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: $v(x) = \frac{x^2-4}{x-2}$

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

$k(x) = 4$

A traffic engineer models the speed of cars on a highway near a bottleneck. For $x ≠ 2$ , the function $v$ defined by $v(x)=\frac{x^2-4}{x-2}$ gives the speed of a car, in miles per hour, when the car is $x$ miles from a reference point along the highway. The engineer is interested in the behavior of $v(x)$ near $x=2$ and in whether the model can be modified to be continuous at $x=2$ .

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

x	v(x)
1.9	3.9
1.99	3.99
2.01	4.01
2.1	4.1

Use the values in Table 1 to estimate $\lim_{x\to 2} v(x)$ . Then write the limit using correct analytic notation.

Find $\lim_{x\to 2} v(x)$ . Show the algebraic work that leads to your answer.

The instantaneous rate of change of speed with respect to position at $x=2$ is defined by a limit. Write an expression for $\lim_{h\to 0}\frac{v(2+h)-L}{h}$ , where $L=\lim_{x\to 2}v(x)$ . Use this limit to find the instantaneous rate of change of speed with respect to position at $x=2$ . Interpret your answer in context, including units.

A new function $w$ is defined by

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

where $k$ is a constant. Find the value of $k$ such that $w$ is continuous at $x=2$ . Justify your answer using the definition of continuity at a point. The function w matches the traffic-speed model v for all x except possibly at x=2 miles. The constant k represents the speed assigned by the modified model exactly at x=2.