AP Calculus AB
Test: Limits and Continuity
Page 1 of 5
January 31, 2022
(print clearly)
Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the
permission of Te
...
AP Calculus AB
Test: Limits and Continuity
Page 1 of 5
January 31, 2022
(print clearly)
Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the
permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
Directions for Free-Response Section (35 Minutes, 30 points total, 15 per
question)
• Write your answer inside the boxes provided (do not write in the margins). Do not
use extra paper. Use a black or blue pen.
• Answer each of the questions fully. Show all your work. Indicate clearly the methods
you use; you’ll be graded on the correctness of your methods as well as the
accuracy of your final answers. Correct answers without supporting work may not
receive credit. Justifications require mathematical (non-calculator, non-computer)
reasons.
• Most questions will be given in several parts. The answers from one section will often
be used in following sections. Missing points in an earlier section does not mean
you’ll lose points in following sections.
• The test corrector will not assume you used proper set-up and methods if you reach
the correct answer. It’s up to you to communicate the methods you used.
Hints and Tips for Free-Response Section
• Take your time reading the questions.
• You have 35 minutes for the free-response questions. Take some time to think about
the question. If you don’t understand the question, you’re very unlikely to find the
right answer.
Scoring
As a general rule, points are divided as follows:
30% Proper set-up of the problem.
30% Proper method employed for solving the problem.
40% Correct answer.
The Calculator
The calculator can be used for specific tasks, but you must show the set-up in standard
mathematical notation. For this test, you may only use your calculator to graph a
function and find the roots of an equation. If you’re asked to find the x values of the
points of intersection between and , you must write before
stating that the solution is {x=.82413, -.82413}.
What Does "Show your Work" Really Mean?
If you’re asked to use limits to find the vertical asymptote for , then you
shouldn’t put x=3 as the answer. You also need to explain that and
.
If the question is worth two points, one point is given for a solid explanation and one
point is given for the correct answer.
Use Units
If units are given in the problem, make sure you give them in your answer.
Answer the Question
Be very careful to answer the question asked. If you’re asked to find the points of
intersection between and , then {x=.82413, -.82413} is the wrong
answer! The correct answer is: points (-.82413, .67919) and (.82413, .67919).
Formulas
Here is a short list of area and volume formulas that you might need:
For a circle of radius r: Area = , Circumference = .
For a sphere of radius r: Volume = , Surface Area = .
For a right circular cylinder with radius r and height h: Volume = , Surface
Area = .
For a right circular cone with radius r and height h: Volume = .
For a 2-liter bottle of cola: Volume = 2 liters.
Answer:
Free-Response Question 1 no calculator (worth 15 points)
Show your work/justify your answers.
f (x)does not exist
Free-Response Question 2 with calculator (worth 15 points)
2. Sketch a graph of each function described.
A. A function that is continuous everywhere except at x=4. At x=4 the function is
continuous from the right but not continuous from the left.
B. A function that has a removable discontinuity at x= -2 and an infinite
discontinuity at x=3, but is continuous elsewhere.
f ( x )={(x-13)2 ,for 5,xall =real -2 numbers
C. A function such that:
D. Graph and determine if f(x) is continuous at 3. Use limits
to justify your answer.
The function is not continuous at 3 because at 3, the limit for the function is
undefined and the left- and right-hand limits do not match.
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