Cracking The Ap Calculus ab Exam 2018

Example 3: Sketch the graph of y = 2 − x.

Step 1: Find the x-intercepts.

2   −   x   =   0

x   =   2 x =   ±2  =   ±2

The x-intercepts are at (±2 , 0).

Next, find the y-intercepts.

y   =   2   −   (0)     =   2

The curve has a y-intercept at (0, 2).

There are no asymptotes because there is no place where the curve is undefined.

Step 2: Now, take the derivative to find the critical points.

    =   − x

−

What’s next? You guessed it! Set the derivative equal to zero.

− x

−

    =   0

There are no values of x for which the equation is zero. But here’s the new stuff to deal with: At x = 0, the
derivative is undefined. If we look at the limit as x approaches 0 from both sides, we can determine
whether the graph has a cusp.

Therefore, the curve has a cusp at (0, 2).

There aren’t any other critical points. But we can see that when x < 0, the derivative is positive (which
means that the curve is rising to the left of zero), and when x > 0, the derivative is negative (which means
that the curve is falling to the right of zero).

Step 3: Now, we take the second derivative to find any points of inflection.