylinder

A 0606 applied maximum/minimum question: form the surface-area function from a volume constraint, then minimise it and prove it is a minimum.

Small Changes and Approximation

Use the derivative as a rate to approximate a small change in y, the standard 0606 'small increments' technique with δy ≈ (dy/dx)·δx.

Stationary Points and the Second Derivative Test

Find both stationary points of a cubic and classify each as a maximum or minimum using the second derivative, a standard 0606 calculus question.

Tangent and Normal to a Cubic Curve

A full 0606 Paper 1 question worked to the mark: differentiate a cubic, then find the tangent and normal at a point, in the exact form examiners want.

Tangent to an Exponential Curve

Find the equation of the tangent to y = e^(2x) at a point, differentiating the exponential and using the chain rule, a 0606 calculus question.

Velocity and Acceleration from Displacement

Differentiate a displacement function to find velocity and acceleration, then find the velocity when acceleration is zero, a 0606 kinematics question.

Where a Function Is Increasing or Decreasing

Find the intervals where a cubic is increasing or decreasing using the sign of dy/dx, a 0606 calculus question that ends in a quadratic inequality.