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: $f(x) = x^2 - 4$ $g(x) = x - 2$
$k(x) = 4$

Water flows through a filter. For $x ≠ 2$ , the flow rate (in liters per minute) is modeled by the function $F$ defined by $F(x)=\frac{x^2-4}{x-2}$ , where $x$ is the setting on the filter’s control dial. The graph of $F$ for $0≤ x≤ 6$ is shown in Figure 1.

$f(x) = x^2 - 4$
$g(x) = x - 2$
$k(x) = 4$

Figure 1. Graph of F on 0 ≤ x ≤ 6, showing a removable discontinuity at x = 2

Estimate the value of $\lim_{x\to 2}F(x)$ using the graph of $F$ in Figure 1.

Represent the limit in part A analytically by writing $\lim_{x\to 2}F(x)$ as the limit of an equivalent expression, and evaluate the limit.

The instantaneous rate of change of the flow rate with respect to the dial setting at $x=2$ is interpreted as a limit of average rates of change. Write a limit expression that represents this instantaneous rate of change, and evaluate the limit.

A new function $G$ is defined by

$G(x)=\begin{cases}\frac{x^2-4}{x-2}, & x≠ 2\\ k, & x=2\end{cases}$

where $k$ is a constant. Find the value of $k$ such that $G$ is continuous at $x=2$ . Justify your answer using the definition of continuity. The function $G$ is defined to match $F$ for all $x≠ 2$ , but assigns a value $k$ at $x=2$ . Continuity at $x=2$ requires that $\lim_{x\to 2}G(x)$ exists and equals $G(2)=k$ .