@?further-maths

Flashcards 2022-02-17 Metadata date: 2022-02-17 17:04 tags: - '@?public' - '@?school' - '@?further-maths' - '@?decision' - '@?year-2' title: Further Maths - Crticial Path Analysis

Flashcards 2022-01-25 What is the cross product useful for?? Finding a normal vector to two other vectors. What is another name for the cross product?? The vector product. Does the cross product give you a scalar or a vector answer?? A vector. How can you work out the direction of the cross product of two vectors?? Using the right hand rule. In the right hand rule for working out the direction of the cross product, what is the first vector $a$ represented by?

Flashcards 2022-01-19 What’s the reducible differential equations topic about?? Transforming complicated differential equations into simpler ones using a substitution. If $z = \frac{y}{x}$, or $y = xz$ what is $\frac{\text{d}y}{\text{d}x}$?? $$ \frac{\text{d}y}{\text{d}x} = z + x\frac{\text{d}z}{\text{d}x} $$ What’s the first stage in doing a first-order reducible differential equations question?? Working out how to substitute the derivative by differentiating. If a reducible differential equations question asks you to substitute $z = \frac{y}{x}$ then what is the first step?

See Also [[Maths - Numerical Methods]]S Try out an interactive visualisation of Euler’s method here: Euler’s method. Flashcards 2021-12-08 How could you summarise Euler’s method for solving first-order differential equations?? Start with some point on the curve and then follow the direction of the curve. If a gradient is given by $\frac{\text{d}y}{\text{d}x}$, how much would you increase the $y$-coordinate for a step size of $h$?? $$ y_1 = y_0 + \frac{\text{d}y}{\text{d}x} h $$

See Also [[Further Maths - L'Hôpital's Rule]]S [[Further Maths - Taylor Series]]S Flashcards 2021-12-05 How would you rewrite $$\lim_{x \to \infty} \frac{2-3x}{1+x}$$ in order to evaluate it without L’Hôpital’s rule?? $$ \frac{\lim_{x \to \infty} 2 - 3x}{\lim_{x \to \infty} 1 + x} $$ How could you evaluate $$\lim_{x \to \pi/2} (x-\frac{\pi}{2})\tan x$$ using a Taylor series?? Approximate $\cot x$ around $\pi/2$ What must you make sure to do when evaluating a limit with a Taylor series?

See Also [[Further Maths - Maclaurin Series]]S [[Further Maths - Limits]]S Flashcards 2021-12-01 What is the Maclaurin series a special case of?? The Taylor series. What is the formula for the Taylor series about $x = a$?? $$ f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + … $$ When is the Taylor series valid for $x = a$?? When $f^{(n)}(a)$ exists and is finite for all natural numbers and for values of $x$ for which the infinite series converges.

Flashcards 2021-12-01 What is the Weierstrass substitution?? The substitution $t = \tan\frac{x}{2}$ used to evaluate integrals. What do you substitute for $\text{d}x$ in the Weierstrass substitution?? $$ \text{d}x = \frac{2}{1 + t^2} \text{d}t $$ Why is the Weierstrass substitution useful?? Because it turns complicated integration of trig functions into a rational function. What technique would you use for evaluating $$\int^{\pi/2}_{pi/3} \frac{1}{1 + \sin x - \cos x} \text{d}x$$?? The Weierstrass substitution. $$\int \csc (x) \text{d}x$$ Can you make the Weierstrass substitution?

Flashcards 2021-11-26 How could you formulate the travelling salesperson problem in terms of Hamiltonian cycles?? Finding the shortest Hamiltonian cycle on a graph. For small graphs, how can you solve the travelling salesperson problem?? Try every Hamiltonian cycle. What is the nearest neighbour algorithm used for?? Finding a Hamiltonian cycle that can be used as an upper bound for the TSP. What are the 4 steps of the nearest neighbour algorithm?? Choose any start vertex Go to the nearest vertex that hasn’t been included Repeat 2 until all vertices have been included Return directly to the start vertex What would be the first step in using the nearest neighbour algorithm on this distance matrix?

See Also [[Further Maths - Limits]]S Flashcards 2021-11-25 What is L’Hôpital’s rule used for?? Finding the limit of two functions divided together. What is $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ equivalent to?? $$ \lim_{x \to a} \frac{f'(x)}{g'(x)} $$ What technique could you use for finding the value of $$\frac{\sin(x)}{x}$$ at $x = 0$?? L’Hôpital’s rule. What are the conditions for applying L’Hôpital’s rule for $$\lim_{x \to a} \frac{f(x)}{g(x)}$$?? $$ \frac{f(x)}{g(x)} = \frac{0}{0} $$

Flashcards 2021-11-24 What is the derivative of $$y = uv$$?? $$ u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x} $$ What is the derivative of $$\frac{\text{d}y}{\text{d}x} = u \frac{\text{d}v}{\text{d}x} + v \frac{\text{d}u}{\text{d}x}$$?? $$ u\frac{\text{d}^2v}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x} + v\frac{\text{d}^2u}{\text{d}x^2} + \frac{\text{d}u}{\text{d}x}\frac{\text{d}v}{\text{d}x} $$ What does Leibnitz’s theorem give a general formula for?? The $n$th derivative of the product of two functions. What is $\frac{\text{d}^0v}{\text{d}x^0}$?? $$ v $$ What is the formula from Leibnitz’s theorem for the $n$th derivative of $y = uv$?

Flashcards What are the three values you should store for each vertex in Dijkstra’s?? Once you’ve labelled the end vertex with a permanent label in Dijkstra’s, what can you do?? Stop and save yourself the work of labelling every vertex. What does Dijkstra’s algorithm do?? Tell you the shortest distance from any node to the start node. At each step of Dijkstra’s algorithm, what should you do?? Look at the verticies connected to the most recently permanently labelled node.

Flashcards What is the $y = x$ equivalent for polar coordinates?? $$ r = \theta $$ What is $x$ in terms of $r$ and $\theta$ for polar coordinates?? $$ x = r\cos\theta $$ What is $y$ in terms of $r$ and $\theta$ for polar coordinates?? $$ y = r\sin\theta $$ What is $r$ in terms of $x$ and $y$ for polar coordinates?? $$ r = \sqrt{x^2 + y^2} $$ What is $\theta$ in terms of $x$ and $y$?