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♾️AP Calculus AB/BC
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♾️AP Calculus AB/BC

FRQs – Graphing calculator required
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Unit 1: Limits and Continuity
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Practice FRQ 1 of 81/8
1. The following functions are defined for this question: k(x)=x2−4x−2k(x)=\frac{x^2-4}{x-2}k(x)=x−2x2−4​ p(x)=x2−4x2−5x+6p(x)=\frac{x^2-4}{x^2-5x+6}p(x)=x2−5x+6x2−4​
The function kkk is defined for x≠2x≠ 2x=2. The function ppp is defined for x≠2,3x≠ 2,3x=2,3. Figure 1 shows the graphs of these functions. A graphing calculator may be used to help analyze the behavior of these functions near their points of discontinuity and as x→∞x\to\inftyx→∞.

Figure 1. Graphs of y = k(x) and y = p(x) with discontinuities highlighted (hole at x = 2 for both; vertical asymptote at x = 3 for p).

Single coordinate-plane graph showing BOTH functions on the same axes. AXES AND SCALE (must be explicit and numeric): - Horizontal axis labeled "x", drawn as a solid line with an arrowhead on the positive end. - Vertical axis labeled "y", drawn as a solid line with an arrowhead on the positive end. - x-axis tick labels shown at every 1 unit from −2 through 6 inclusive: −2, −1, 0, 1, 2, 3, 4, 5, 6. - y-axis tick labels shown at every 1 unit from −4 through 6 inclusive: −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6. - The visible plotting window is exactly x from −2 to 6 and y from −4 to 6. - No grid is required; if grid is drawn, it must align exactly with the integer tick marks. CURVE 1: y = k(x) Mathematical identity to enforce the correct shape: k(x) = (x^2 − 4)/(x − 2) simplifies to the straight line y = x + 2 for all x ≠ 2. - Draw k(x) as a straight, solid line with slope 1 (rises 1 unit for every 1 unit moved right). - The line must pass exactly through these visible reference intercepts: • y-intercept at x = 0: the line crosses the y-axis at y = 2. • x-intercept at y = 0: the line crosses the x-axis at x = −2. - Mark the removable discontinuity at x = 2 as an OPEN CIRCLE located on the same line at the y-value the line would have there: y = 4. • The open circle must be centered precisely on the intersection of x = 2 and y = 4. • The line must be drawn continuously on both sides of x = 2 but with a visible gap at the open circle (do NOT fill it). - Place a text label "y = k(x)" next to this line in the left half of the graph (for example near x = −1 to 0 region), with a small leader line or positioned so it clearly refers to the line. CURVE 2: y = p(x) Mathematical identity to enforce the correct shape: p(x) = (x^2 − 4)/(x^2 − 5x + 6) = (x + 2)/(x − 3) for x ≠ 2,3. Key required features: 1) Vertical asymptote at x = 3 - Draw a vertical dashed line exactly through the x-axis tick labeled "3". - Label this dashed line "x = 3" near the top of the plotting window. - The p(x) curve must approach this line without touching it: • As x approaches 3 from the left, the curve drops downward without bound (falls toward negative y-values below the window). • As x approaches 3 from the right, the curve rises upward without bound (rises toward y-values above the window). 2) Removable discontinuity (hole) at x = 2 - Since p(x) matches (x + 2)/(x − 3) wherever defined, the missing point at x = 2 must lie at y = −4. - Place an OPEN CIRCLE exactly at the intersection of x = 2 and y = −4. - The curve must approach this open circle smoothly from both sides (no corner), but the point itself is not filled. 3) Horizontal asymptote y = 1 (end behavior) - Draw a horizontal dashed line at y = 1 across the full width of the window. - Label this dashed line "y = 1" near the right side of the window. - The right-hand branch of p(x) (for x > 3) must decrease toward this dashed line from above as x moves right toward 6. - The left-hand branch of p(x) (for x < 3) must also approach this dashed line as x moves left toward −2, staying near but not identical to it. 4) Exact reference points to lock the curve shape - Ensure p(x) crosses the x-axis at x = −2 (because the numerator is zero at x = −2 and the denominator is nonzero there). • The curve must pass through the x-axis exactly at the tick labeled −2 (filled point, because it is defined there). - Ensure p(x) crosses the y-axis at x = 0 with y = −2/3. • Plot this crossing below 0 and above −1, closer to −1 than to 0, consistent with −0.666… . - Ensure p(x) passes through x = 4 with y = 6 (since (4+2)/(4−3) = 6). • Plot a point on the right branch at x = 4 exactly at y = 6, and draw the curve through it. 5) Overall shape of p(x) by interval (must match rational-function behavior) - For x from −2 up to just less than 3: draw a continuous branch that • passes through (x = −2, y = 0), • crosses the y-axis at y = −2/3, • has an open circle at (x = 2, y = −4), • then plunges downward toward negative infinity as it nears the vertical asymptote x = 3 from the left. - For x from just greater than 3 to 6: draw a second continuous branch that • starts very high (above the top of the window) near x = 3+, • goes through the point (x = 4, y = 6), • then decreases while flattening out and approaching the horizontal asymptote y = 1 by the time it reaches the right edge x = 6. STYLING AND LABELING (to remove ambiguity): - Use two clearly distinct styles: • k(x): solid medium-thick line. • p(x): solid medium-thick curve in a different color OR a different line style (e.g., solid black for k and solid dark gray for p), but both must remain clearly readable. - Open circles (holes) must be drawn as unfilled circles with a visible outline. - The dashed asymptote lines (x = 3 and y = 1) must be thinner than the function graphs. - Place a text label "y = p(x)" adjacent to the p(x) curve on the right-hand branch (for example near x = 4 to 5, y between 2 and 5) so it clearly refers to that curve. Everything listed (ticks, labels, holes, asymptotes) must be visible within the window x ∈ [−2, 6], y ∈ [−4, 6].
A. Use correct limit notation to represent lim⁡x→2k(x)\lim_{x\to 2}k(x)limx→2​k(x). Then find the value of the limit. Show the work that leads to your answer.
B. Let xxx approach 2 through the values 1.9, 1.99, 2.01, and 2.1.
(i) Use a calculator to approximate the average rate of change of kkk on the interval [1.9,2.1][1.9,2.1][1.9,2.1].
(ii) Use a calculator to approximate the average rate of change of kkk on the interval [1.99,2.01][1.99,2.01][1.99,2.01].
(iii) Based on your results in parts (i) and (ii), estimate the instantaneous rate of change of kkk at x=2x=2x=2. Justify your reasoning.
C. Write a limit expression that describes the end behavior of p(x)p(x)p(x) as x→∞x\to\inftyx→∞. Then evaluate the limit using limit theorems.
D. Define a new function qqq by q(x)={p(x),x≠2a,x=2q(x)=\begin{cases}p(x),& x≠ 2\\ a,& x=2\end{cases}q(x)={p(x),a,​x=2x=2​ Find the value of the constant aaa such that qqq is continuous at x=2x=2x=2. Justify your answer using the definition of continuity at a point.
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