AP Calculus AB/BC FRQ Practice

♾️AP Calculus AB/BC

Guided Practice

Practice FRQ 1 of 81/8

1. The following functions are defined for this question:

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

k(x) = 4

A temporary channel is built to move water away from a construction site. For

x≠ 2

, the function

f

defined by

f(x)=\frac{x^2-4}{x-2}

models the flow rate of water, in cubic meters per minute, at position

x

meters along the channel. A graph of

y=f(x)

is shown in Figure 1.

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

Figure 1. Graph of y = f(x) for f(x) = (x^2 − 4)/(x − 2), x ≠ 2 (flow rate vs. position).

Single 2D graph on a clean coordinate plane (no grid).

Axes (all required items):
- Horizontal axis labeled exactly: "x (meters)". The x-axis runs from −1 to 5 with tick marks every 1 unit, labeled −1, 0, 1, 2, 3, 4, 5. The origin is labeled "0" at the axes intersection. Put an arrow on the positive end of the x-axis.
- Vertical axis labeled exactly: "f(x) (cubic meters per minute)". The y-axis runs from 0 to 8 with tick marks every 1 unit, labeled 0 through 8. The origin label "0" is at the intersection (shared with x-axis). Put an arrow on the positive end of the y-axis.

Curve shape description (path and key features):
- The graph is a straight line with positive slope everywhere it is drawn, matching the linear rule f(x) = x + 2 for all x-values shown except the single excluded x-value where the hole occurs.
- On the left edge of the window (at x = −1), the line is at y = 1. From there it rises uniformly (constant slope) as x increases.
- The line passes exactly through the y-axis at y = 2 (so the y-intercept is 2).
- The line continues rising through x = 1 at y = 3, then reaches the excluded x-value at x = 2.

Discontinuity (removable hole) with exact numeric placement:
- At x = 2, do NOT draw a filled point. Instead, leave a small gap in the line and draw an open circle exactly on the line directly above the tick labeled 2 on the x-axis.
- The open circle is placed at y = 4 (aligned horizontally with the y-axis tick labeled 4), indicating the value the line would have there.
- The straight line is drawn on both sides of x = 2 as two separate solid segments: one segment from x = −1 up to but not including x = 2, and a second segment starting just to the right of x = 2 through the right edge of the window.

Right-side continuation:
- After the open circle location, the line continues with the same constant slope upward, passing through x = 3 at y = 5, through x = 4 at y = 6, and reaching y = 7 at the right edge x = 5.

Curve behavior summary (segment-by-segment, as required):
- Segment for x < 2: perfectly straight solid line, increasing left-to-right, constant slope; no curvature (neither concave up nor concave down).
- At x = 2: removable discontinuity shown by a single open circle at y = 4 with a visible break in the line; no filled point at that x-value.
- Segment for x > 2: perfectly straight solid line, increasing left-to-right with the same constant slope as the left segment; no curvature.

Styling:
- Draw the line in solid black, medium thickness.
- Draw the open circle in black with a white interior, clearly visible and centered exactly at the intersection of x = 2 and y = 4 alignment.
- No extra annotations, no equation text inside the plotting area, and no legend.

A. Use the definition of average rate of change to find the average rate of change of

f

over the interval

1≤ x≤ 3

. Show the setup for your calculations.

B. Use average rates of change over intervals containing

x=2

to estimate the instantaneous rate of change of

f

x=2

. Show the setup for your calculations.

C. Write a limit expression that represents the value of

f(x)

x

approaches

2

, and evaluate the limit.

The function

g

matches

f

for all

x≠ 2

but assigns the value

k

x=2

to remove the discontinuity.

D. A new function

g

is defined by

g(x)=\begin{cases}f(x),&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.

♾️AP Calculus AB/BC

Guided Practice

Practice FRQ 1 of 81/8

1. The following functions are defined for this question:

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

k(x) = 4

A temporary channel is built to move water away from a construction site. For

x≠ 2

, the function

f

defined by

f(x)=\frac{x^2-4}{x-2}

models the flow rate of water, in cubic meters per minute, at position

x

meters along the channel. A graph of

y=f(x)

is shown in Figure 1.

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

Figure 1. Graph of y = f(x) for f(x) = (x^2 − 4)/(x − 2), x ≠ 2 (flow rate vs. position).

A. Use the definition of average rate of change to find the average rate of change of

f

over the interval

1≤ x≤ 3

. Show the setup for your calculations.

B. Use average rates of change over intervals containing

x=2

to estimate the instantaneous rate of change of

f

x=2

. Show the setup for your calculations.

C. Write a limit expression that represents the value of

f(x)

x

approaches

2

, and evaluate the limit.

The function

g

matches

f

for all

x≠ 2

but assigns the value

k

x=2

to remove the discontinuity.

D. A new function

g

is defined by

g(x)=\begin{cases}f(x),&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.