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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?

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?

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 2021-10-09 What is the $t$-substitution ($t = …$)?? $$ t = \tan^{-1}\left(\frac{\theta}{2}\right) $$ What is $\sin\theta$ in terms of $t$?? $$ \sin\theta = \frac{2t}{1 + t^2} $$ What is $\cos\theta$ in terms of $t$?? $$ \cos\theta = \frac{1 + t^2}{1 - t^2} $$ What is $\tan\theta$ in terms of $t$?? $$ \tan\theta = \frac{2t}{1 - t^2} $$ What triangle can you imagine for deriving the $t$-formulae?? 2021-10-12 What $t$-substitution could you make other than $t = \tan\left(\frac{\theta}{2}\right)$ in order to rewrite $\sin 2\theta$?

Flashcards Parabolas How can you form a parabola from a cone?? Slice it parallel to its slope. Why must you slice a cone PARALLEL to the slope to form a parabola?? Otherwise you’d either get an ellipse or intersect the cone twice and get a hyperbola. What is the parametric equation that defines a parabola?? $$ x = at^2 $$ $$ y = 2at $$ When thinking about parabolas as a conic section, is it better to think of them symmetrical around the $x$-axis or $y$-axis?

Flashcards Backlinks [[Further Maths - Syllabus]]S Metadata date: 2021-10-05 16:31 tags: - '@?public' - '@?school' - '@?further-maths' - '@?inequalities' - '@?further-pure-1' - '@?safe-to-post-online' title: Further Maths - Inequalities