Violeta CIT

Determine if various combinations of vectors are defined or not.

Simple Indefinite Integrals

Integration by susbtitution. Use the letter C for the constant.

Find the dot product and the angle between two vectors

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

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Multiplication of $2 \times 2$ matrices.

Integration by Substitution

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

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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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Find the solution of a first order separable differential equation of the form $a\sin(x)y'=by\cos(x)$.

Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.

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

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

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Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$

Find the dot product and the angle between two vectors

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Practical questions

Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Solving a system of three linear equations in 3 unknowns using Gauss Elimination in 4 stages. Solutions are all integral.

Linear combinations of $2 \times 2$ matrices. Three examples.

Cofactors Determinant and inverse of a 3x3 matrix.

Find the determinant and inverse of three $2 \times 2$ invertible matrices.

Find the determinant of a $3 \times 3$ matrix.

Elementary Exercises in multiplying matrices. 

Multiplication of $2 \times 2$ matrices.

Simple Indefinite Integrals

Simple Indefinite Integrals