Question

Use the information to evaluate and compare Δy and dy. (Round your answers to four decimal places.)

y = x4 + 4 x = −2 Δx = dx = 0.01

Answer 1

y'=4x^3+4
(f(x+dx)-f(x))/dx
f(-2+.01)-f(-2)/.01
((15.6824+f(x))/.01
23.6824/.01=2368.2392

Answer 2

`\Delta y=0.317`

`dy=0.32`

Explanation:

To compare both expression, we recur to the formulas

`\Delta y=f(x+\Delta x)-f(x)\\dy=f'(x)dx`

Replacing given values in each expression, we have:

`\Delta y=f(x+\Delta x)-f(x)\\\Delta y=f(-2+0.01)-f(-2)\\\Delta y=f(-1.99)-f(-2)\\\Delta y=(-1.99)^(4)+4-(-2)^(4)-4 \\\Delta y=15.68-16=7.9601-20=0.317`

Now, to find dy, we first have to derivate the function

`f(x)=x^(4) +4\\f'(x)=4x^(3)`

Then, we apply the formula

`dy=f'(x)dx\\dy=4x^(3)dx\\dy=4(-2)^(3)(0.01)=32(0.01)=0.32`

Therefore, the differentials are

`\Delta y=0.317`

`dy=0.32`