I want to verify the claim, that the steepest descent direction $-\nabla f(x^k)$ is the one along which $f\in C^2(\mathbb R^n)$ decreases most rapidly.
Therefore, I considered the Taylor expansion $$f(x^k+\alpha p)=f(x^k)+\alpha p^T\nabla f(x^k)+\frac 12\alpha^2p^T\nabla^2f(x^k+tp)p\;\;\;\text{for some }t\in (0,\alpha)\;,$$ where $p$ is any search direction and $\alpha$ is the step-length parameter. I've read the following:
The rate of change in $f$ along $p$ at $x^k$ is the coefficient of $\alpha$, namely, $p^T\nabla f(x^k)$
Why? What's meant by "rate of change" at all? I thought they may mean $$\frac{f(x^k+\alpha p)-f(x^k)}{\alpha\left\|p\right\|}=\frac{p^T}{\left\|p\right\|}\left(\nabla f(x^k)+\frac 12\alpha\nabla^2f(x^k+tp)p\right)\;,$$ but that seems to be wrong. So, what's meant exactly?
That statement is only correct (or at least only clear) if $p$ is a unit vector. In that case, you can drop $\|p\|$ in your expression. What they mean by the rate of change is not that, but its limit for $\alpha\to0$.