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A Level Pure 2 Essential Skills
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Differentiation
Find the gradient at a point
\[ \text{Gradient of } y = x^3 – 2x \text{ at } x = 2 \]
Find the derivative and evaluate it at a given x-coordinate.
Find stationary points
\[ \frac{dy}{dx} = 0 \]
Set the derivative equal to zero and solve to find stationary points.
Integration
Integrate x^n (n ≠ -1)
\[ \int x^4 \, dx \]
Integrate a power of x where the power is not -1.
Integrate e^(kx)
\[ \int e^{2x} \, dx \]
Integrate an exponential function with base e.
Integrate 1/x
\[ \int \frac{3}{x} \, dx \]
Integrate expressions of the form k/x.
Integrate sin(kx) and cos(kx)
\[ \int \cos(3x) \, dx \]
Integrate sine or cosine with a linear argument.
Evaluate a definite integral
\[ \int_1^3 2x \, dx \]
Calculate the exact value of a definite integral with given limits.
Integration by substitution (linear)
\[ u = 2x + 1 \]
Use a given linear substitution to evaluate an integral.
Integration by parts
\[ \int x e^x \, dx \]
Apply the integration by parts formula to evaluate an integral.
Find area under a curve
\[ \text{Area} = \int_a^b y \, dx \]
Calculate the area under a curve between two x-values.
Exponentials and Logarithms
Use laws of logarithms
\[ \ln(8) + \ln(3) \]
Apply the laws of logarithms to simplify or combine expressions.
Solve exponential equations
\[ e^{2x} = 7 \]
Solve equations involving e^x by taking natural logarithms.
Solve logarithmic equations
\[ \ln(2x + 1) = 4 \]
Solve equations involving ln(x) by exponentiating.
Trigonometry
Convert between degrees and radians
\[ 60° \to \text{radians} \]
Convert angles from degrees to radians or vice versa.
Use exact trigonometric values
\[ \sin\left(\frac{\pi}{3}\right) \]
State exact values of sin, cos, or tan for standard angles.
Use the identity sin²x + cos²x = 1
\[ \sin^2 x + \cos^2 x = 1 \]
Apply the Pythagorean identity to find one trig ratio from another.
Use compound angle formulae
\[ \sin(75°) \]
Apply sin(A±B) or cos(A±B) formulae to evaluate expressions.
Use double angle formulae
\[ \cos(2x) = 2\cos^2 x – 1 \]
Apply double angle formulae for sin, cos, or tan.
Solve trigonometric equations (basic)
\[ \sin(x) = 0.5 \]
Solve a basic trigonometric equation in a given interval.
Solve trigonometric equations (transformed)
\[ \sin(2x) = \frac{\sqrt{3}}{2} \]
Solve trigonometric equations with a coefficient of x.
Solve sin(x) = -k
\[ \sin(x) = -\frac{1}{2} \]
Solve trig equations with negative values.
Sequences and Series
Use binomial expansion (positive integer n)
\[ (1 + 2x)^5 \]
Expand (1+bx)^n using the binomial theorem for positive integer n.
Use binomial expansion (fractional/negative n)
\[ (1 + x)^{-2} \]
Expand (1+x)^n for fractional or negative n.
Find the range of validity
\[ |x| < \frac{1}{3} \]
Determine the values of x for which a binomial expansion converges.
Numerical Methods
Show a root exists in an interval
\[ f(a) \text{ and } f(b) \text{ have opposite signs} \]
Use sign change to demonstrate existence of a root.
Use iterative formulae
\[ x_{n+1} = \sqrt{5 – x_n} \]
Apply a given iterative formula repeatedly to find successive approximations.
Use the trapezium rule
\[ \frac{h}{2}[y_0 + 2(y_1 + …) + y_n] \]
Apply the trapezium rule formula to estimate a definite integral.
Proof
Disprove by counter-example
\[ n=0: 0^2 \not> 0 \]
Find a counter-example to disprove a mathematical statement.
Prove by deduction (algebraic)
\[ n + (n+1) = 2n + 1 \]
Construct an algebraic proof of a general statement.
Parametric Equations
Eliminate the parameter
\[ x = t + 1, \; y = t^2 \]
Convert parametric equations to a Cartesian equation.
Find dy/dx for parametric curves
\[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
Use the chain rule to find the gradient of a parametric curve.
Differential Equations
Solve dy/dx = f(x)
\[ \frac{dy}{dx} = 3x^2 \]
Solve a simple differential equation by direct integration.
Separate variables
\[ \frac{dy}{dx} = xy \]
Solve a separable first-order differential equation.
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