@?further-maths

See Also [[Further Maths - Mean Value of a Function]]S [[Further Maths - Improper Integrals]]S [[Further Maths - Hyperbolic Functions]]S Flashcards $$\frac{d}{dx}(\sin^{-1}(x))$$ What is this equal to?? $$ \frac{1}{\sqrt{1 - x^2}} $$ $$\frac{d}{dx}(\cos^{-1}(x))$$ What is this equal to?? $$ -\frac{1}{\sqrt{1 - x^2}} $$ $$\frac{d}{dx}(\tan^{-1}(x))$$ What is this equal to?? $$ \frac{1}{1 + x^2} $$ $$\int \frac{1}{\sqrt{1-x^2}} dx$$ What is this equal to?? $$ \sin^{-1}(x) + c $$

See Also [[Maths - Integration]]S [[Maths - Integration by Substitution]]S Flashcards What two things mean an integral is improper?? One or both of its limits are infinite It is undefined anywhere in $[a, b]$. What do you call an improper integral that exists (has a defined value)?? Convergent. What do you call an improper integral that does not exist?? Divergent. $$\int^\inft_0 e^{-x} dx$$ What would you write instead to see if it has a defined value?

See Also [[Maths - Integration]]S [[Maths - Integration by Substitution]]S Flashcards What do you divide the definite integral between $[a,b]$ in order to find the mean value?? $$ b - a $$ Backlinks [[Further Maths - Integrating and Differentiating Inverse Trig Functions]]S [[Further Maths - Syllabus]]S Metadata date: 2021-04-27 16:20 tags: - '@?further-maths' - '@?school' - '@?public' - '@?methods-in-calculus' title: Further Maths - Mean Value of a Function

See Also [[Maths - Integration]]S Flashcards $$\int (x+2)^5 dx$$ What subsitution could you make in terms of $u$?? $$ u = x + 2 $$ If $u = x + 2$, what is $\frac{du}{dx}$?? $$ \frac{du}{dx} = 1 $$ If $u = x + 2$ and $\frac{du}{dx} = 1$, what is $dx$?? $$ du $$ $$\int (x+2)^5 dx$$ If $u = x+2$ and $du = dx$, how could you rewrite the integral?

See Also [[Maths - Differentiation]]S [[Maths - Product Rule]]S [[Maths - Chain Rule]]S Flashcards What is the Quotient Rule used for?? Finding the derivative of two things being divided. $$\frac{d}{dx}\left(\frac{u}{v}\right)$$ What is this equal to?? $$ \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} $$ What is important you remember about the product rule?? The order. Where does the $v^2$ bit come from in the product rule?? It’s actually the product rule with $v^{-1}$, which goes to $\frac{1}{v^2}$

See Also Flashcards $$\sinh x$$ What is the definition?? $$ \frac{e^x - e^{-x}}{2} $$ $$\cosh x$$ What is the definition?? $$ \frac{e^x + e^{-x}}{2} $$ $$\tanh x$$ What is the definition?? $$ \frac{e^{2x} - 1}{e^{2x} + 1} $$ What function is this?? $$ \sinh $$ What function is this?? $$ \cosh $$ What function is this?? $$ \tanh $$ $$y = \sinh x$$ What does the graph look like?? $$y = \cosh x$$ What does the graph look like?

Flashcards $$|z^4| = 16$$ What is $|z|$?? $$ 2 $$ $$\arg z^4 = \frac{\pi}{2}$$ What is $\arg z$?? $$ \frac{\pi}{8} + \frac{2\pi n}{4} $$ $$\arg z^3 = 0$$ What is $\arg z$?? $$ \frac{2\pi n}{3} $$ $$z = \sqrt[3]{4 + 4i\sqrt{3}}$$ How could you rewrite this?? $$ z = 4 + 4i\sqrt{3} $$ If the modulus of $z^3$ is $8$, what must the modulus of $z$ be?? $$ 2 $$ If the argument of $z^3$ is $\frac{\pi}{3}$, what must the argument of $z$ be?

Flashcards What is $\left(z + \frac{1}{z}\right)$?? $$ 2\cos\theta $$ What is $\left(z - \frac{1}{z}\right)$?? $$ 2i\sin\theta $$ What is $\left(z - \frac{1}{z}\right)^4$?? $$ 16\sin^4\theta $$ What is $\left(z + \frac{1}{z}\right)^n$?? $$ 2^n\cos^n\theta $$ What is $\left(z^5 + \frac{1}{z^5}\right)$?? $$ 2\cos 5\theta $$ What is $\left(z^n - \frac{1}{z^n}\right)$?? $$ 2i\sin n\theta $$ If you’re trying to find out an identity for $\sin^4\theta$, what should you do?? $$ \left(z - \frac{1}{z}\right)^4 $$ If you’re trying to find out an identity for $\sin 4\theta$, what should you do?

See Also [[Further Maths - Complex Numbers]]S Sergeant further-maths/textbooks/year-2/chapter-1-complex-numbers/ex1a Flashcards What is Euler’s relation?? $$ e^{i\theta} = cos \theta + i \sin \theta $$ Why can you rewrite $e^{i\theta}$ as $\cos\theta + i\sin\theta$?? Because the Macluarin series of $\sin x$, $\cos x$ and $e^x$ match up. How can you write a complex number with argument $\theta$ and moudlus $r$ in exponential form?? $$ re^{i\theta} $$ $$e^{\pi i} = -1$$ What is this identity a special case of?

What is $\csc\theta$?? $$ \frac{1}{\sin\theta} $$ What is $\sec\theta$?? $$ \frac{1}{\cos\theta} $$ What is $\cot\theta$?? $$ \frac{1}{\tan\theta} $$ What is $\csc\theta$ in terms of triangle sides?? $$ \frac{\text{hyp}}{\text{opp}} $$ What is $\sec\theta$ in terms of triangle sides?? $$ \frac{\text{hyp}}{\text{adj}} $$ What is $\cot\theta$ in terms of triangle sides?? $$ \frac{\text{adj}}{\text{opp}} $$ 2021-02-22 $$\cos^2 x + \sin^2 x = 1$$ What do you get if you divide both sides by $\cos^2 x$?? $$ 1 + \tan^2 x = \sec^2 x $$

How could you imagine any function $f(x)$ could be written as a polynomial?? $$ f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ If you wanted to work out $a_0$, what could you set $x$ equal to?? $$ 0 $$ $f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + … + a_r x^r$ What happens if you substitute in $x = 0$?

See Also [[Maths - Differentiation]]S [[Maths - Chain Rule]]S Flashcards If $y = uv$, what is $\frac{dy}{dx}$?? $$ u\frac{dv}{dx} + v\frac{du}{dx} $$ If $f(x) = g(x)h(x)$, what is $f'(x)$?? $$ g(x)h'(x) + h(x)g'(x) $$ How would you explain the product rule in English?? To find the derivative of two things multiplied together, add the products of the pairs of each function and its opposite’s derivative. This diagram represents $h(x) = f(x)g(x)$.