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Chapter 9: Problem 13

Compare the values of \(d y\) and \(\Delta y\). \(y=x^{4}+1 \quad x=-1 \quad \Delta x=d x=0.01\)

Short Answer

Expert verified
The differential of y, \(d y = -0.04\), and the difference in y, \(\Delta y = 0.000401\). Thus, \(d y \neq \Delta y\).

Step by step solution

01

Find the derivative of the function

Given \(y = x^{4} + 1\), the derivative of y with respect to x (\(dy/dx\)) can be obtained using the power rule. The derivative is \(dy/dx = 4x^{3}\).
02

Calculate \(d y\)

Next, using the derivative from Step 1, \(d y\) can be calculated by multiplying \(dy/dx\) with \(d x\). This gives \(d y = (dy/dx) * d x = 4*(-1)^{3} * 0.01 = -0.04\).
03

Find \(y_1\) and \(y_2\)

Now, \(\Delta y\) can be understood as the difference in y-values for a small change in x. Here, the change in x (\(\Delta x\)) is 0.01. So, calculate the values of y at x and at x + \(\Delta x\). This gives, \(y_1 = (-1)^{4} + 1 = 2\) and \(y_2 = ([-1 + 0.01]^{4}) + 1 = 2.000401\).
04

Compute \(\Delta y\)

We can calculate \(\Delta y\) as \(y_2 - y_1\). This gives \(\Delta y = 2.000401 - 2 = 0.000401\).

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