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Flashcards 2021-12-05 In what form can you express $a\sin x \pm b\cos x$?? $$ R\sin(x \pm \alpha) $$ In what form can you express $a\cos x \pm b\sin x$?? $$ R\cos(x \mp \alpha) $$ You want to express $$a\sin x \pm b\cos x$$ as $$R\sin(x \pm \alpha)$$ How can you calculate $R$?? $$ R = \sqrt{a^2 + b^2} $$ You want to express $$a\sin x \pm b\cos x$$ as $$R\sin(x \pm \alpha)$$ How can you calculate $\alpha$?

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 2021-12-01 What do Kepler’s laws describe?? The motion of orbits. What is Kepler’s First Law?? All of the planets move in elliptical orbits with the sun at one focus. What is Kepler’s Second Law?? The radius vector sweeps out equal areas in equal times. What is the “radius vector” for Kepler’s Second Law?? The imaginary line between the sun and the planet. Which of Kepler’s laws does this show?? Kepler’s second law. What is Kepler’s Third Law?

Flashcards 2021-11-25 How would you write the probability that $B$ occurs, given that $A$ has already occurred?? $$ P(B|A) $$ What is true about $P(A|B)$ for independent events?? $$ P(A|B) = P(A|B') = P(A) $$ How would you calculate $P(A | B)$?? $$ \frac{0.15}{0.15 + 0.25} $$ How would you calculate $P(A | A \cup B)$?? $$ \frac{0.15}{0.4 + 0.15 + 0.25} $$ How would you calculate $P(A' | B')$?? $$ \frac{0.2}{1-0.15-0.25} $$ Why do you have to be careful dealing with conditional probabilities involving things like “given $B'$”?

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 2021-11-20 What is the formula for the maximum friction?? $$ F_\text{max} = \mu R $$ What is $\mu$ when talking about friction?? The coefficient of friction. What is $\mu$ if a surface is described as “smooth”?? $$ 0 $$ If a block is being pushed with a $10\text{N}$ force but there is a $20\text{N}$ max friction, what will happen?? The block will not move. What is the situation called when a block is being pushed with a $20\text{N}$ force but there is a $20\text{N}$ max friction?

Flashcards 2021-11-20 What is the angle beneath the red block?? $$ 3\text{kg} $$ What is the resultant force in this diagram, assuming no resistive forces?? $$ 3g\sin(30^\circ) $$ What is the resultant force in this diagram?? $$ 2g\sin(20^\circ) - 2 $$ What is the resultant force in this diagram?? $$ 5g\cos(60^\circ) - mg\sin(30^\circ) $$ 2022-01-20 What will happen to the direction of friction for an object moving up a slope but accelerating downwards?? It will flip direction when the object flips direction.

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 are the 5 assumptions made for an ideal gas (VVETE)?? There are lots and lots of particles with random velocities. The individual particles have much smaller volumes than the volume of the gas. The collisions with container walls are perfectly elastic. The duration of collisions is much smaller than the duration of not-collisions. The electrostatic forces are negligible expect for during collisions. Why is the assumption that collisions between particles and container walls being perfectly elastic useful?

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 $y$?

See Also [[Further Maths - Radians]]S Flashcards What is the small angle approximation for $\sin \theta$ and $\tan \theta$?? $$ \sin \theta \approx \tan \theta \approx \theta $$ What is the small angle approximation for $\cos \theta$?? $$ 1 - \frac{\theta^2}{2} $$ What is $$\frac{\tan 2\theta}{\sin \theta}$$ for $\theta \approx 0$?? $$ 2 $$ Backlinks [[Maths - Syllabus]]S Metadata date: 2021-11-09 15:30 tags: - '@?public' - '@?

Flashcards What does Avagadro’s constant represent?? The number of atoms in $12$ grams of Carbon-12. What is one mole of a substance?? An amount in which the number of atoms/molecules is equal to Avogadro’s constant. What is the molar mass of a substance?? The mass per mole of a substance. How do you find the mass of one atoms from its mass number/nucleon number?? Divide the mass number by Avogadro’s constant. What is a nucleon?? A proton or a neutron.