AP Calculus AB Continuity and the Intermediate Value Theorem Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Continuity and the Intermediate Value Theorem – Maths PDF

Cover of the book AP Calculus AB Math Tests with Answer Keys (PDF)

Download as a PDFThe AP Calculus AB tests book with answer keysPrintable PDFSee the book ›

Sommaire

This continuity test answers for AP Calculus AB covers continuity and the intermediate value theorem (Common Core AP CALC AB 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16), with a free PDF to print.

Use this answer key to score the continuity 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 1.10, AP CALC AB 1.11, AP CALC AB 1.12, AP CALC AB 1.16

Continuity test answers: worked solutions

Question 1 – Three conditions [2 pts]

\(f(3)=5\) is defined; the two one-sided limits are both \(5\), so the limit exists and equals \(5\); the limit equals \(f(3)\). So f is continuous.

\(g(3)\) is undefined, so condition 1 fails: g is not continuous at 3.

Scoring: 1 point for f with all three conditions; 1 point for g.

Question 2 – Classifying discontinuities [2 pts]

For \(g\): the one-sided limits are \(3\) and \(6\), so it is a jump discontinuity.

For \(h\): \(\frac{(x-5)(x+5)}{x-5}=x+5\), so the limit is \(10\) but \(h(5)\) is undefined: removable.

Graph of g near x = 2.

Scoring: 1 point for each classification with a reason.

Question 3 – Repairing a function [2 pts]

\(\frac{(x-4)(x+3)}{x-4}=x+3\), so the limit is \(7\).

Set \(k=7\). Final answer: \(k=7\).

Scoring: 1 point for the limit; 1 point for k.

Question 4 – Continuity at two points [3 pts]

At \(x=-1\): \(-a+b=3\). At \(x=2\): \(2a+b=9\).

Subtract: \(3a=6\), so \(a=2\) and \(b=5\).

Final answer: \(a=2,\ b=5\).

Scoring: 1 point for each equation; 1 point for solving.

Question 5 – IVT and a root [3 pts]

\(f\) is a polynomial, so it is continuous on \([1,2]\).

\(f(1)=1-3-1=-3\) and \(f(2)=8-6-1=1\).

Since \(0\) is between \(-3\) and \(1\), the IVT guarantees a \(c\) in \((1,2)\) with \(f(c)=0\).

Graph of f(x) = x³ − 3x − 1.

Scoring: 1 point for continuity; 1 point for both function values; 1 point for the conclusion citing the IVT.

Question 6 – IVT guarantees [2 pts]

Yes: \(5\) is between \(-4\) and \(8\) and \(f\) is continuous.
No guarantee: \(10\) is outside \([-4,8]\).

Scoring: 1 point per part.

Question 7 – Parking fee steps [3 pts]

\(3\) and \(5\).
No; the one-sided limits differ, so it is a jump discontinuity.
\(t=1\) and \(t=2\)

Fee in dollars for t hours of parking.

Scoring: 1 point per part.

Question 8 – Temperature and the IVT [3 pts]

Yes. \(T\) is continuous on the time interval, \(T(6)=52\) and \(T(12)=81\), and \(70\) is between them. By the IVT, \(T(c)=70\) for some \(c\) between 6:00 a.m. and noon.
No. \(90\) is not between \(52\) and \(81\), so the IVT does not apply.

Scoring: (a) 1 point for continuity and endpoint values, 1 point for the conclusion; (b) 1 point.

Total score

Total

20 points

Key facts behind this continuity test answers

Key facts

Continuity needs three things: f(a) exists, the limit exists, and they are equal.
Removable discontinuities are holes; jump discontinuities have different one-sided limits; infinite discontinuities are vertical asymptotes.
To make a piecewise function continuous, set the one-sided limits equal at the boundary.
The IVT guarantees a value, but only if the function is continuous on the whole closed interval.

How to use this continuity 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.

Download the continuity test answers as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF