diff --git a/mechanical/mmme1026_maths_for_engineering/calculus.md b/mechanical/mmme1026_maths_for_engineering/calculus.md index 386f648..674c648 100755 --- a/mechanical/mmme1026_maths_for_engineering/calculus.md +++ b/mechanical/mmme1026_maths_for_engineering/calculus.md @@ -40,6 +40,9 @@ Its range has to exclude all those values of $x$ where $g(x) = 0$. +
+ + ### Inverse Functions Consider the function $f(x) = y$. @@ -48,6 +51,12 @@ we can define the inverse $f^{-1}$ as: $$f^{-1}(y) = f^{-1}(f(x)) = x$$ + +
+ +
+ + ### Limits Consider the following: @@ -64,6 +73,8 @@ $$\lim_{x \rightarrow a} f(x)$$ to be the limiting value, if it exists, of $f(x)$ as $x$ gets approaches $a$. + + #### Limits from Above and Below Sometimes approaching 0 with small positive values of $x$ gives you a different limit from @@ -79,12 +90,16 @@ $$\lim_{x \rightarrow a^-} f(x)$$ If the two limits are equal, we simply refer to the *limit*. + +
+ ## Important Functions ### Exponential Function $$f(x) = e^x = \exp x$$ +
It can also be written as an infinite series: