Reading Assignments for Calculus 1
    Fall 2010, Math 101

    October, 2010



    Be sure to check back often, because assignments may change!
    (Last modified: Thursday, October 28, 2010, 11:21 AM )


    All section and page numbers refer to sections from Calculus: Early Transcendental Functions, 3rd Edition, by Smith and Minton.

    I'll use Maple syntax for some of the mathematical notation on this page. (Paying attention to how I type various expressions is a good way to absorb Maple notation). I will not use it when I think it will make the questions too difficult to read.


    Due Friday 10/1 at 8am

    Project 1 (continued)

    To read: You should have read through the entire letter before lab on Thursday. Before class on Friday, re-read the letter, and review the work you did in lab on Thursday.

    No Reading Questions Today

    Reminder:


    Due Monday 10/4 at 8am

    Section 2.5: The Chain Rule

    To read: Through Example 2.5.4. In Example 2.5.3, pay particular attention to how rewriting h(x) in various ways allows you to think of different approaches to differentiating the function.

    Reading questions:

    1. Consider the "simple examples" that open the section, on page 189.
      1. Why would you speculate that the derivative of (x2+1)4 is 4(x2+1)3(2x)?
      2. If, as the book suggests toward the bottom of page 189, we let g(x)=x2+1 and f(x)=x4, then what is f '(x)? f '[g(x)]? g'(x)? How are these pieces related to the conjectured derivative above?
    2. When do you use the chain rule to differentiate?

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    Reminder:


    Due Wednesday 10/6 at 8am

    Bring Questions for Exam 1

    No Reading Questions Today
    Reminders:


    Due Friday 10/8 at 8am

    Section 2.5: The Chain Rule

    To read: Review the chain rule.

    No Reading Questions Today

    Reminder:


    Due Monday 10/11 at 8am

    Enjoy October Break!


    Due Wednesday 10/13 at 8am

    Section 2.6: Derivatives of Trigonometric Functions
    Section 2.7: Derivatives of Exponential and Logarithmic Functions (for Thursday)

    To read: All. In Section 2.6, pay particular attention to the clever steps in the proof of Lemma 2.6.4, and to the use of trigonometric identities and factoring in the proof of Theorem 2.6.1. In Section 2.7, pay particular attention to the way the chain rule comes into play wiht exponential functions in Examples 2.7.2 and 2.7.3. Also, just be aware that we will be seeing a slightly different proof of Theorem 2.7.3.

    Reading questions:

    1. What is lim h → 0 sin(h)h?
    2. What is lim h → 0 (1-cos(h))h?
    3. Why are the limits in the first two questions so important in this section?
    4. Use the derivative of cosine and rules from Section 2.4 or 2.5 to differentiate 1cos(x).

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    Reminder:


    Due Friday 10/15 at 8am

    Section 2.6 and Section 2.7

    To read: Review both sections.

    No Reading Questions Today

    Reminder:


    Due Monday 10/18 at 8am

    Section 2.9: The Mean Value Theorem

    To read: All. As the text suggests, be sure to try drawing functions where f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), but doesn't have a horizontal tangent anywhere on (a,b).

    Reading questions:

    1. Suppose f(x) is continuous on [a,b] and differentiable on (a,b). Does Rolle's Theorem tell you how to find the point c in (a,b) such that f '(c)=0?
    2. Why do we say that Rolle's Theorem is a special case of the Mean Value Theorem?
    3. In the proof of Theorem 2.9.5, why does showing that f(b)=f(a) for the arbitrary points a and b allow us to conclude that f is constant on the interval I?

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    Reminder:


    Due Wednesday 10/20 at 8am

    Introduction to Chapter 3
    Section 3.2: Indeterminate Forms and l'Hôpital's Rule

    To read: All

    Reading questions:

    1. Does l'Hôpital's Rule apply to limx→ ∞ x2ex ? Why or why not?
    2. Does l'Hôpital's Rule apply to lim x→ ∞ x2 sin(x) ? Why or why not?

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    Reminders:


    Due Friday 10/22 at 8am

    Section 3.3: Maximum and Minimum Values

    To read: All

    Reading questions:

    1. For each of the intervals below, please answer the following question: If f is continuous on the interval, must f have an absolute maximum and an absolute minimum on that interval? If not, can it have an absolute maximum or an absolute minimum on that interval?
      1. [-5,5]
      2. (-∞, ∞)
    2. Does the Extreme Value Theorem help with finding absolute extrema, when it applies?
    3. Suppose that c is a critical number for f. Must f have a local extremum at x=c?
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    Reminders:


    Due Monday 10/25 at 8am

    Section 3.4: Increasing and Decreasing Functions

    To read: All. As always, pay attention to the marginal graphs, comments, and charts -- most people find the charts that help determine where a function is increasing and decreasing to be very helpful. Also play close attention to the last paragraph in Example 3.4.1, as well as the paragraph in between Example 3.4.1 and 3.4.2.

    Reading questions:

    1. The graph of a function f '(x) from x=-10 to x=10 is shown below. Estimate the intervals on which f is increasing. How about decreasing?

    2. Use the Calculus methods from this section to determine on what intervals the function g(x), where g(x) is given below, is increasing and decreasing.
      g(x)=6x4-3x2+5000

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    Reminders:


    Due Wednesday 10/27 at 8am

    Section 3.5: Concavity and the Second Derivative Test

    To read: All. Once again, and as usual, pay attention to all marginal comments, graphs, and charts. The charts for keeping track of concavity and inflection points are just as useful as those for determining direction and extrema.

    Reading questions:

    1. On page 287, consider Figures 3.5.55 a and b, and 3.5.56 a and b. In figure 3.5.55a, the graph is getting steeper, while in Figure 3.5.56a, the graph is getting flatter -- yet in both cases, the authors write that the slopes of the tangent lines are increasing. Similarly, in Figure 3.5.55b, the graph is getting flatter, while in Figure 3.5.56b, the graph is getting steeper -- yet in both cases, the authors say that the slopes of the tangent lines are decreasing. What is going on here?
    2. Are all points where f ''=0 or is undefined automatically inflection points?
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    Reminder:



    Due Friday 10/29 at 8am

    Section 3.6: Overview of Curve-Sketching

    To read: All

    Reading questions:

    1. What is the point of learning how to sketch curves, given how available graphing tecnology now is?
    2. In Example 3.6.4,
      1. How do the authors conclude that f'(x)<0 for all x ≠ a, -1?
      2. The authors say we can disregard the factor of x2 +3x+3 in the numerator of f ''(x) because it is always positive. What is one way that you can figure out that it is always positive?
      3. Why does the factor x2+3x+3 being always positive mean that we can disregard it when discussing the sign of f ''(x)?

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    Reminder:


    Here ends the reading for October
    Go to the reading assignments for November!
    Janice Sklensky
    Wheaton College
    Department of Mathematics and Computer Science
    Science Center, Room 101A
    Norton, Massachusetts 02766-0930
    TEL (508) 286-3973
    FAX (508) 285-8278
    jsklensk@wheatonma.edu


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