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: $$f^{\mathrm{\prime }\mathrm{\prime }}(x)=(f^{\mathrm{\prime }}(x){)}^{\mathrm{\prime }}=f^{(2)}(x)=\frac{d^{2}f}{d{x}^2}=\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{d}{dx}(f(x))\right)=\underset{h\to 0}{lim}\frac{f^{\prime }(x+h)-f^{\prime }(x)}{h}$$

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 $f$: $$f^{″}(x)=\underset{h\to 0}{lim}\frac{f^{\prime }(x+h)-f^{\prime }(x)}{h}$$ Hence, all your table work (forward, backward, centered difference formulas) still applies, it's just that we use values for $f^{\prime }$ to do our calculations.

• $f(x)=x^2$: $f^{\prime }(x)=2x$, and $f^{″}(x)=2$: $f^{″}(x)>0⟶$ concave up:

• $f(x)=-x^2$: $f^{\prime }(x)=-2x$, and $f^{″}(x)=-2$: $f^{″}(x)<0⟶$ 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: