AP Calculus AB/BC FRQ practice

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♾️AP Calculus AB/BC

Course

Practice FRQ 1 of 21/2

1. The function $f$ is defined by

f(x) = \begin{cases} \dfrac{x^2 - 5x + 6}{x - 3}, & x < 3 \\ 2x - 5, & x ≥ 3 \end{cases}

The function $g$ is continuous on the closed interval $0 ≤ x ≤ 8$ . Selected values of $g(x)$ are given in the table shown.

$x$	$0$	$2$	$4$	$6$	$8$
$g(x)$	$-3$	$-1$	$2$	$5$	$4$

Find $\lim_{x \to 3^-} f(x)$ and $\lim_{x \to 3^+} f(x)$ . Is $f$ continuous at $x = 3$ ? Justify your answer using the definition of continuity.

Use the average rate of change of $g$ over the interval $2 ≤ x ≤ 4$ to approximate the instantaneous rate of change of $g$ at $x = 3$ . Show the work that leads to your answer.

Must there be a value $c$ , for $0 < c < 2$ , such that $g(c) = -2$ ? Must there be a value $k$ , for $6 < k < 8$ , such that $g(k) = 4.5$ ? Justify your answers.

A new function $h$ is defined by

h(x) = \begin{cases} \dfrac{x^2 - 5x + 6}{x - 3}, & x < 3 \\ a x^2 - 4, & x ≥ 3 \end{cases}

where $a$ is a constant. Find the value of $a$ for which $h$ is continuous at $x = 3$ . Justify your answer using the definition of continuity.

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