Exponential function: tangents, inflection and Newton's method

State the value of $\lim\limits_{x \to -\infty} g(x)$.

The derivative, $g'(x)$, can be expressed in the form $g'(x) = k \times 2^x$.

Find the real number $k$.

Let $a$ be a real number. Find, in terms of $a$, the equation of the tangent to $g$ at the point $(a, g(a))$.

Hence, or otherwise, find the equation of the tangent to $g$ that passes through the origin, correct to three decimal places.

Let $h : \R \to \R, h(x) = 2^x - x^2$.

Find the coordinates of the point of inflection for $h$, correct to two decimal places.

Find the largest interval of $x$ values for which $h$ is strictly decreasing.

Give your answer correct to two decimal places.

Apply Newton's method, with an initial estimate of $x_0 = 0$, to find an approximate $x$-intercept of $h$.

Write the estimates $x_1$, $x_2$ and $x_3$ in the table below, correct to three decimal places.

| | | |---|---| | $x_0$ | $0$ | | $x_1$ | | | $x_2$ | | | $x_3$ | |