Linear Approximations and Differentials

The tangent line as proxy

Error control

Differentials

Toward higher-order approximation

Which expression is the linear approximation of f(x) near the base point a?

The error bound for linear approximation is M/2 times |x - a| squared. If the distance from the base point is halved, what happens to the bound?

In the language of differentials, the quantity dy is defined in terms of f prime of x and dx. What is the definition?

A quantity Q equals the square root of a measurement x. Using differentials, the relative error dQ/Q equals what multiple of the relative error dx/x?

When approximating the cube root of 8.1 by linear approximation, why is a = 8 the right base point?

For a function y = f(x), what is the precise difference between the increment Delta y and the differential dy?

Linear approximation of f(x) = x^(1/3) at a = 8 gives f prime of 8 = 1/12. What approximation does it give for the cube root of 8.1? Answer with a number rounded to four decimals.

For f(x) = x^2, the differential dy equals a simple expression in x and dx. What is dy? Answer like 2x dx.

The area of a square is A = s^2. A measurement of the side has relative error ds/s. What is the relative error dA/A, as a multiple? Answer with a number.

The error bound for linear approximation is M/2 times (x - a)^2 where M bounds the second derivative. If M = 6 and x - a = 0.01, what is the bound? Answer with a number.