AP Calculus AB Introducing Limits Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Introducing Limits – Maths PDF

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This limits intro test answers for AP Calculus AB covers introducing limits (Common Core AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4), with a free PDF to print.

Use this answer key to score the limits test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 1.1, AP CALC AB 1.2, AP CALC AB 1.3, AP CALC AB 1.4

Limits intro test answers: worked solutions

Question 1 – Estimating from a table [2 pts]

Outputs: \(11.41, 11.94, 12.06, 12.61\). They approach \(12\).

Final answer: \(12\).

Scoring: 1 point for a correct table; 1 point for the correct limit.

Question 2 – Reading a graph [2 pts]

\(5\): the left piece approaches height 5.
\(4\): the right piece starts at height 4.
\(f(2)=1\): the filled point.
does not exist because \(5\neq 4\).

Graph of f near x = 2.

Scoring: 1 point for parts (a) and (b) both correct; 1 point for parts (c) and (d) both correct.

Question 3 – Piecewise function [2 pts]

Left: \(4-(-1)^{2}=3\). Right: \(2(-1)+5=3\).

Both one-sided limits equal \(3\), so the limit exists and equals 3.

Scoring: 1 point for both one-sided limits; 1 point for the conclusion with justification.

Question 4 – Parameter for a limit [3 pts]

Right-hand limit: \(9-5=4\). Left-hand limit: \(3k-2\).

Set \(3k-2=4\), so \(k=2\). The common value is \(4\).

Final answer: \(k=2\), limit \(4\).

Scoring: 1 point for the right-hand limit; 1 point for the equation; 1 point for k and the limit.

Question 5 – Instantaneous speed [2 pts]

\(s(2)=20\). Averages: \(\frac{s(2.1)-20}{0.1}=20.5\), \(20.05\), \(20.005\) meters per second.

They approach \(20\). Final answer: \(20\text{ m/s}\).

Scoring: 1 point for the three correct averages; 1 point for the estimate with units.

Question 6 – Unbounded behavior [3 pts]

The outputs increase without bound from both sides.
No. The outputs do not approach a single real number; the limit is unbounded (\(\infty\)).
\(k(0.9)=\frac{1}{0.01}=100\) and \(k(1.1)=\frac{1}{0.01}=100\).

The graph of k(x) = 1/(x − 1)².

Scoring: 1 point per part.

Question 7 – Taxi fare jump [3 pts]

\(\lim_{d\to 1^-}F(d)=8\) and \(\lim_{d\to 1^+}F(d)=11\) dollars.
No; the one-sided limits differ, so the fare jumps and the two-sided limit does not exist.

Scoring: 1 point for each one-sided limit; 1 point for the justified conclusion.

Question 8 – Table and conclusion [3 pts]

Left-hand limit \(-1\); right-hand limit \(1\).
Does not exist because \(-1\neq 1\).

Values of h near x = 0.

Scoring: 1 point for each one-sided limit; 1 point for the justified conclusion.

Total score

Total

20 points

Key facts behind this limits intro test answers

Key facts

A limit describes nearby behavior, not the value at the point itself.
The two-sided limit exists only if the left-hand and right-hand limits are equal.
Tables and graphs give estimates; algebra gives exact values.
A limit fails to exist because of a jump, unbounded growth or oscillation.

How to use this limits intro test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

All AP Calculus AB chapters: AP Calculus AB math.

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