notes/uni/mmme/1026_maths_for_engineering/calculus.md

author	date	title	tags
Alvie Rahman	\today	MMME1026 // Calculus	uni nottingham mechanical engineering mmme1026 maths calculus

Calculus of One Variable Functions

Key Terms

Function

A function is a rule that assigns a unique value f(x) to each value x in a given domain.

The set of value taken by f(x) when x takes all possible value in the domain is the range of f(x).

Rational Functions

A function of the type

 \frac{f(x)}{g(x)} 

where f and g are polynomials, is called a rational function.

Its range has to exclude all those values of x where g(x) = 0.

Inverse Functions

Consider the function f(x) = y. If f is such that for each y in the range there is exactly one x in the domain, we can define the inverse f^{-1} as:

f^{-1}(y) = f^{-1}(f(x)) = x

Limits

Consider the following:

f(x) = \frac{\sin x}{x}

The value of the function can be easily calculated when x \neq 0, but when x=0, we get the expression \frac{\sin 0 }{0}. However, when we evaluate f(x) for values that approach 0, those values of f(x) approach 1.

This suggests defining the limit of a function

\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 approaching with small negative values of x.

The limit you get from approaching 0 with positive values is known as the limit from above:

\lim_{x \rightarrow a^+} f(x)

and with negative values is known as the limit from below:

\lim_{x \rightarrow a^-} f(x)

If the two limits are equal, we simply refer to the limit.

Important Functions

Exponential Functions

f(x) = e^x = \exp x

It can also be written as an infinite series:

\exp x = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...

The two important limits to know are:

• as x \rightarrow + \infty, \exp x \rightarrow +\infty (e^x \rightarrow +\infty)
• as x \rightarrow -\infty, \exp x \rightarrow 0 (e^x \rightarrow 0)

Note that e^x > 0 for all real values of x.

Hyperbolic Functions (sinh and cosh)

The hyperbolic sine (\sinh) and hyperbolic cosine function (\cosh) are defined by:

\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}

Some key facts about these functions:

• \cosh has even symmetry and \sinh and \tanh have odd symmetry
• as x \rightarrow + \infty, \cosh x \rightarrow +\infty and \sinh x \rightarrow +\infty
• \cosh^2x - \sinh^2x = 1
• $\tanh$'s limits are -1 and +1
• Derivatives:
  • \frac{\mathrm{d}}{\mathrm{d}x} \sinh x = \cosh x
  • \frac{\mathrm{d}}{\mathrm{d}x} \cosh x = \sinh x
  • \frac{\mathrm{d}}{\mathrm{d}x} \tanh x = \frac{1}{\cosh^2x}

Natural Logarithm

\ln{e^y} = \ln{\exp y} = y

Since the exponential of any real number is positive, the domain of \ln is x > 0.

Implicit Functions

An implicit function takes the form

f(x, y) = 0

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.