diff --git a/mechanical/mmme1026_maths_for_engineering/calculus.md b/mechanical/mmme1026_maths_for_engineering/calculus.md
index 674c648..d081bc9 100755
--- a/mechanical/mmme1026_maths_for_engineering/calculus.md
+++ b/mechanical/mmme1026_maths_for_engineering/calculus.md
@@ -95,16 +95,16 @@ If the two limits are equal, we simply refer to the *limit*.
## Important Functions
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+
### Exponential Function
$$f(x) = e^x = \exp x$$
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-
-
-It can also be written as an infinite series:
+It can also be written as an infinite series:
+
$$\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...$$
@@ -116,6 +116,11 @@ The two important limits to know are:
Note that $e^x > 0$ for all real values of $x$.
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+
+
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### Hyperbolic Functions (sinh and cosh)
The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defined by:
@@ -123,6 +128,8 @@ The hyperbolic sine ($\sinh$) and hyperbolic cosine function ($\cosh$) are defin
$$\sinh x = \frac 1 2 (e^x - e^{-x}) \text{ and } \cosh x = \frac 1 2 (e^x + e^{-x})$$
$$\tanh = \frac{\sinh x}{\cosh x}$$
+
+
](./images/Sinh_cosh_tanh.svg)
Some key facts about these functions:
@@ -136,17 +143,33 @@ Some key facts about these functions:
- $\frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x$
- $\frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}$
-## Natural Logarithm
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+
+
+
+
+### Natural Logarithm
$$\ln{e^y} = \ln{\exp y} = y$$
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+
Since the exponential of any real number is positive, the domain of $\ln$ is $x > 0$.
-## Implicit Functions
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+
+
+
+
+### Implicit Functions
An implicit function takes the form
$$f(x, y) = 0$$
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+
To draw the curve of an implicit function you have to rewrite it in the form $y = f(x)$.
There may be more than one $y$ value for each $x$ value.
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+