Chapter 3.11, Problem 16E

a.

To determine

To calculate: The approximation for the function $f\left(x\right)={\left(1-x\right)}^6$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.

b.

To determine

To calculate: The approximation for the function $f\left(x\right)=\frac{2}{1-x}$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.

c.

To determine

To calculate: The approximation for the function $f\left(x\right)=\frac{1}{\sqrt{1+x}}$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.

d.

To determine

To calculate: The approximation for the function $f\left(x\right)=\sqrt{2+x^2}$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.

e.

To determine

To calculate: The approximation for the function $f\left(x\right)={\left(4+3x\right)}^{1/3}$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.

f.

To determine

To calculate: The approximation for the function $f\left(x\right)=\sqrt[3]{{\left(1-\frac{x}{2+x}\right)}^2}$ near $x=0$ using the linear approximation ${\left(1+x\right)}^k\approx 1+kx$.