Calculus¶

A non linear function will never have a constant slop line a straight line.
“A branch of mathematics concerned with determining the slope of a function at any point. Knowledge of differential calculus enables us to easily maximize and minimize functions and also graph complicated functions.”

Motivation for Differential Calculus¶

It is necessary in businesses to know some minimum and maximum point of a function defining a system.
“The maximum or minimum value of a function is the point at which the slope of the curve is zero.”
The slope of the function at \(x\) can be approximated by the slope of the line joining \((x, f(x))\) and \((x+\Delta x, f(x + \Delta x))\), as \(\Delta x\) tends to \(0\).
Making \(\Delta x\) approach to 0 is equivalent to taking first order derivative of the function \(f(x)\).

“The slope of a function at a given point can be shown visually by drawing a line such that it touches the function only at that point. That line is called tangent.”
“When the two points come together, Δx becomes zero, the line becomes a tangent line, and the slope of the function at that now-single point is revealed.”
“The point where the function shifts from increasing values of y to decreasing values of y (or from decreasing values of y to increasing values of y) is where the slope of the tangent line equals zero — i.e., where the tangent line is horizontal.”

\(dy/dx\) is used as notation for derivative of \(y\) with respect to \(x\).
\(dy/dx = y' = f'(x)\)
If \(f'(x) > 0\), the function is increasing at \(x\), and that if \(f'(x) < 0\), the function is decreasing at \(x\).
Rule 1: If \(f(x) = k\) then \(y' = 0\).
Rule 2: If \(f(x) = x^n\), where \(n\) is a any number, then \(y' = nx^{n-1}\)
Rule 3: If \(f(x)=kg(x)\), then \(y' = kg'(x)\) where \(g'(x)\) is the derivative of \(g(x)\).
Rule 4: If \(f(x) = g(x) + h(x)\), then \(f'(x) = g'(x) + h'(x)\).

Sometimes it is necessary to maximize a function or minimize a function.
Concave functions can be easily maximized while convex functions can be easily minimized.
To determine whether a function is convex or concave, second derivative (\(y''\)) is used.
\(d^2y/d^2x = y'' = f''(x)\).
Second derivative is the derivative of first derivative of a function.
“When a function’s second derivative is positive, the function’s first derivative is increasing; when a function’s second derivative is negative, the function’s first derivative is decreasing.”
“A function \(f(x)\) is convex if for all values of \(x\), \(f"(x) ≥ 0\).”
“A function \(f(x)\) is concave if for all values of \(x\), \(f"(x) ≤ 0\).”

“A concave function attains its maximum value at any point where \(f'(x) = 0\).”
**”A convex function attains its minimum value at any point where \(f'(x) = 0\).”

“A point where a curve changes from being convex to concave or concave to convex is an inflection point.”
An infletion point exist at \(f''(x)=0\).