Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by=0$. Complex roots.

Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.

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A quadratic is and a graph of it is given. A tangent is also sketched. The equation of the tangent line is asked for.

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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Linear program described in words. Student must write out constraints as equations in standard form, then identify the optimal solution in a finished simplex tableau.

$f(x)= ae^{-bt}+c$ is given and plotted. A few points are plotted on the curve. $x$-coordinates are provided for two of them and $y$-coordinate provided for third. Student is required to determine other coordinates.

A quadratic is and a graph of it is given. A tangent is also sketch. The equation of the tangent line is asked for.

Solve: $\displaystyle \frac{d^2y}{dx^2}+2a\frac{dy}{dx}+a^2y=0,\;y(0)=c$ and $y(1)=d$.  (Equal roots example).

A quadratic equation (equivalent to $(x+a)^2-b$) is given and sketched. Three equations are given that can be solved using the graph. There is a chance there will only be one solution.

A few quadratic equations are given, to be solved by completing the square. The number of solutions is randomised.

Solve the equation  5x - 8 = 32

student is to re-arrange the equations $pV=nRT$ and $mgh = \frac{1}{2}mv^2$

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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Cramers Rule applied to 3 simultaneous equations

A quadratic is and a graph of it is given. A tangent is also sketch. The equation of the tangent line is asked for.

Question is to calculate the area bounded by a cubic and the $x$-axis. Requires finding the roots by solving a cubic equation. Calculator question

Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.

Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.

rebelmaths

Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.

rebelmaths

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

Cramers Rule applied to 3 simultaneous equations

Write down the Newton-Raphson formula for finding a numerical solution to the equation $e^{mx}+bx-a=0$. If $x_0=1$ find $x_1$.

Included in the Advice of this question are:

6 iterations of the method.

Graph of the NR process using jsxgraph. Also user interaction allowing change of starting value and its effect on the process.