Reading Questions

Bradley and Smith, 2nd Edition

  • Review Materials
  • Sections 3.1, 3.5, 3.6 September 4
  • Chapter 5
  • Section 5.1, September 9
  • Section 5.2, September 10
  • Section 5.3, September 11
  • Section 5.4, September 16
  • Section 5.5, September 23
  • Section 5.6, September 25
  • Section 5.7, September 27
  • Section 5.8, September 30
  • Chapter 6
  • Section 6.1, October 2
  • Section 6.2, October 4
  • Section 6.3, October 9
  • Section 6.4, October 11
  • Section 6.5, October 15
  • Chapter 7
  • Section 7.1, October 18
  • Section 7.2, October 22
  • Section 7.3, October 25
  • Section 7.4, October 30
  • Section 4.8, November 1
  • Section 7.5, November 4
  • Section 7.6, November 5
  • Section 7.7, November 6
  • Section 7.8, November 8
  • Chapter 8
  • Section 8.1, 8.2, 8.3 November 11
  • Section 8.4, November 12
  • Section 8.5, November 15
  • Section 8.6, November 22
  • Section 8.7, November 27
  • Section 8.8, December 3

    • Sections 3.1, 3.5, 3.6

    1. From geometry we know it takes two points to determine a line. How is it possible to find a tangent line if we only use the single point (a, f(a))?
    2. Why is the chain rule useful?
    3. Explain the word `implicit' in implicit differentiation.

    Section 5.1

    1. What is the definition of an antiderivative function?
    2. Using the idea of slope fields, explain why a single function can have many different antiderivatives.
    3. For a given positive function f(x) with domain [1,5], describe the graphical meaning of A(4) where A is the area function associated with the function f.

    Section 5.2

    1. What is a `partition of an interval'?
    2. What is the `index of summation' of a summation written in sigma notation?
    3. Limits are used in the definition of `area' in this section of the text. Explain why the limit in the blue box on page 332 represents the exact area of the region under the curve y=f(x) over the interval [a,b].

    Section 5.3

    1. What is the definition of the `norm of the partition P' of the interval [a,b]?
    2. Why is Theorem 5.5 important?
    3. What is the difference between the distance travelled and the displacement of an object over the time interval [a,b]?

    Section 5.4

    1. What's important about the first fundamental theorem of calculus?
    2. What's important about the second fundamental theorem of calculus?
    3. Explain why the variable used in a definite integral is called a `dummy' variable.

    Section 5.5

    1. Which differentiation rule does the method of substitution reverse?
    2. How do you adjust the limits of integration in a definite integral after (or during) a substitution?
    3. What specific line (or lines) of this section are the least clear to you?

    Section 5.6

    1. What is an orthogonal trajectory of a family of curves?
    2. What is the slope field of a differential equation?
    3. Why is a separable differential equation called ``separable''?

    Section 5.7

    1. Explain the graphical meaning of the ``average value'' of a continuous function f on the interval [a,b].
    2. The Intermediate Value Theorem is referenced in the proof of Theorem 5.11 (Mean Value Theorem for Integrals). State the Intermediate Value Theorem.
    3. Specify the part of this section you found most difficult to understand.

    Section 5.8

    1. Discuss the Trapezoidal error and Simpson's error. Specifically, explain why the formulas for those errors indicate Simpson's Rule should be thought of as a more accurate approximation technique than the Trapezoidal Rule.
    2. Why don't the authors choose n = -58 for their answer in Example 4?
    3. What is roundoff error and does it occur in your calculator?

    Section 6.1

    1. If two curves intersect one or more times, how do you use the techniques of this section to find the area of the region between them?
    2. What is meant by the phrase `leading curve' is the discussion on area by horizontal strips?
    3. Explain the meaning of each of the functions in the net earnings formula P(x)=R(x)-C(x).

    Section 6.2

    1. What does the `cross sectional area' function A(x) measure.
    2. Does the disk method find volumes by using cross sectional areas or does it use some other technique?
    3. How does the washer method differ from the disk method for finding volumes?

    Section 6.3

    1. Can you use the method of cylindrical shells to find the volume of any solid or will it only work for special types of solids?
    2. Can you use the method of cylindrical shells to find the volume of a solid of revolution if that solid is obtained by revolving a region about a horizontal axis of revolution or is it only restricted to solids obtained by revolving about a vertical axis of revolution?
    3. The expression 2p x occurs in the formulas on page 426, what is the geometric reason for this expression occurring in the formulas?

