A-level Mathematics/OCR/C3/Formulae

By the end of this module you will be expected to have learnt the following formulae:

1. ${\displaystyle y=-f\left(x\right)\,}$ is a reflection of ${\displaystyle y=f\left(x\right)\,}$ through the x axis.
2. ${\displaystyle y=f\left(-x\right)\,}$ is a reflection of ${\displaystyle y=f\left(x\right)\,}$ through the y axis.
3. ${\displaystyle y={\begin{vmatrix}f\left(x\right)\end{vmatrix}}}$ is a reflection of ${\displaystyle y=f\left(x\right)\,}$ when y < 0, through the x-axis.
4. ${\displaystyle y=f\left({\begin{vmatrix}x\end{vmatrix}}\right)}$ is a reflection of ${\displaystyle y=f\left(x\right)\,}$ when x < 0, through the y-axis.
5. ${\displaystyle y=f^{-1}\left(x\right)\,}$ is a reflection of ${\displaystyle y=f\left(x\right)\,}$ through the line y = x.
   Note: ${\displaystyle f^{-1}\left(x\right)}$ exists only if ${\displaystyle f\left(x\right)}$ is bijective, that is, one-to-one and onto.

1. ${\displaystyle y=af\left(x\right)\,}$ is stretched toward the x-axis if ${\displaystyle 0<a<1\,}$ and stretched away from the x-axis if ${\displaystyle a>1\,}$. In both cases the change is by a units.
2. ${\displaystyle y=f\left(bx\right)\,}$ is stretched away from the y-axis if ${\displaystyle 0<b<1\,}$ and stretched toward the y-axis if ${\displaystyle b>1\,}$. In both cases the change is by b units.

1. ${\displaystyle y=f\left(x-h\right)\,}$ is a translation of f(x) by h units to the right.
2. ${\displaystyle y=f\left(x+h\right)\,}$ is a translation of f(x) by h units to the left.
3. ${\displaystyle y=f\left(x\right)+k\,}$ is a translation of f(x) by k units upwards.
4. ${\displaystyle y=f\left(x\right)-k\,}$ is a translation of f(x) by k units downwards.

1. ${\displaystyle e^{\ln x}=\ln e^{x}=x\,}$
2. ${\displaystyle y\left(t\right)=y_{0}e^{kt}\,}$, where y(t) is the final value, ${\displaystyle y_{0}}$ is the initial value, k is the growth constant, t is the elapsed time.
3. ${\displaystyle k=-{\frac {\ln 2}{half-life}}}$, k for calculations involving half-lives.

• ${\displaystyle \sec \theta \equiv {\frac {1}{\cos \theta }}}$
• ${\displaystyle \operatorname {cosec} \ \theta \equiv {\frac {1}{\sin \theta }}}$
• ${\displaystyle \cot \theta \equiv {\frac {1}{\tan \theta }}\equiv {\frac {\cos \theta }{\sin \theta }}}$
• ${\displaystyle \sec ^{2}\theta \equiv 1+\tan ^{2}\theta }$
• ${\displaystyle \operatorname {cosec} ^{2}\ \theta \equiv 1+\cot ^{2}\theta }$

• ${\displaystyle \sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)\,}$
• ${\displaystyle \cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)\,}$
• ${\displaystyle \tan(A\pm B)={\frac {\tan(A)\pm \tan(B)}{1\mp \tan(A)\tan(B)}}}$

Note: The sign ${\displaystyle \mp }$ means that if you add the angles (A+B) then you subtract in the identity and vice versa. It is present in the cosine identity and the denominator of the tangent identity.

• ${\displaystyle \sin 2A\equiv 2\sin A\cos A}$
• ${\displaystyle \cos 2A\equiv \cos ^{2}A-\sin ^{2}A\equiv 1-2\sin ^{2}A\equiv 2\cos ^{2}A-1}$
• ${\displaystyle \tan 2A\equiv {\frac {2\tan A}{1-\tan ^{2}A}}}$

Using radians r = amplitute α = phase. ${\displaystyle r={\sqrt {a^{2}+b^{2}}}}$

${\displaystyle a\sin x+b\cos x=r\cdot \sin(x+\alpha )\,}$

where

${\displaystyle \alpha =\arcsin {\frac {b}{r}}}$

${\displaystyle a\sin x+b\cos x=r\cdot \cos(x-\alpha )\,}$

where

${\displaystyle \alpha =\arccos {\frac {b}{r}}}$

• If ${\displaystyle y=\operatorname {e} ^{kx}\,}$, then ${\displaystyle {\frac {dy}{dx}}=k\operatorname {e} ^{kx}}$
• If ${\displaystyle y=\ln x\,}$, then ${\displaystyle {\frac {dy}{dx}}={\frac {1}{x}}}$
• If ${\displaystyle y=f(x).g(x)\,}$, then ${\displaystyle {\frac {dy}{dx}}=f^{'}(x)g(x)+g^{'}(x)f(x)}$
• If ${\displaystyle y={\frac {f(x)}{g(x)}}}$, then ${\displaystyle {\frac {dy}{dx}}={\frac {f^{'}(x)g(x)-g^{'}(x)f(x)}{\left\{g(x)\right\}^{2}}}}$
• ${\displaystyle {\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}}$
• If ${\displaystyle y=f[g(x)]\,}$, then ${\displaystyle {\frac {dy}{dx}}=f^{'}[g(x)].g^{'}(x)}$
• ${\displaystyle {\frac {dy}{dt}}={\frac {dy}{dx}}.{\frac {dx}{dt}}}$

• ${\displaystyle \int \operatorname {e} ^{kx}\,dx={\frac {1}{k}}\operatorname {e} ^{kx}+c}$
• ${\displaystyle \int {\frac {1}{x}}\,dx=\ln \left|x\right|+c}$

For volumes of revolution:

• ${\displaystyle V_{x}=\pi \int _{a}^{b}y^{2}\,dx}$
• ${\displaystyle V_{y}=\pi \int _{c}^{d}x^{2}\,dy}$

Simpson's Rule ${\displaystyle \int _{a}^{b}ydx\approx {\frac {1}{3}}h\left\{\left(y_{0}+y_{n}\right)+4\left(y_{1}+y_{3}+\ldots +y_{n-1}\right)+2\left(y_{2}+y_{4}+\ldots +y_{n-2}\right)\right\}}$

where${\displaystyle h={\frac {b-a}{n}}}$ and n is even