Question

Compare the values of \(d y\) and \(\Delta y\). \(y=2 x+1 \quad x=2 \quad \Delta x=d x=0.01\)

Short answer

The values of \(dy\) and \(\Delta y\) are both equal to 0.02 for \(dx = 0.01\).

Step-by-step solution

Find the derivative of \(y\)

The function given is a simple linear function, \(y = 2x + 1\). It's known that the derivative of \(x\) is just 1 and constants disappear. It means the derivative \(dy/dx\) of the function \(y\) is \(2 * 1 = 2\). So \(dy = 2 dx\).

Calculate \(dy\)

To calculate \(dy\), we will substitute \(dx = 0.01\) into our equation from Step 1, \(dy = 2 dx\). Doing this gives \(dy = 2 * 0.01 = 0.02\).

Calculate \(\Delta y\)

By definition, \(\Delta y\) is the actual change in \(y\) when \(x\) changes by an amount \(dx\). So we calculate \(\Delta y\) by substituting \(x + dx\) into the equation \(y = 2x + 1\) and then subtracting the original value of \(y\) when \(x = 2\). This gives us \(\Delta y = y(x + dx) - y(x)\) which simplifies to \(\Delta y = 2*(2 + 0.01) + 1 - (2*2 + 1) = 0.02\).

Comparing \(dy\) and \(\Delta y\)

From Steps 2 and 3, we see that \(dy = 0.02\) and \(\Delta y = 0.02\). So the differential \(dy\) and the increment \(\Delta y\) of \(y\) are equal to each other for \(dx = 0.01\).