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

I'll use WebWork and/or 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 WebWork/Maple notation). I will not use it when I think it will make the questions too difficult to read.

Due Wednesday 5/2 at 8:30am

Section 9.5: Calculus and Polar Coordinates

Reading Questions:
Consider the polar "rectangle" described by α ≤ θ ≤ β and 0 ≤ r ≤ R.

1. What is the shape of this "rectangle"?

   RD says:
   

   The "rectangle" is a radial segment of a circle with radius R, going from θ=α to θ=β.

2. What is the area of this "rectangle"?

   JC says:
   

   The area is
   [^(β-α)⁄_2π] πR²

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Reminder:

• I'll answer questions on WW 13 either Thursday or Friday.

Due Monday 4/30 at 8:30am

Section 9.4: Polar Coordinates

Reading Questions:

1. Why might we use polar coordinates rather than rectangular coordinates?

   KS says:
   

   The polar coordinate system allows us to plot graphs that don't pass the vertical line test on a cartesian plane like circles and similar curves.

2. Give two different representations in polar coordinates of the point with rectangular coordinates (2sqrt(2), 2sqrt(2))

   FS says:
   

   [2√2]²+[2√2]²=r²
   r²=16
   r=4 or -4

   tan(θ)=^2√2⁄_2√2=1
   a=π/4 or 5π/4
   Polar Coordinates: (4,π/4), (-4,5π/4)

3. What is the relationship between the graphs y=sin(x) and y=sin(2x) in rectangular coordinates?

   LH says:
   

   In rectangular coordinates, y=sin(2x) differs from y=sin(x) in terms of a horizontal shrink by a factor of 2.

4. What is the relationship between the graphs r=sin(θ) and r=sin(2θ) in polar coordinates?

   BL says:
   

   In polar coordinates, r=sin(θ) is a simple circle whereas r=sin(2θ) is a four-leaf rose.

Due Friday 4/27 at 8:30am

Section 8.8: Applications of Taylor Series

Reading Questions:

1. If you wanted to find e^sqrt(3) using a Taylor series, what would you use as a base point?

   • EC says: I would use 0. This is because we know that the value [of f^(k)] at 0 is 1 [for all k] whereas with most any other number it would get more complicated.
   • JC says: I would use ln(6) as my starting point. I know the value [of f^(k)] at the value is 6 [for all k], and ln(6) is very close to sqrt(3), with a difference of only about .06.

Due Friday 4/20 at 8:30am

Section 8.7: Taylor Series

Reading Questions:

1. Give two good reasons for writing a known function ( such as cos(x) ) as a power series.

   JC, EL, and KS say:
   

   1. It is then possible to approximate difficult values of that function, such as cos(.6) using a partial sum of the series.
   2. It provides us with an alternate form of a function, which is sometimes more convenient to work with, as it is simply an (infinite) polynomial with adidition and multiplication.
      • This form can allow us to determine whether some complicated limits converge or diverge
      • We can antidifferentiate functions we wouldn't be otherwise be able to.
      • We can also tell exactly what is going on in the function being modeled without dealing with its transcendental features.

Due Wednesday 4/18 at 8:30am

Section 8.6: Power Series

Reading Questions:

1. How do power series differ from the series we have looked at up to this point?

   MB says:
   

   Power series have variables.

2. What is the interval of convergence of a power series? Why have we not discussed the interval of convergence before? Explain in your own words.

   HA says:
   

   The interval of convergence of a power series is exactly as the name describes; the interval over which the series converges.

   We haven't discussed this interval yet because none of the other types of series that we have discussed converge over specific intervals. The reason power series converge over specific intervals is because they involve functions of x.

3. Suppose the power series ∑b_k(x-c)^k has a non-trivial and non-infinite interval of convergence. Where will the midpoint of that interval be?

   RD says:
   

   The midpoint of the interval will be at x = c.

4. As you have read, we can differentiate and integrate a convergent power series term-by-term. Why is this not obvious?

   EL says:
   

   This is not obvious because a series is not a sum but a limit of a sum. So the results of the term-by-term differentiation or integration are (it turns out) true for power series but nor for series with variables in general.

5. Why might you want or need to use a power series?

   NI says:
   

   Because a power series allows us to write functions in a different form. The power series allows us to compute derivatives and integrals easily.

Due Monday 4/16 at 8:30am

Section 8.5: Absolute Convergence and the Ratio Test

Reading Questions:

1. Explain in a couple of sentences why the Ratio Test makes sense.

   SB, EB and MC say:
   

   The ratio test makes sense because either the series can be compared to a convergent geometric series or compared to a series that we can tell diverges using the nth term test. The ratio test incorporates the comparison and nth term test.

   Essentially what's going on is that if lim | ^a_k+1⁄_ak| < 1, the terms approach 0 quickly enough for the series to converge.

   Specifically, when lim | ^a_k+1⁄_ak| < 1, the ratio test makes sense because there will always be a number r between L and 1 where r is greater than | ^a_k+1⁄_ak| for all k > N for some N. Therefore, | a_k+1| < r | a_k|. This leads to the fact that |a_N+k| < r^k|a_N| for all k greater than N. ∑ r^k is a geometric series that converges absolutely, so a_k converges absolutely too.

