AP Calculus AB/BC FRQ Practice

♾️AP Calculus AB/BC

Practice FRQ 1 of 81/8

1. The following functions are defined for this question: $k(x)=\frac{x^2-4}{x-2}$ $p(x)=\frac{x^2-4}{x^2-5x+6}$

The function $k$ is defined for $x≠ 2$ . The function $p$ is defined for $x≠ 2,3$ . Figure 1 shows the graphs of these functions. A graphing calculator may be used to help analyze the behavior of these functions near their points of discontinuity and as $x\to\infty$ .

Figure 1. Graphs of y = k(x) and y = p(x) with discontinuities highlighted (hole at x = 2 for both; vertical asymptote at x = 3 for p).

Use correct limit notation to represent $\lim_{x\to 2}k(x)$ . Then find the value of the limit. Show the work that leads to your answer.

Let $x$ approach 2 through the values 1.9, 1.99, 2.01, and 2.1.

(i) Use a calculator to approximate the average rate of change of $k$ on the interval $[1.9,2.1]$ .

(ii) Use a calculator to approximate the average rate of change of $k$ on the interval $[1.99,2.01]$ .

(iii) Based on your results in parts (i) and (ii), estimate the instantaneous rate of change of $k$ at $x=2$ . Justify your reasoning.

Write a limit expression that describes the end behavior of $p(x)$ as $x\to\infty$ . Then evaluate the limit using limit theorems.

Define a new function $q$ by Continuity of a piecewise-defined function at $x=2$ is enforced by matching the function value to the two-sided limit.

$q(x)=\begin{cases}p(x),& x≠ 2\\ a,& x=2\end{cases}$

Find the value of the constant $a$ such that $q$ is continuous at $x=2$ . Justify your answer using the definition of continuity at a point.

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