Cracking The Ap Calculus ab Exam 2018

Now, plug in to the formula.

f(x +   ∆x) ≈   f(x)    +   f′(x)∆x

Now, if we plug in x = 9 and ∆x = + 0.01.

(0.01) ≈ 3.001666666

If you enter into your calculator, you get: 3.001666204. As you can see, our answer is a pretty

good approximation. It’s not so good, however, when ∆x is too big. How big is too big? Good question.

Example 2: Use differentials to approximate .

Let x = 9, ∆x = +.5, f(x) = and plug in to what you found in Example 1.

(.5) ≈ 3.083333333

However, equals 3.082207001 on a calculator. This is good to only two decimal places. As the

ratio of grows larger, the approximation gets less accurate, and we start to get away from the actual

value.

There’s another approximation formula that you’ll need to know for the AP Exam. This formula is used to
estimate the error in a measurement, or to find the effect on a formula when a small change in
measurement is made. The formula is:

dy  =   f′(x)   dx

Note    that    this    equation    is  simply  a   rearrangement   of      =   f′(x).

This notation may look a little confusing. It says that the change in a measurement dy, due to a differential
dx, is found by multiplying the derivative of the equation for y by the differential. Let’s do an example.

Example 3: The radius of a circle is increased from 3 to 3.04. Estimate the change in area.