linear approximation differentials?

Question

Let f(x) = sqrt(225-x^2) Use differentials (linear approximation) to estimate f(12.1). Base your approximation at x = 12 . Give your answer to at least 4 significant figures or use fractions.

Answer 1

df = f'(x) dx

f' = (1/2)(225-x²)^(-1/2) (-2x) = -x (225-x²)^(-1/2)

dx = 12.1 - 12 = 0.1

So, evaluating df at x=12, dx = 0.1,

df = [-12(225-144)^(-1/2)] 0.1 = (-12/9)(0.1) = -2/15

Therefore, f(12.1) ≈ f(12) + df = 9 - 2/15 = 133/15 ≈ 8.867

The exact answer, correct to 4 significant figures, is 8.865

Answer 2

you're able to try this as a Taylor sequence boost of the function f(x)=(a million+x)^6. A Taylor sequence boost is given via, f(x) = f(0)+ f'(0)*x +(a million/2)f''(0)*x^2+... retaining in basic terms the 1st 2 words of the upward push delivers the linear approximation of the function. Now, all you may desire to do it calculate the spinoff and evaluate the unique function and the spinoff at 0. you may get that the linear approximation is, f(x)=a million+6*x this might carry for small values of x and in basic terms small values of x. Now, substitute x=0.01 and you gets the approximate answer to the difficulty.