Linear approximation and Taylor polynomials

Use a linear approximation of $f(x) = \sqrt{x}$ at $x = 25$ to estimate $\sqrt{26}$.

The linear approximation of a function $f$ about $x = 2$ is $L(x) = 4x - 7$. What are $f(2)$ and $f'(2)$?

The third-order Maclaurin polynomial of a function $f$ is given below. Find $f'''(0)$.

Use a linear approximation to estimate $(3.002)^{4}$.

Use a linear approximation to estimate $\arctan(0.9)$, using the fact that $\arctan 1 = \dfrac{\pi}{4}$.

A function $f$ has $f(1) = 3$, $f'(1) = -2$ and $f''(1) = 6$. Use the quadratic approximation of $f$ centred at $x = 1$ to estimate $f(1.1)$.

Find the coefficient of $x^{3}$ in the third-order Maclaurin polynomial of $f(x) = (1 + 2x)^{-1/2}$.

A function $f$ has third derivative $f'''(x) = \dfrac{\cos x}{4 + x^{2}}$. If $f(0.5)$ is approximated by its second-order Maclaurin polynomial $T_{2}(0.5)$, find the best bound on the error $|f(0.5) - T_{2}(0.5)|$ that the Taylor remainder formula gives.

The equation $y^{3} + xy = 10$ defines $y$ implicitly as a function of $x$ near the point $(1, 2)$. Use the tangent-line approximation at that point to estimate $y$ when $x = 1.1$.

Use the second-order Taylor polynomial of $f(x) = \sqrt{x}$ about $x = 16$ to estimate $\sqrt{17}$, and use the Taylor remainder to give a bound on the error of your estimate.