Pure Mathematics 3

Worked examples (5)

• 1Sketch $y = |f(x)|$ for a quadratic (5 marks, P3)
• 2Sketch $y = f(|x|)$ vs $y = |f(x)|$ (5 marks, P3)
• 3Solve $|2x + 1| = |x - 3|$ (5 marks, P3)
• 4State domain and range of a piecewise modulus function (5 marks, P3)
• 5Combined transformation involving modulus (6 marks, P3)

Worked examples (5)

• 1Solve $e^{2x + 1} = 7$ giving an exact answer (3 marks, P3)
• 2Exponential decay — half-life problem (6 marks, P3)
• 3Linearise $y = a x^{n}$ via a $\\ln y$ vs $\\ln x$ plot (6 marks, P3)
• 4Linearise $y = ab^{x}$ from a straight-line plot (6 marks, P3)
• 5Solve $e^{2x} - 5 e^{x} + 6 = 0$ (5 marks, P3)

Worked examples (5)

• 1Solve a modulus equation $|2x - 3| = x + 4$ (5 marks, P3)
• 2Solve a modulus inequality $|3x + 2| u003e 4$ (4 marks, P3)
• 3Composite function $fg(x)$ and domain reasoning (6 marks, P3)
• 4Find an inverse function with domain/range (7 marks, P3)
• 5Reflection $y = f^{-1}(x)$ in $y = x$ (4 marks, P3)

Worked examples (5)

• 1Prove $\\tan^{2}\\theta + 1 = \\sec^{2}\\theta$ from first principles (3 marks, P3)
• 2Solve $\\sec^{2}\\theta = 3 \\tan\\theta - 1$ for $0 \\leq \\theta \\leq 2\\pi$ (6 marks, P3)
• 3Prove the identity $\\cot\\theta + \\tan\\theta \\equiv 2\\csc(2\\theta)$ (5 marks, P3)
• 4Range of $\\arcsin$ and an inverse-trig evaluation (4 marks, P3)
• 5Sketch $y = \\sec x$ and identify asymptotes (5 marks, P3)

Worked examples (5)

• 1Exact value of $\\sin 75°$ using the addition formula (3 marks, P3)
• 2Derive $\\cos 2A$ in three forms and prove an identity (5 marks, P3)
• 3Express $3\\sin\\theta + 4\\cos\\theta$ in the form $R\\sin(\\theta + \\alpha)$ (4 marks, P3)
• 4Use the R-formula to solve and find a maximum (7 marks, P3)
• 5Use $\\tan(A + B)$ to find an exact angle (4 marks, P3)

Worked examples (6)

• 1Differentiate $y = \\sin(2x^{2} + 3)$ (4 marks, P3)
• 2Differentiate $y = e^{3x^{2} - 1}$ (3 marks, P3)
• 3Differentiate $y = \\ln(\\sin x)$ (3 marks, P3)
• 4Prove $\\dfrac{d}{dx}(\\sec x) = \\sec x \\tan x$ (4 marks, P3)
• 5Parametric curve — find $\\dfrac{dy}{dx}$ at a parameter value (5 marks, P3)
• 6Connected rate of change — expanding sphere (6 marks, P3)

Worked examples (6)

• 1Integrate $e^{3x + 2}$ and $\\dfrac{1}{4x - 1}$ (4 marks, P3)
• 2Integrate $\\sin^{2}x$ using double-angle (5 marks, P3)
• 3Integration by substitution: $\\int x \\sqrt{x^{2} + 1}\\,dx$ (5 marks, P3)
• 4Integration by parts: $\\int x e^{x}\\,dx$ (4 marks, P3)
• 5Iterated integration by parts: $\\int x^{2} \\sin x\\,dx$ (6 marks, P3)
• 6Definite integral via substitution (6 marks, P3)

Worked examples (5)

• 1Locate a root by sign change (3 marks, P3)
• 2Fixed-point iteration: rearrange and iterate (6 marks, P3)
• 3Newton–Raphson method (5 marks, P3)
• 4Explain when Newton–Raphson fails (3 marks, P3)
• 5Interpret a cobweb / staircase diagram (4 marks, P3)