Chapter 3 - Section 3.10 - Linear Approximations and Differentials. - 3.10 Exercises - Page 259: 26

$\Delta x = dx = 0.5$ $\Delta y = 0.6487$ $dy = 0.5$ We can see a sketch below.

$y = e^x$ $\Delta x = dx = 0.5$ When $x = 0$, then $y = e^0= 1$ When $x = 0.5$, then $y = e^{0.5} = 1.6487$ $\Delta y = 1.6487-1 = 0.6487$ We can find the value of $dy$: $\frac{dy}{dx} = e^x$ $dy = e^x~dx$ $dy = e^0~(0.5)$ $dy = (1)~(0.5)$ $dy = 0.5$ We can see a sketch below.