   When lim | ^a_k+1⁄_ak| > 1, there is a number N > zero where for k≥ N, | ^a_k+1⁄_ak| > 1. This means |a_k+1|> |a_k| > 0 for all k≥ N. This means the limit of a_k cannot be 0, and that a_k diverges according to the kth-term test.

   For L=1, neither of these apply and there is no conclusion.

Due Friday 4/13 at 8:30am

Section 8.5: Absolute Convergence and the Ratio Test

Reading Questions:

1. Give an example of a series that is conditionally convergent. Explain.

   MS says:
   

   A conditionally convergent series is one that converges, while the absolute value series does not converge.

   An example of this is ∑^(-1)k+1⁄ _k^1/4 from k=1...infinity.

   This series is conditionally convergent because ∑¹⁄ _k^1/4 diverges because it's a p-series with p<1, but the original series converges because lim[¹⁄ _k^1/4, k→ ∞] = 0 and ¹⁄ _k^1/4 is a positive, decreasing function.

2. Give an example of a series that is absolutely convergent. Explain.

   EB says:
   

   ∑C_k= ∑^sin(k)⁄ _k³ is a series that is absolutely convergent:

   The series ∑|C_k| ≤ ∑ ¹⁄ _k³ which converges, so the original series converges absolutely.

Due Wednesday 4/11 at 8:30am

Section 8.4: Alternating Series

Reading Questions:

1. Does sum( ^(-1)k⁄_sqrt(k), k=1..∞) converge or diverge?

   JC says:
   

   If lim_k→∞ a_k = 0 and 0 ≤ a_k+1 ≤ a_k for all k ≥ 1, then the series ∑(-1)^ka_k converges.

   • a_k = ¹⁄_sqrt(k).
     As k→ ∞, a_k approaches 0, since the denominator approaches infinity.
   • 0 ≤ ¹⁄_sqrt(k+1) ≤ ¹⁄_sqrt(k):
     • ¹⁄_sqrt(k+1) can never be less than 0 since sqrt(k+1) cannot be negative
     • sqrt(k+1) is always greater than sqrt(k), so ¹⁄_sqrt(k+1) is always less than ¹⁄_sqrt(k).

   Therefore, the alternating series converges.

2. How closely does S₁₀₀ approximate the series sum((-1)^k (1/k), k=1 .. ∞) ? Why?

   CB says:
   

   It approximates within ± 0.0099 because |S-S₁₀₀| ≤ S₁₀₁=¹⁄₁₀₁ ≈ 0.0099

Due Monday 4/9 at 8:30am

Guide to Writing Mathematics
Checklist
Section 8.3: The Integral Test and Comparison Tests

Consider the series ∑ ¹⁄_(k⁴+5).

1. Show that the series converges.

   k⁴+5 ≥ k⁴ for all k ≥ 1.

   Thus ¹⁄_(k⁴+5) ≤ ¹⁄_k⁴ for all k ≥ 1, so
   

   ∑_k=1^{∞ 1}⁄_(k⁴+5) ≤ ∑_k=1^∞1⁄_k⁴, which is a p-series with p=4, and so is convergent.

   Thus by the comparison test, our smaller but still non-negative-term series converges.

2. Estimate the limit of the series within 0.01.

   Want: The remainder of our series, R_N to be less than or equal to 0.01.

   That is, we need to find N so that
   ∑_k=N+1^{∞ 1}⁄_(k⁴+5) ≤ 0.01.

   From our comparison above, we know that
   ∑_k=N+1^{∞ 1}⁄_(k⁴+5) ≤ ∑_k=N+1^∞1⁄_k⁴.

   Thus if we find N so that ∑_k=N+1^∞1⁄_k⁴ ≤ 0.01, R_N will be also.

   Unlike with geometric series, there's no nice expression for ∑_k=N+1^∞1⁄_k⁴.

   However, we can integrate the associated function.

   By the Integral Test,
   ∑_k=N+1^∞1⁄_k⁴ ≤ ∫_k=N^∞1⁄_x⁴dx, so
   R_N ≤ ∫_k=N^∞1⁄_x⁴dx = ... = ¹⁄_3N³.

   Thus if we find N so that ¹⁄_3N³ ≤ 0.01, R_N will be also.

   Solving that inequality for N, we find that we need N ≥ 3.2.

   Thus R₄ ≤ 0.01, and so S₄ is within 0.01 of ∑_k=1^{∞ 1}⁄_(k⁴+5).

   Using Maple, we can find that S₄ is roughly 0.2297, so S is within 0.01 of 0.2297.

3. Is your estimate an over- or under- estimate?

   Because our series is a non-negative term series, S ≥ S_N for any N.

   In particular, S ≥ S₄.

   Thus 0.2297 is an under-estimate.

