This is the Taylor's Theorem
$f(x)=f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+R_2$
I am not sure how it relates to the formulas below, which is shown in Numerical Optimization book by Jorge Nocedal and Stephen J. Wright.
$f(x+p)=f(x)+\nabla f(x+tp)^\top p$
for some $t\in(0,1)$, and if $f$ is twice continuously differentiable, we can have
$\nabla f(x+p)=\nabla f(x)+\int_0^1\nabla^2f(x+tp)p~dt$
thus we can have
$f(x+p)=f(x)+\nabla f(x)^\top p+\frac{1}{2}p^\top\nabla^2f(x+tp)p$
Why is the function becomes $f(x+p)$? and what is the use of $t$.
Thank you very much.