Posted: May 7th, 2015

Calculus 1

MTH 151: Calculus 1
Writing Assignment 2: KEY CONCEPTS OF CALCULUS
Due: Friday, May 8
. The questions cover (most of) the key concepts we learned throughout Calculus. They are meant to help
you summarize what you learned in the course. It is also my hope that this helps a little in your preparation
for the nal exam.

the paper should be as long as needed to fully answer the questions. You will proba-
bly need to devote a paragraph (or more) to answer each question completely. In your
paper, please headline
each question with the concept, and then provide your answer. You do not need to restate the question, but
you may do so if you like. You must hand in a typed, physical copy to me by the due date (or earlier). Feel free
to ask me any questions you have regarding the assignment or your paper specically. Additionally, I can give
a quick proofread of your paper before you hand it in.
Anytime I use the words “explain” or “describe”, I mean for you to do so in your own words. You may
want to include graphs to help explain the concepts, even when I don’t explicitly ask for a graph. You may
sketch the graphs by hand provided that they are neat! Here are the exercises:
(1) Limit Give a description of the notation in the equation lim
x!a
f(x) = L. Explain what this equation means
and illustrate with a sketch and/or table of values. Is it possible for this statement to be true if f(a) 6= L? How
about if lim
x!a??
f(x) 6= lim
x!a+
f(x)? Explain.
(2) Asymptote Describe the meaning of the limits: lim
x!a+
f(x) = 1, lim
x!1
f(x) = L. In each case, does
the limit exist? Explain.
(3) Continuity Explain what it means for a function f to be continuous at a number a. What are the
three conditions that f must satisfy in order to be continuous at a? If f fails one (or more) of these conditions,
then f is said to be discontinuous at a. For each of the three conditions, sketch the graph of some function that
fails the condition (and that hence is discontinuous at a).
(4) Derivative Explain the meaning of lim
x!a
f(x) ?? f(a)
x ?? a
, or alternatively, lim
h!0
f(x + h) ?? f(x)
h
. What does
this quantity represent graphically? You may wish to explain the denition from the perspective of solving the
Tangent Line Problem. What are the three situations in which a function is not dierentiable? How does each
case fail the denition of dierentiability? Sketch three separate graphs illustrating the three cases of nondier-
entiability.
(5) Maximum/Minimum Value Explain the dierence between an absolute maximum value and a local
maximum value. Is every local maximum value also an absolute maximum value? Is every absolute maximum
value also a local maximum value? Illustrate with a sketch.
(6) Integral Explain the meaning of lim
n!1
Xn
i=1
f(xi
)x. What does this quantity represent graphically? You
may wish to explain the denition from the perspective of solving the Area Problem. What is the dierence
betweeen the denite and the indenite integral? What does each represent? Explain.
(7) Connection Describe what you think is the connection between the limit, derivative, and denite inte-
gral.
1
Grading Rubric
This paper:
Answers all questions completely
Includes graphs when prompted
Is well-organized, coherent, well-written, and demonstrates a command of the English language
Explains concepts in the author’s own words
Explains concepts in a clear, succinct manner
Shows mastery of the various Calculus concepts