5564620
Descripción
Fichas por Daniel Cox, actualizado hace más de 1 año
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Creado por Daniel Cox
hace casi 8 años
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| Pregunta | Respuesta |
| How would you split the following expression into partial fractions? \[\frac{f(x)}{(ax+b)(cx+d)}\] | \[\frac{A}{ax+b}+\frac{B}{cx+d}\] |
| How would you split the following expression into partial fractions? \[\frac{f(x)}{(ax+b)(cx+d)^2}\] | \[\frac{A}{ax+b}+\frac{B}{cx+d}+\frac{C}{(cx+d)^2}\] |
| \[\sin(A\pm B)=?\] (Given in formulae booklet) | \[\sin(A\pm B)=\sin A \cos B \pm \cos A \sin B\] |
| \[\cos(A\pm B)=?\] (Given in formulae booklet) | \[\cos(A\pm B)=\cos A \cos B \mp \sin A \sin B\] |
| \[\tan(A\pm B)=?\] (Given in formulae booklet) | \[\tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \] |
| \[\sin 2A = ?\] | \[\sin 2A = 2\sin A \cos A\] |
| Give the 3 identities for \(\cos 2A\) | \[\begin{align*} \cos 2A=&\cos^2 A - \sin^2 A \\ \cos 2A=&2\cos^2 A - 1 \\ \cos 2A=&1-2\sin^2 A \end{align*} \] |
| \[\tan 2A=?\] | \[\tan 2A=\frac{2\tan A}{1-\tan^2 A}\] |
| How do you find \[\int \sin^2 x \, dx \] | (1) use the identity \(\cos 2x = 1-2\sin^2 x\) (2) rearrange to get \(\sin^2 x =\frac{1}{2}(1-\cos 2x)\) (3) work out \(\int \frac{1}{2}(1-\cos 2x) \, dx \) |
| How do you find \[\int \cos^2 x \, dx \] | (1) use the identity \(\cos 2x = 2\cos^2 x-1\) (2) rearrange to get \(\cos^2 x =\frac{1}{2}(1+\cos 2x)\) (3) work out \(\int \frac{1}{2}(1+\cos 2x) \, dx \) |
| How do you find \[\int \tan^2 x \, dx \] | (1) use the identity \(1+\tan^2 x=\sec^2 x\) (2) rearrange to get \(\tan^2 x=\sec^2 x-1\) (3) work out \(\int \sec^2 x-1 \, dx \) |
| What is meant by a Cartesian equation? | An equation where the only variables are \(x\) and/or \(y\) |
| When applying the \(R, \alpha\) method, if \(R\sin \alpha =a\) and \(R\cos \alpha =b\), how would you find \(R\) and \(\alpha\)? | \[R=\sqrt{a^2+b^2}\] \[\alpha = \tan^{-1} \left( \frac{a}{b} \right)\] |
| What is the greatest value of \(R\sin (x+\alpha)\) and find a value of \(x\) for which this occurs | Greatest value is \(R\). Happens when \(x+\alpha=90^{\circ}\), i.e. when \(x=90^{\circ} -\alpha\) |
| What is the least value of \(R\sin (x+\alpha)\) and find a value of \(x\) for which this occurs | Least value is \(-R\). Happens when \(x+\alpha=270^{\circ}\), i.e. when \(x=270^{\circ} -\alpha\) |
| If \(x=f(t)\) and \(y=g(t)\), how would you find \(\frac{\text{d}y}{\text{d}x}\)? | \[\frac{\text{d}y}{\text{d}x}=\frac{\left(\frac{\text{d}f}{\text{d}t}\right)}{\left(\frac{\text{d}g}{\text{d}t}\right)}\] |
| If the rate of increase of \(P\) over time is directly proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\). | \[\frac{\text{d}P}{\text{d}t}=k\times f(P)\] |
| If the rate of decrease of \(P\) over time is directly proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\). | \[\frac{\text{d}P}{\text{d}t}=-k\times f(P)\] |
| If the rate of increase of \(P\) over time is inversely proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\). | \[\frac{\text{d}P}{\text{d}t}=\frac{k}{f(P)}\] |
| If the rate of decrease of \(P\) over time is inversely proportional to \(f(P)\), write down a differential equation that is satisfied by \(P\). | \[\frac{\text{d}P}{\text{d}t}=-\frac{k}{f(P)}\] |
| What is the formula for the volume of revolution of a curve about the \(x\)-axis? | \[V=\int_a ^b \pi y^2 \, \text{d}x\] |
| What is the formula for integration by parts? (given in formulae book) | \[\int u \frac{\text{d}v}{\text{d}x} \, \text{d}x=uv-\int v \frac{\text{d}u}{\text{d}x} \, \text{d}x\] |
| \[\int \frac{1}{ax+b} \, \text{d}x=?\] | \[\int \frac{1}{ax+b} \, \text{d}x=\frac{1}{a}\ln {|ax+b|}+c\] |
| When \(n \neq -1\), \[\int (ax+b)^n \, \text{d}x=?\] | \[\int (ax+b)^n \, \text{d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+c\] |
| What is a unit vector? | A vector of length \(1\) |
| What is the magnitude of vector \(a\mathbf{i}+b\mathbf{j}+c\mathbf{k}\) | \[|a\mathbf{i}+b\mathbf{j}+c\mathbf{k}|=\sqrt{a^2+b^2+c^2}\] |
| If vectors \(\mathbf{a}\) and \(\mathbf{b}\) are parallel, then...? | \[\mathbf{a}=k\mathbf{b}\] They are multiples of each other |
| If \(\mathbf{a}\) and \(\mathbf{b}\) are the position vectors of points \(\mathbf{A}\) and \(\mathbf{B}\), how would you find \(\mathbf{AB}\)? | \[\mathbf{AB}=\mathbf{b}-\mathbf{a}\] |
| A line passes through the point with position vector \(\mathbf{a}\) and is in the direction of \(\mathbf{b}\). What is the vector equation of the line? | \[\mathbf{r}=a+\lambda \mathbf{b}\] (any parameter will do - here, I used \(\lambda \)) |
| What is the formula for the scalar product of vectors \(\mathbf{a}\) and \(\mathbf{b}\)? | \[\mathbf{a} \cdot \mathbf{b}=|\mathbf{a}||\mathbf{b}|\cos \theta\] where \(\mathbf{a} \cdot \mathbf{b}=a_1b_1+a_2b_2+a_3b_3\) |
| If vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular, \(\mathbf{a} \cdot \mathbf{b}=?\) | \[\mathbf{a} \cdot \mathbf{b}=0\] |
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