(a) Find the domain of the function $g(x)$ defined by the following formula. $$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$ Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations. (b) $g(1000)$. (c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope. (d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope. (e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.
(a) Find the domain of the function $g(x)$ defined by the following formula.
$$g(x) = \int_{10}^{x} \log_{10}\left(\log_{10}\left(t^2 - 1000t + 10^{1000}\right)\right) dt$$
Calculate the quantities below. You may give an approximate answer where necessary, but clearly state which answers are exact and which are approximations.
(b) $g(1000)$.
(c) $x$ in $[10, 1000]$ where $g(x)$ has the maximum possible slope.
(d) $x$ in $[10, 1000]$ where $g(x)$ has the least possible slope.
(e) $\lim_{x \rightarrow \infty} \frac{\ln(x)}{g(x)}$ if it exists.