Antiderivatives

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Definite Intgerals

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Find the general solution of $y''+2py'+(p^2-q^2)y=x$ in the form  $Ae^{ax}+Be^{bx}+y_{PI}(x),\;y_{PI}(x)$ a particular integral.

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Find the solution of $\displaystyle x\frac{dy}{dx}+ay=bx^n,\;\;y(1)=c$

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Find the solution of $\displaystyle \frac{dy}{dx}=\frac{1+y^2}{a+bx}$ which satisfies $y(1)=c$

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Solve 4 first order differential equations of two types:$\displaystyle \frac{dy}{dx}=\frac{ax}{y},\;\;\frac{dy}{dx}=\frac{by}{x},\;y(2)=1$ for all 4.

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This is a homework assignment. Students may complete this assignment during a maths lesson with the support from the teacher. 

• Passing mark 60%
• Difficulty level 3 Moderat/Average (C+, B-, B Level).

Practice of basic transpositions. Doesn't include roots

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Antiderivatives

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Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

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Critical points, absolute minimum, local maximum and minimum points, increasing and decreasing, concavity

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Critical points, absolute minimum, local maximum and minimum points, increasing and decreasing, concavity

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Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

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Match the question with the correct formula.

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Laplace of constants and powers of t

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Laplace transform of e^{at}

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Indices

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Converting a decimal number to binary.

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Converting Binary to Decimal

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Converting a number from binary to decimal.

Equations which can be written in the form

\[\dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(y),   \dfrac{\mathrm{d}y}{\mathrm{d}x} = f(x)f(y)\]

can all be solved by integration.

In each case it is possible to separate the $x$'s to one side of the equation and the $y$'s to the other

Solving such equations is therefore known as solution by separation of variables

Indices

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Integration by Substitution

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Expand $(ax+b)(cx+d)$.

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Expansion of two brackets

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