Reading Questions

Hass, Weir, and Thomas

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    Section 1.1

    1. What is the set notation for the graph of the function f whose domain is the interval [1,2] and where, for each x in that domain, f(x)=5x?
    2. What is a rational function?
    3. Are there functions that are neither even nor odd?

    Section 1.2

    1. How do you change the equation y=f(x) so that the graph of the resulting equation is shifted 3 units to the left, one unit up and is then reflected across the x-axis?
    2. What is the center of an ellipse?
    3. What is the standard equation for a circle centered at the point (h,k) and with radius r?

    Section 1.3

    1. Which trigonometric functions are positive in the second quadrant?
    2. What is the law of cosines?
    3. What is the equation whose graph is the result of compressing the sine function to have period 2 and reflecting the graph about the x-axis?

    Section 1.4

    1. What is special about the number e?
    2. If n is a positive integer what does (15)^{1/n} mean? [The symbol "^" represents exponentiation.]
    3. Is y=(1/2)^x a valid exponential equation?

    Section 1.5

    1. Can a function that is not one-to-one have an inverse function?
    2. Simplify 2ln(e^6).
    3. What is the domain of the inverse cotangent function? Why isn't this domain the set of all real numbers?

    Section 1.6

    1. What type of calculator do you have? Do you have the manual?
    2. Does your calculator default to a square viewing window?
    3. Does your calculator graph both branches of y=x^{1/3}?
    4. How are you supposed to know when your calculator makes a mistake (like only graphing the positive branch of y=x^{1/3}?

    Section 2.1

    1. What does "delta y" represent in Example 1?
    2. What is the relationship beteen the average rate of change of a function and the secant lines to the graph of that function?
    3. If it exists, what is an "instantaneous rate of change" of a function f?

    Section 2.2

    1. If the functions f and g both have limits as x goes to c, does the function fg also have such a limit and, if so, what does it equal?
    2. Which limit rules from Theorem 1 (Limit Laws) are used in computing the limit in part (c) of Example 5?
    3. Why did the authors do so much algebra in working Example 9?

    Section 2.3

    1. What do L, epsilon, and delta represent in the Definition of limit on page 75?
    2. In your own words, explain the "Challenge-Response" figures on page 76.
    3. Examples 6 and 7 prove two results about the behavior of limits. Why do the authors go to this effort instead of just saying "the limit of the sum is the sum of the limits" and "the limit of a dominated function is dominated by the limit of the dominating function"?

    Section 2.4 (one-sided limits)

    1. What is the relationship amongst the limit of a function and the two corresponding one-sided limits of that function?
    2. Give an example of a function f and a number c for which both one-sided limits of f as x goes to c exist but the limit of f as x goes to c does not.
    3. Does it bother you that 1 is the limit as x goes to 2 of the function f in Example 2? Why or why not?

    Section 2.4 (limits at infinity)

    1. Does the infinity symbol represent a number?
    2. What is really meant by the phrase "as x approaches infinity"?
    3. What is a horizontal asymptote for the graph of a function?

    Section 2.5

    1. What is a vertical asymptote of the graph of a function?
    2. Why doesn't the function f(x)=(x^2-x)/(x-1) have a vertical asymptote at x=1?
    3. In Example 9 the function f has one oblique and one vertical asymptote. Is it possible for a function to have exactly one oblique and exactly one horizontal asymptote?

    Section 2.6

    1. Explain, in your own words, what it means when we say a function f is continuous at the input c.
    2. What is a removable discontinuity?
    3. Is it possible to have functions f and g that are both continuous at every real number and where the composition of f with g is continuous but the composition of g with f is not? Why or why not?

    Section 2.7

    1. What do secants have to do with whether or not a curve y=f(x) has a slope at the point P(x_0, f(x_0)?
    2. According to our authors (with whom I agree), the derivative is one of the two most important mathematical concepts in calculus. What is the name of the other one?
    3. What is the definition of the "tangent at P" of a curve y=f(x)?

    Section 3.1

    1. What is the difference in meaning between the word "derivative" and the word "differentiation"?
    2. What can we learn from the graph of y=f'(x)?
    3. The text gives four different graphical situations where a function f fails to be differentiable at a point P. What are those situations and how do they differ?

    Section 3.2

    1. Did you read the Appendix on the method of Mathematical Induction and did you study it in a previous mathematics class?
    2. What does y''' mean when talking about derivatives?
    3. Do you understand how to use the different notations for the nth derivative of a function y=f(x)?

    Section 3.3

    1. What is "jerk"?
    2. Since "at an instant" there is no time for an object to move, what does instantenous velocity really mean?
    3. What is the difference between the speed and velocity of a moving object?

    Section 3.4

    1. Do you understand the proof that the derivative function of sin(x) is cos(x)? If not, at which line of the proof on page 157 do you "get lost"?
    2. At which values of x are the trigonometric functions differentiable?
    3. What is an example of simple harmonic motion?

    Section 3.5

    1. At which values of x is the absolute value function y=|x| differentiable? What is the formula for this derivative function?
    2. What is the formula for the derivative of the sine function if angles are measured in degrees?
    3. Part (a) of Example 9 gives a parametric curve. The parameter values t=0, t=pi/2 and t=pi correspond to three points on the graph of this curve. What are the coordinates of those three points?

    Section 3.6

    1. What is the difference between an implicitly defined function and a function that is not defined implicitly (an explicitly defined function)?
    2. Which derivative rule is always applied when using implicit differentiation?
    3. What is the "normal" to a lens surface at the point where light enters the lens?

    Section 3.7

    1. How do you parametrize an inverse function?
    2. Explain the method of logarithmic differentiation.
    3. What limit in this section is equal to the number "e"?

    Section 3.8

    1. Why isn't the derivative of the inverse tangent function equal to 1 divided by the derivative of the tangent function?
    2. What is the identity relating any inverse trigonometric function to its inverse cofunction?
    3. Why is the restriction |u|<1 necessary for the derivative of the inverse sine function?

    Section 3.9

    1. What is a related rates problem?
    2. Which one of the six steps in the Related Rates Problem Strategy do you think is the hardest, in general, to implement? Why?
    3. In Example 3, why do the authors differentiate the equation s^2=x^2+y^2 rather than solving for s and then differentiating the result?

    Section 3.10

    1. What is the "linearization" of a differentiable function f at x=a?
    2. If y=f(x) is differentiable what is the differential dy?
    3. What is the differential approximation formula?

    Section 3.11

    1. What is the definition of the hyperbolic tangent function tanh(x)?
    2. What is the difference in the formulas of the derivative of the inverse sine function and the derivative of the inverse hyperbolic sine function?
    3. What is your calculator and can it compute sinh(x) or arcsinh(x)? [arcsinh(x) is a notation for the inverse hyperbolic sine function.]

    Section 4.1

    1. What is the difference between a relative minimum and a global minimum of a function?
    2. What conditions on a function f guarantee that it will have both a global minimum and a global maximum?
    3. What is the difference between a critical point of a function f and a point c where f'(c)=0?

    Section 4.2

    1. Does the Mean Value Theorem apply to every function that is continuous on a closed interval [a,b]?
    2. Do you understand how the Mean Value Theorem guarantees that the function f in Corollary 1 must be a constant function?
    3. Doe the five graphs in Figure 4.19 look to you like they differ by a constant? Explain how you can use the graphs to show any two of the functions that are graphed differ by a constant.

    Section 4.3

    1. Does Corollary 3 tell us that if a function f is increasing on an interval [a,b] then f'(x)>0 at each point x in the interval (a,b)?
    2. What is a local extremum for a function?
    3. Review question: What is a critical point for a function f?

    Section 4.4

    1. Using the first derivative of f as part of your explanation, tell why having f''(x) > 0 for every x in the interval (a,b) forces the function f to be concave up on that interval.
    2. What does the Second Derivative Test tell us about a number x=c where f'(c) = 12 and f''(c) < 0?
    3. How do you check for horizontal asymptotes of a rational function? [See Example 7 for one such function.]

    Section 4.5

    1. Explain why the paragraph near the bottom of page 272 that begins "Notice from the graph that ..." shows that the function A in Example 2 must have a global minimum.
    2. Using derivatives, explain why, in Figure 4.40, the maximum profit must occur at a point where the slope of the Cost curve is the same as the slope of the Revenue curve.
    3. What is the Law of Refraction?

    Section 4.6

    1. How many different indeterminate forms are mentioned in this section? What are they?
    2. Why did combining the fractions using a common denominator in Example 6 allow the author's to compute the given limit?
    3. Why is "0/0" called an indeterminate form?

    Section 4.7

    1. In the box on page 292 the authors say "Use the first approximation to get a second ..." Fill in the details on this process.
    2. What is the intuitive meaning of the idea of the convergence of the approximations x_n?
    3. Will Newton's method always converge to the root you seek no matter what starting value you use?

    Section 4.8

    1. What is the most general antiderivative for sin(kx)?
    2. What is an initial value problem?
    3. What is the indefinite integral of a function f?

    Section 5.1

    1. What is the midpoint rule for estimating areas bounded by the graphs of functions?
    2. What is an antiderivative of a function (see Section 4.8).
    3. What is the difference between displacment and distance travelled.

    Section 5.2

    1. Explain why the Constant Multiple Rule for Finite Sums is valid.
    2. What do we mean by the limiting value of a finite sum? Do all finite sums have limiting values?
    3. What is a Riemann sum for a function f on the interval [a,b]?

    Section 5.3

    1. In a definite integral, what is the integrand?
    2. Use the idea of area bounded by a non-negative function to explain, in your own words, why item 6 (Max-Min Inequality) in Table 5.3 is true.
    3. Do all functions have definite integrals? If not, what functions are we sure are integrable?

    Section 5.4

    1. What does the Mean Value Theorem for Integrals have to do with the average value of a function?
    2. What is the total area of region bounded by the graph of a function y=f(x) and the x-axis over the interval [a,b]?
    3. Why do most people like Part 2 of the Fundamental Theorem?

    Section 5.5

    1. How can you check if an integration using the substitution rule is correct?
    2. What is the difference between an indefinite and a definite integral?
    3. Is it possible that two different substitutions can be used to solve a single integration problem?

    Section 5.6

    1. What is the definition of an even function?
    2. Why is the definite integral of an odd function over the interval [-a,a] equal to zero?
    3. How do you find the area of the region between two functions, f and g, if f(x) is greater than or equal to g(x) for every x in the interval?

    Section 5.7

    1. What is the definition of the number e?
    2. Explain what ln(3) is as an area.
    3. Explain how you would use the information in chapter 5 to approximate ln(3) to 4 decimal places of accuracy. DO NOT do the approximation.

    Section 6.1

    1. What is Cavalieri's Principle?
    2. How does the Washer Method differ from the Disk Method?
    3. Both the Disk and Washer Methods use the "method of slicing". Can the method of slicing be used for any solids other than solids of revolution?

    Section 6.2

    1. Does the Shell Method only apply to volumes of solids of revolution or can it be used to find the volumes of other types of solids?
    2. What is the "thickness variable" if the solid of revolution is obtained by revolving a region about a horizontal line?
    3. Is it possible that finding the volume of a solid of revolution will be easier using the shell method than using the disk method?

    Section 6.3

    1. On what page of our text is "parametric curve" defined? Give an example of a parametric curve that is not listed in section 6.3 of the book.
    2. How do the authors convert a continuously differentiable function y=f(x) into a parametrized curve?
    3. What does "ds" equal?

    Section 6.4

    1. What is a frustum of a cone?
    2. Why is formula (2) on page 416 not a Riemann Sum?
    3. What is a surface of revolution?

    Section 6.5

    1. What is the difference between exponential growth and exponential decay? How is this difference reflected in the rate constant?
    2. What is a separable differential equation?
    3. What is the definition of the half-life of a radioactive substance?

    Section 6.6

    1. No reading assignment.

    Section 6.7

    1. What is the system torque of a physical system of masses on a rigid axis?
    2. Are you comfortable with the development of the center of mass for masses distributed on a rigid x-axis?
    3. What is a centroid?

    Section 7.1

    1. What derivative formula is reversed by the Integration by Parts formula?
    2. What is the goal of using integration by parts?
    3. Do you understand the method used in Example 4?

    Section 7.2

    1. Which trigonometric identity mentioned in this section is most unfamiliar to you?
    2. In the Products of Powers of Sines and Cosines subsection, why does having m an odd integer lead to a useful substitution?
    3. In the Products of Sines and Cosines subsection, would you rather memorize the three identities or have to do double integration by parts? Why?

    Section 7.3

    1. If x=a tan(z) what does "a |sec(z )|" equal?
    2. In example 3, why did the authors factor the 25 out of the square root?
    3. For what values of theta does the function x=a sin(theta) have an inverse function?

    Section 7.4

    1. What is the purpose of the method of partial fractions?
    2. In using partial fractions, what is the first step if the degree of the numerator is greater than or equal to the degree of the denominator?
    3. If (x^2+x+1)^3 was a factor of the denominator, then you should assign the sum of how many partial fractions to this factor?

    Section 7.5

    1. What is the purpose of a reduction formula?
    2. What are nonelementary integrals?
    3. What is a CAS?

    Section 7.6

    1. What do we mean by the error when approximating a definite integral using either the Trapezoid or Simpson's Rules?
    2. Is the Trapezoid or Simpson's Rule likely to give a better approximation when estimating a definite integral?
    3. What is "M" in the eror bound for Simpson's Rule?

    Section 7.7

    1. In an improper integral of type II, what type of asymptote does the function f(x) have?
    2. What is the purpose of the tests for convergence in the book?
    3. Does the Direct Comparison Test use the Domination Rule of Table 5.3?

    Sections 8.1

    1. What does it mean for a sequence to diverge to negative infinity?
    2. Give an example of a recursively defined sequence that is not in this section of the text.
    3. Explain, in your own words, why the Nondecreasing Sequence Theorem is valid.

    Section 8.2

    1. What is an infinite series and how does it relate to its sequence of partial sums?
    2. Explain how the nth-Term Test for Divergence follows from Theorem 7.
    3. Must the sum of two divergent infinite series diverge? If so, site a pertinent theorem. If not, give an example of two divergent infinite series whose sum converges. [Hint: subtracting is really adding the negative.]

    Section 8.3

    1. Can we apply the integral test (Theorem 9) to the infinite series that sums all of the terms in the sequence {a(n)} where a(n)=(cos(n)?
    2. Give two values of p for which the corresponding p-series diverge and one value of p for which the corresponding p-series converges.
    3. Estimate the sum in Example 4 but use the value n=100. Include as many significant digits as your calculator will compute. [Do not show my your computations, just your answer.]

    Section 8.4

    1. Does the comparison test apply to an infinite series in which there is a negative term? Can you figure out how to adapt the Comparison Test so it can be used for an infinite series in which every term is negative?
    2. Do parts 2 and 3 of the Limit Comparison Test for infinite series on page 530 make intuitive sense to you? Would you be able to show that there are two analogous parts that can be added to the Limit Comparison Test for integrals in section 7.7?
    3. What is the advantage of comparing (or limit comparing) a series to a geometric series or p-series?

    Section 8.5

    1. Explain why, in the proof of the Ratio Test, a_{N+2} < r^2 a_N. (In words, why is the N+2 term of sequence a(n) less than the Nth term of the sequence times the square of r?)
    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. The proof of Theorem 16 uses the fact that the difference of two convergent series is convergent. Use this fact to determine if the sum of two series, one convergent and one divergent, is convergent.
    3. Does it bother you that any conditionally convergent series can be rearranged to converge to any number you choose?

    Section 8.7

    1. Do you understand the proof of Theorem 18?
    2. What are the possibilities for the convergence of a power series?
    3. How do you take the derivative of a power series?

    Section 8.8

    1. What is the difference between a Taylor Series generated by a function f at x=a and a Maclaurin Series generated by f?
    2. Is it possible for a function to have a convergent Taylor Series where that series does not converge to the function?
    3. What is the coefficient for the n th term of the Taylor Series generated by the function f at x=1?

    Section 8.9

    1. Why does showing the remainder term of order n limits to zero for every x in an interval I imply that the Taylor Series generated by the function f converges to f(x) for each x in the interval I?
    2. In the Remainder Estimation Theorem, what does M represent? Give your answer in English (not mathematical symbols).
    3. If the result of evaluating the fourth derivative of a function f at the number 2 is 7, what is the coefficient of the 4th term in the Taylor Series for f?

    Section 8.10

    1. What are the values of x for which the binomial series converges?
    2. If m=7, how many terms are there in the binomial series?
    3. Is it possible to use the binomial series to estimate the square root of 2?

    Section 9.1

    1. What are the Cartesian coordinates [the (x,y) coordinates] of a point with polar coordinates (3,5pi/3)?
    2. Is there only one way to designate a point in the plane using polar coordinates?
    3. What geometric shape has polar equation r=6sin(theta)?

    Section 9.2

    1. What is the Cartesian r-theta plane?
    2. Why isn't the slope of a polar curve r=f(theta) given by the derivative of f with respect to theta?
    3. Do you understand the technique for graphing outlined on page 585?

    Section 9.3

    1. Given a circle of radius r, what is the area of a sector of that circle subtended by an angle of θ radians?
    2. Given the equations in polar coordinates for two curves, is solving the two equations simultaneously guaranteed to give all points of intersection of the curves?
    3. What is the standard parametrized form of a curve whose polar equation is r=f(θ)?