Cracking The Ap Calculus ab Exam 2018

3.y =   ex  at  y   =   e

4.y =   x   +   x^3     at  y   =   −2

5.y =   4x  −   x^3     at  y   =   3

6.y =   ln  x   at  y   =   0

DIFFERENTIALS

Sometimes this is called “linearization.” A differential is a very small quantity that corresponds to a
change in a number. We use the symbol ∆x to denote a differential. What are differentials used for? The
AP Exam mostly wants you to use them to approximate the value of a function or to find the error of an
approximation.

Recall the formula for the definition of the derivative.

f′(x)   =   

Replace h with ∆x, which also stands for a very small increment of x, and get rid of the limit.

f′(x)   ≈   

Notice that this is no longer equal to the derivative, but an approximation of it. If ∆x is kept small, the
approximation remains fairly accurate. Next, rearrange the equation as follows:

f(x +   ∆x) ≈   f(x)    +   f′(x)∆x

This is our formula for differentials. It says that “the value of a function (at x plus a little bit) equals the
value of the function (at x) plus the product of the derivative of the function (at x) and the little bit.”

Example 1: Use differentials to approximate .

You can start by letting x = 9, ∆x = +0.01, f(x) = . Next, we need to find f′(x).

f′(x)   =