Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.

Integrate the product of two functions by the method of integration by parts.

Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$. Solution involves $\arctan$.

Other method. Find $p,\;q$ such that $\displaystyle \frac{ax+b}{cx+d}= p+ \frac{q}{cx+d}$. Find the derivative of $\displaystyle \frac{ax+b}{cx+d}$.

Solve for $x$ and $y$:  \[ \begin{eqnarray} a_1x+b_1y&=&c_1\\   a_2x+b_2y&=&c_2 \end{eqnarray} \]

The included video describes a more direct method of solving when, for example, one of the equations gives a variable directly in terms of the other variable.

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Students are given six graphs, corresponding to curves $\gamma(t)$. They must match each with its signed curvature function, $\kappa(t)$.

The graphs are generated by calculating $\theta(t)=\int \kappa(t) \mathrm{d}t$ (by hand: these are given to the question as functions of a variable '#', in string form), and solving $x^{\prime}=\cos(\theta(t)-\theta(0))$ and $y^{\prime}(t)=\sin(\theta(t)-\theta(0))$ numerically (using the RKF method) with a JavaScript extension.

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

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.

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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.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Practice dividing polynomials using the long division method.

Find a% of b using a calculator. Suggested method to use decimal equivalent.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

This is the question for Lent Term week 7 of the MA100 course at the LSE. It looks at material from chapters 33 and 34.

The following is a description of parts a and b. In particular it describes the varaibles used for those parts.

This question (parts a and b) looks at optimisation problems using the langrangian method. parts a and b of the question we will ask the student to optimise the objective function f(x,y) = y + (a/b)x subject to the constraint function r^2 = (x-centre_x)^2 + (y-centre_y)^2.

The variables centre_x and centre_y take values randomly chosen from {6,7,...,10} and r takes values randomly chosen from {1,2,...,5}.

We have the ordered set of variables (a,b,c) defined to be randomly chosen from one of the following pythagorean triplets: (3,4,5) , (5,12,13) , (8,15,17) , (7,24,25) , (20,21,29). The a and b variables here are the same as those in the objective function. They are defined in this way because the minimum will occur at (centre_x - (a/c)*r , centre_y - (b/c)*r) with value centre_y - (b/c)r + (a/b) * centre_x - (a^2/bc)*r , and the maximum will occur at (centre_x + (a/c)*r , centre_y + (b/c)*r) with value centre_y + (b/c)r + (a/b) *centre_x + (a^2/bc)r. The minimisation problem has lambda = -c/(2br) and the maximation problem has lambda* = c/(2br).

We can see that all possible max/min points and values are nice rational numbers, yet we still have good randomisation in this question. :)

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Practice dividing polynomials using the long division method.

Questions on the subtleties and pitfalls of methods of data collection.

Questions on the subtleties and pitfalls of methods of data collection.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.

Questions on integration using various methods such as parts, substitution, trig identities and partial fractions.