AP Calculus AB/BC FRQ practice

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♾️AP Calculus AB/BC

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Practice FRQ 1 of 21/2

1. A vibration sensor is attached to a bridge cable. For $x > 0$ , the function $D$ defined by $D(x) = x^2 \sin\left(\dfrac{1}{x}\right)$ models the displacement, in millimeters, of the cable at a distance of $x$ meters from the anchor point. The expression is not defined at $x = 0$ . Table 1 gives selected values of $D(x)$ for $x$ near 0, rounded to six decimal places.

Selected values of D(x)

$x$	$0.2$	$0.1$	$0.05$	$0.02$	$0.01$
$D(x)$	$-0.038357$	$-0.005440$	$0.002282$	$-0.000105$	$-0.000051$

Use the values in Table 1 to estimate $\lim_{x\to 0^+} D(x)$ . Write your answer using correct limit notation, and give a reason based on the table for your estimate.

Use the inequality $-1 ≤ \sin\left(\dfrac{1}{x}\right) ≤ 1$ for $x > 0$ and the Squeeze Theorem to find $\lim_{x\to 0^+} D(x)$ . Show the work that leads to your answer.

Write a limit expression that describes the end behavior of $\dfrac{D(x)}{x}$ as $x \to 0^+$ , and evaluate the limit. Use the Squeeze Theorem to justify your answer.

A new function $h$ is defined by

h(x)=\begin{cases} D(x), & x>0 \\ kx + 4, & x ≤ 0 \end{cases}

where $k$ is a constant. Find all values of $k$ , if any, that make $h$ continuous at $x = 0$ . Justify your answer using the definition of continuity at a point.

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