    Section 6.4

    1. What is the geometric meaning of the first expression involving the square root in the blue box on page 434?
    2. Does the formula for surface area apply to any surface or does it only apply to certain ones?
    3. What portion of this section was least clear to you?

    Section 6.5

    1. What is Hooke's Law for springs?
    2. The Riemann Sum for computing the work done by a variable force is explicitly written on page 442. On what page, and where on that page (which line) is the Riemann Sum used in solving Example 3 written?
    3. Which of the three applications (Modeling fluid pressure and force, modelling the centroid of a plane region, Volume theorem of Pappus) would you most like to see more details about?

    Section 7.1

    1. Does the large number of integrals in the Appendix D integral table, mean you will never find an integral that requires the use of substitution?
    2. What is a reduction formula?
    3. What substitution does the book suggest for an integral involving fractional exponents?

    Section 7.2

    1. Integration by parts reverses which rule of differentiation?
    2. Since applying integration by parts to a given integral results in a different integral, what do we hope to gain?
    3. How do we compute a definite integral using integration by parts?

    Section 7.3

    1. Why does the book suggest we substitute u = tan(x) when an integral has an an even power of sec(x) function?
    2. What is the first step in solving a problem where an integral contains even powers of cosine as well as even powers of the sine?
    3. What does it mean to `complete the square' of a quadratic?

    Section 7.4

    1. What does it mean to say that a rational function is reduced?
    2. If the denominator of a reduced rational function is (x-3)^3(x+5)^4, how many fractions would there be in the partial fractions decomposition of the rational function?
    3. Suppose the denominator of a reduced rational function is (x-1)^2(x^2+4)^2. List the four denominators of the fractions obtained by a a partial fraction decomposition of the original rational function.

    Section 7.5

    1. What is the `Weierstrass substitution'?
    2. How many different forms of integrals does the book say you should look for when you classify an integration by parts integral?
    3. Which is easier to do: integration or differentiation? Why?

    Section 7.6

    1. What does it mean to say the function I(x) is an `integrating factor' for a first order linear differential equation?
    2. In the logistic equation of example 3, B denotes the `carrying capacity' of the logistic model. What does this mean?
    3. In the dilution model of example 4, why does the author set S'(t) = (inflow rate) - (outflow rate)?

    Section 4.8

    1. What does it mean to say a limit has an `indeterminate form'?
    2. What does it mean to say a limit does not have an indeterminate form?
    3. How many indeterminate forms are listed in the examples of this section?

    Section 7.7

    1. What does it mean to say an improper integral diverges?
    2. What is the difference between an improper integral of the first type and an improper integral of the second type?
    3. Does the example of Gabriel's Horn make sense to you?

    Section 7.8

    1. Are any of the six hyperbolic functions even? Which? Are any odd? Which?
    2. What is the derivative of cosh(2x)?
    3. What topic in this section would you like to have explained in greater detail?

    Sections 8.1, 8.2, 8.3

    1. What does it mean for a sequence to be eventually monotonic?
    2. What is a telescoping series?
    3. For what values of p does the p-series S(1/n)p converge? For what values of p does the sequence an = (1/p)n converge?

    Section 8.4

    1. What does it mean to say a series is dominated by a second series?
    2. If a given series dominates a divergent series what can we say about the convergence or divergence of the first series?
    3. The proof of the Direct Comparison Test requires that the terms of both series consist of non-negative terms. Why?

    Section 8.5

    1. What can you conclude about a series for which the ratio test ends with a limit of L=1?
    2. Why is the ratio test ``most useful with series involving factorials or exponentials''?
    3. Would the root test work well with series involving factorials? Would it work well with series involving exponentials?

    Section 8.6

    1. Why can't you use a comparison test on an alternating series?
    2. Do you understand the conclusion of Theorem 8.19? Specifically, if S is the infinite sum of an alternating series, do you understand why the error made in using the n'th partial sum of the series to approximate the infinite sum must be less than the (n+1)'st term of the series?
    3. What does it mean to say a series is conditionally convergent?

    Section 8.7

    1. What is the radius of convergence of a power series that converges for all values of x?
    2. How do you take the derivative of a power series?
    3. Must a power series converge at the endpoints of its interval of convergence?

    Section 8.8

    1. The book has formulas for the Taylor and Maclaurin series associated with the function f(x). What is the coefficient of the x2 term in the Maclaurin series for that function?
    2. Example 7 shows it is possible to use a Taylor polynomial to approximate ln(2) to an accuracy of 0.000005. Is it possible to use a Taylor polynomial to approximate ln(2) to 1000-place accuracy?
    3. If you had the power to say yes or no to reading assignments and questions in your next mathematics class, which way would you decide? Why?