3. Calculus

Let $f(x)$ be a function defined in an open interval containing $a$. The derivative of a function $f(x)$ at $a$ is denoted by $f^{\prime }(x)$ and is defined by:

$f^{\prime }(a)={\lim }_{h\to 0}\frac{f(a+h)-f(a)}{h}$

Differentiation Rules:

• Constant rule
• Power rule
• Sum rule
• Difference rule
• Constant multiple rule
• Product rule
• Quotient rule

Chain Rule If we have $h(x)=f(g(x))$ then $h^{\prime }(x)=f^{\prime }(g(x))g^{\prime }(x)$

Derivatives of Exponentials & Logarithms

$\frac{d}{dx}(b^{g(x)})=b^{g(x)}{g}^{\prime }(x)ln(b)$

$\frac{d}{dx}(lo{g}_{b}g(x))=\frac{g^{\prime }(x)}{g(x)ln(b)}$

Numerical Differentiation

• One-sided (forward) difference:

$\frac{f(a+h)-f(a)}{h}$

• One-sided (backward) difference:

$\frac{f(a)-f(a-h)}{h}$

• The central difference:

$\frac{f(a+h)-f(a-h)}{2h}$

Fermat’s Theorem

If $f$ has a local extremum at $c$ and $f$ is differentiable at $c$, then $f^{\prime }(c)=0$.

Extreme Value Theorem: A continuous function over a closed, bounded interval has an absolute maximum and minimum. Each extremum occurs at a critical point or an endpoint.

Rolle’s Theorem

Let $f$ be a continuous function over the closed interval $[a,b]$ and differentiable over the open interval $(a,b)$ such that $f(a)=f(b)$. There then exists at least one $c$ $ϵ(a,b)$ such that $f^{\prime }(c)=0$.

Mean Value Theorem

Let $f$ be continuous over the closed interval $[a,b]$ and differentiable over the open interval $(a,b)$. Then, there exists at least one point $c$ $ϵ(a,b)$, such that:

$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

1. If $f^{\prime }(x)>0$ for all x$$\epsilon (a,b), then $f$ is an increasing function over $[a,b]$
2. If $f^{\prime }(x)<0$ for all x$$\epsilon (a,b), then $f$ is an decreasing function over $[a,b]$

Convex vs Concave Functions

Feature	Convex	Concave
Visual Shape	Cup / Smile ($\cup$)	Cap / Frown ($\cap$)
Midpoint Test	Line is above the curve	Line is below the curve
Optimization	Easy to find the minimum	Easy to find the maximum
Second Derivative	$f^{\prime \prime }(x)\ge 0$	$f^{\prime \prime }(x)\le 0$