Chapter 3 - Section 3.10 - Linear Approximations and Differentials. - 3.10 Exercises - Page 259: 31

$(1.999)^4 \approx 15.968$

Let $y = x^4$ $\frac{dy}{dx} = 4x^3$ $dy = 4x^3~dx$ Let $x = 2$ and let $dx = -0.001$ $dy = 4(2)^3~(-0.001)$ $dy = 4(2)^3~(-0.001)$ $dy = -0.032$ We can find an approximation for $(1.999)^4$ $(1.999)^4 \approx (2)^4-0.032$ $(1.999)^4 \approx 16-0.032$ $(1.999)^4 \approx 15.968$