Cracking The Ap Calculus ab Exam 2018

    =   24x −   168

At  the end of  the day,    no  matter  how complex the math    might

get,    if  a   problem is  based   on  a   real    world   example,    like    this

cardboard   box,    then    the answer  will    make    sense   in  reality.

At x = 3.4,

= −86.4

So, the volume of the box will be maximized when x = 3.4.

Therefore, the dimensions of the box that maximize the volume are approximately: 11.2 in. × 17.2 in. ×
3.4 in.

Sometimes, particularly when the domain of a function is restricted, you have to test the endpoints of the
interval as well. This is because the highest or lowest value of a function may be at an endpoint of that
interval; the critical value you obtained from the derivative might be just a local maximum or minimum.
For the purposes of the AP Exam, however, endpoints are considered separate from critical values.

Example 5: Find the absolute maximum and minimum values of y = x^3 − x on the interval [−3, 3].

Take the derivative and set it equal to zero.

    =   3x^2    −   1   =   0

Solve for x.

x   =   ±   

Test the critical points.

    =   6x

At x = , we have a minimum. At x = − , we have a maximum.

At  x   =   − , y   =       ≈   0.385