Calculus

You will be able to

• Explain how the gradient of a chord tends to the gradient of the tangent as the chord shortens, and what the derived function f′(x) measures.
• Use the derivatives of xⁿ, sin x, cos x, eˣ and ln x, with the chain rule for simple composites, to find the gradient of a curve at a point.
• Find the equations of the tangent and the normal to a curve at a given point.

You will be able to

• Identify whether an expression is a product, a quotient or a composite function, and so which rule to apply first.
• Use the chain, product and quotient rules, with the derivatives of xⁿ, eˣ, ln x, sin x, cos x and tan x, to differentiate an expression.
• Evaluate a derivative at a point and check it against the gradient of the tangent there.

You will be able to

• Find the coordinates of the stationary points of a curve by solving dy/dx = 0, giving exact values where they are exact.
• Determine the nature of each stationary point using the sign of d²y/dx², or the sign of dy/dx either side of it.
• Explain why d²y/dx² = 0 does not classify a point, and use the first-derivative test to identify a stationary point of inflection.

You will be able to

• Use the chain rule to connect rates of change, such as dr/dt = dr/dV × dV/dt, and evaluate them at a given instant.
• Use the approximation δy ≈ (dy/dx)δx to estimate a small change in y, or an approximate value such as √4.1.
• Explain why the approximation improves as δx gets smaller, using the tangent and the curve.

You will be able to

• Form an expression for a quantity to be maximised or minimised in terms of one variable, and state the valid interval for that variable.
• Find the maximum or minimum value by solving dV/dx = 0, rejecting any solution outside the valid interval with a reason.
• Justify that a stationary value is a maximum or a minimum using the sign of the second derivative.

You will be able to

• Evaluate a definite integral exactly as [F(x)] from a to b, using integration as the reverse of differentiation.
• Explain why a region below the x-axis gives a negative integral, and find a total area by splitting the integral at the roots.
• Find the area enclosed between a straight line and a curve as the integral of (top − bottom) between their intersections.

You will be able to

• Find velocity and acceleration by differentiating displacement or velocity with respect to time, and displacement by integrating velocity with a constant found from the initial conditions.
• Determine when a particle is instantaneously at rest by solving v = 0, and find its acceleration at that moment.
• Calculate the total distance travelled when the velocity changes sign, and explain why it differs from the displacement.