AP Calculus AB/BC FRQ practice

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♾️AP Calculus AB/BC

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Practice FRQ 1 of 81/8

1. A sensor measures the strength of a signal at a distance of $x$ meters from a transmitter, for $x ≥ 0$ . The signal strength, in volts, is modeled by the function $S$ defined by $S(x)=\dfrac{\sqrt{x+5}-3}{x-4}$ for $x ≠ 4$ . The expression is not defined at $x=4$ . Table 1 gives selected values of $S(x)$ for $x$ near 4, rounded to five decimal places.

Selected values of S(x) near x = 4

$x$	3.9	3.99	3.999	4.001	4.01	4.1
$S(x)$	0.16713	0.16671	0.16667	0.16666	0.16662	0.16621

Use the values in Table 1 to estimate $\lim_{x\to 4} S(x)$ . Write your answer using correct limit notation, and give a reason based on the table for your estimate.

Find $\lim_{x\to 4} \dfrac{\sqrt{x+5}-3}{x-4}$ by rewriting the expression as an equivalent expression for $x ≠ 4$ and using limit theorems. Show the work that leads to your answer.

Write a limit expression that represents the instantaneous rate of change of $S$ at $x=5$ in terms of average rates of change of $S$ over intervals containing $x=5$ . Then use your calculator to approximate this instantaneous rate of change to three significant digits. Include units.

In a different scenario, a new function $h$ is defined by

h(w)=\begin{cases}\dfrac{2w^2-7w-4}{w-4}, & w<4\\ kw-3, & w≥ 4\end{cases}

where $k$ is a constant. Find the value of $k$ that makes $h$ continuous at $w=4$ . Justify your answer using the definition of continuity at a point.

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