Calculus I
&
Engineering Mathematics 2

Workshop 1


A Couple of Important Functions

Function Inverse Domain Range
$f(x)=e^x$ $f^{-1}(x)=\ln x$ $(-\infty,\infty)$ $(0,\infty)$
$g(x)=\ln x$ $g^{-1}(x)=e^x$ $(0,\infty)$ $(-\infty,\infty)$

These functions are inverses to each other since:

\( e^{\ln x}=x \quad (x>0), \qquad \ln(e^x)=x \)



Transformations of Trigonometric Functions

General form: \[ y = A\sin\big[B(x + C)\big] + D \]

Parameter Effect
$A$ Amplitude $=|A|$
$B$ Period $=\dfrac{2\pi}{|B|}$
$C$ Horizontal shift
$D$ Vertical shift

Function Inverse Domain Range
$f(x)=e^x$ $f^{-1}(x)=\ln x$ $(-\infty,\infty)$ $(0,\infty)$
$g(x)=\ln x$ $g^{-1}(x)=e^x$ $(0,\infty)$ $(-\infty,\infty)$

\(y = A\sin\big[B(x + C)\big] + D,\,\) $|A|=\,$Amplitude, $\frac{2\pi}{B}=\,$Period, $C=\,$Horizontal shift, $D=\,$Vertical shift

Trigonometric
$\sin^2 x + \cos^2 x = 1$
$1 + \tan^2 x = \sec^2 x$
$\tan x=\dfrac{\sin x}{\cos x}$
$\sin x=\dfrac{1}{\csc x},\;\cos x=\dfrac{1}{\sec x}$
$\sin(x\pm y) = \sin x\cos y \pm \cos x \sin y$
$\cos(x\mp y) = \cos x \cos y \pm \sin x \sin y$

Hyperbolic Trigonometric
$\sinh x=\dfrac{e^x-e^{-x}}{2},\; \cosh x=\dfrac{e^x+e^{-x}}{2}$
$\cosh^2 x - \sinh^2 x = 1$
$\tanh x=\dfrac{\sinh x}{\cosh x}$
$\sech x=\dfrac{1}{\cosh x},\;\csch x=\dfrac{1}{\sinh x}$
$\sinh(x\pm y) = \sinh x\cosh y \pm \cos x \sin y$
$\cosh(x\pm y) = \cosh x \cosh y \pm \sinh x\sinh y$

Credits