Use the information to evaluate and compare Δy and dy

Question

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

y = x⁴      x = 1      Δx = dx = 0.1

Δy	=
dy	=

Answer 1

Take the derivative of y = x⁴

dy/dx = 4 x³

dy = 4 x³ dx

plug in x = 1    dx = 0.1 in the equation above

Answer 2

Δy is the actual change in the value of y = f(x) if some change Δx is made to x. This is to say, we find the value of f at x + Δx and subtract away the value of f at x.

On the other hand, dy is the approximate change in the value of y = f(x) if some change dx = Δx is made to x, where the approximation is the linear approximation of f at x. This is to say, we find the derivative of f at x, giving the instantaneous slope of f at x. Then we multiply this slope (which, remember, is rise/run) by the amount we have "run" from x, which is precisely Δx, thus giving us the approximate amount that we have "risen."

Here's an image explaining this: https://imgur.com/1HZysT9

We can formally describe these ideas by defining Δy and dy to be functions of x and Δx. Δy is given by f(x + Δx) - f(x). Meanwhile, dy is given by f'(x)*dx = f'(x)*Δx. Note that dx is identically equal to Δx, but it's used just to make it clear that we're talking about the linear approximation. We can use these to fill out the charts:

y = x⁴ ⇒ y' = 4x³, so:

Δy = f(x + Δx) - f(x) = (1 + 0.1)⁴ - (1)⁴ = 1.4641 - 1 = 0.4641 ≈ 0.464 (rounding to three decimal places)

dy = f'(x)*dx = 4(1)³ * (0.1) = 4 * 0.1 = 0.400