Day 16, Mat128: Calculus A

I'll summarize before we move on to 1.8.

If a function's derivative is a function, does that derivative function have a derivative?

The derivative of the derivative of a function \(f\) is called the second derivative of \(f\), and frequently denoted \(f''\). More notation: \[ f^{\prime\prime}(x) = (f^\prime(x))^\prime = f^{(2)}(x)=\frac{d^2f}{dx^2}=\frac{d^2y}{dx^2}=\frac{d}{dx}\left(\frac{d}{dx}\left( f(x) \right) \right) =\lim_{h \to 0} \frac{f'(x+h)-f'(x)}{h} \]

Several of these forms remind us that differentiation is itself a function (we call it an "operator", since it "operates" on functions): it takes a function (from its domain of functions) and returns another function in its range of functions -- the derivative function.

But most importantly for us, we realize that the second derivative can be thought of as the derivative of the derivative of \(f\): \[ f''(x)=\lim_{h\to 0}{\frac{f'(x+h)-f'(x)}{h}} \] Hence, all your table work (forward, backward, centered difference formulas) still applies, it's just that we use values for \(f'\) to do our calculations.

• $f(x)=x^2$: $f'(x)=2x$, and $f''(x)=2$: $f''(x)>0 \longrightarrow$ concave up:

• $f(x)=-x^2$: $f'(x)=-2x$, and $f''(x)=-2$: $f''(x)<0 \longrightarrow$ concave down:

• Identify the Function, First, and Second Derivatives
• Derivatives and the Shape of a Graph

• Activity 1.6.2 (we started talking about this one last time)
• Activity 1.6.3: more potatoes! I explore this one with Mathematica.

This is our poster child for inflection: