Question

Let Y - 5x² + 4x + 3. If Ax = 0.4 At X = 5, Use Linear Approximation To Estimate ∆y

∆y = _______________

Answer 1

To estimate ∆y using linear approximation, you find the derivative of the function at the point of interest and multiply by ∆x. Evaluating the derivative of Y at x = 5 and multiplying it by ∆x = 0.4 gives an estimated ∆y of approximately -18.4.

Step-by-step explanation:

The question asks us to use linear approximation to estimate ∆y when given a function Y and a change in x (∆x) at a specific point. To begin, we need to understand that linear approximation involves finding the derivative of the function at the point of interest, which can be used to approximate changes in Y for small changes in x.

Let's denote our function as Y = -5x² + 4x + 3 and we want to approximate ∆y when x = 5 and ∆x = 0.4. First, we find the derivative of Y, which is dY/dx = -10x + 4. Evaluating this derivative at x = 5 gives us the slope of the tangent line at that point, which is -10(5) + 4 = -50 + 4 = -46. The linear approximation formula is ∆y ≈ (dY/dx)|x=a × ∆x.

Using the calculated slope (-46) and the given ∆x (0.4), we get:
∆y ≈ (-46) × (0.4) = -18.4.

Therefore, the estimated change in Y, ∆y, is approximately -18.4.