   0.2297 ≤ S ≤ 0.2297+0.01

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Reminders:

• Finish the mathematics of the project before class. If you run into difficulties, come to my office hours Monday.
• Use the Guide to Writing Mathematics and the checklist to help you write your project.

Due Friday 4/6 at 8:30am

Section 8.3: The Integral Test and Comparison Tests

Reading questions:

1. Explain in a couple of sentences why the Comparison Test makes sense. That is, explain the idea behind it.

   MC says:
   

   The convergence test makes sense because of the sizes of the terms. If the series with larger terms converges, then the series with terms smaller than it must also converge because it is too small to diverge.

   Likewise, if the series with smaller terms diverges, then the series with terms larger than it must also diverge because it is too large to converge.

2. Show that the series ∑ ¹⁄_(2+3^j) converges.

   RD says:
   

   Compare ∑ ¹⁄_(2+3^j) to ∑ ¹⁄_3^j, which converges as j approaches infinity.

   Since 2+3^j > 3^j, we determine that ¹⁄_(2+3^j) < ¹⁄_3^j (for all j), and is therefore bounded above by it.

   By employing the Comparison Test, we see that since ∑ ¹⁄_3^j converges, then ∑ ¹⁄_(2+3^j) must also converge.

Due Friday 3/30 at 8:30am

Section 8.3: The Integral Test and Comparison Tests

Reading Questions:

1. Explain in a couple of sentences (in your own words, of course) why the integral test makes sense. That is, explain the big ideas behind the integral test.

   AS says:
   

   The point of the integral test is to determine whether a series converges or diverges by comparing it to an improper integral that is monotone decreasing on the interval [1,inf] where f(x) is greater than or equal to 0 for all x greater than or equal to 1.

   When using right-hand Reimann sums, the series is less than the integral, so it will converge if the integral converges. By using the left hand Reimann sums, the series is larger than the integral, so if the integral diverges, so will the series!

Due Friday 3/23 at 8:30am

Section 8.2: Infinite Series

Reading Questions:

1. There are (at least) two sequences associated with every series. What are they?

   YZ says:
   

   One is {a_n} [the sequence of terms], and ths second one is {S_n} [the sequence of partial sums ]

2. Does the geometric series ∑(¹⁄₄)^k converge or diverge? Why?

   KZ says:
   

   It converges, because 1/4<1.

   It converges to 4/3 [assuming the sum goes from 0 to infinity].

3. What does the kth Term Test tell you about each of the following series? Explain.
   1. ∑sin(k)

      ZJ says:
      

      lim _x→∞ sin(k) does not equal 0, so ∑sin(k) diverges.

   2. ∑¹⁄_k

      LW says:
      

      As k approaches infinity, 1/k approaches 0. So the nth term test says it might diverge, might converge.

      However, the series is always bigger than the integral of 1/k when k is real. Since the integral of 1/k diverges, the series diverges.

Due Wednesday 3/21 at 8:30am

Section 3.2: Indeterminate Forms and l'Hôpital's Rule
Section 8.1 Sequences of Real Numbers

Reading Questions:

1. Does l'Hôpital's Rule apply to lim_{(x -> infty)} ^x2 ⁄ _e^x ? Why or why not?

   LW says:
   

   L'Hopital's Rule does apply to the limit as x approaches infinity of (x^2/e^x), but the rule must be applied twice:

   lim(x->inf) (x^2/e^x) =(inf/inf)

   lim(x->inf) d/dx(x^2/e^x) =lim(x->inf) (2x/e^x) (inf/inf)

   lim(x->inf) d/dx(2x/e^x) =lim(x->inf) (2/e^x)=0

2. Does l'Hôpital's Rule apply to lim_{(x -> infty)} ^x2 ⁄ _sin(x) ? Why or why not?

   NI says:
   

   No, because this limit does not have the indeterminate form of ∞/∞ or 0/0.

3. Find a symbolic expression for the general term a_k of the sequence {0, 3, 6, 9, 12, 15, . . . }

   MB says:
   

   {ak}={k*3}(k=0 to infinity)

4. Does the following sequence converge or diverge? (You may assume that the obvious pattern that you see is in fact the pattern the sequence will continue to follow!) Be sure to explain your answer.
   {1, 3, 5, 7, 9, 11, 13, . . .}

   HA says:
   

   Diverge: this function is of the form 2n+1 which is not bounded by any value. Furthermore the function does not approach any finite number as it increases in size/input.

5. In Section 8.1, Example 1.10, what two methods of demonstrating that the sequence is increasing were used? Which did you prefer, in that particular situation?

   SB says:
   

   The first method used was looking at the ratio of two successive terms. The second method was taking the derivative of the function. I prefered taking the derivative and it took less steps to find the answer.

   Must note, however, that while if it *is* positive, that means the sequence *is* increasing, if the derivative is sometimes positive and sometimes negative then the sequence might still be increasing, if it dips between integers and is always back on the rise at the integers.

6. Give an example of a sequence (other than the one in Example 1.13) that is bounded and monotonic. (Keep it simple. You may find it helpful to think of a function with a horizontal asymptote at infinity.)

   CZ says:
   

   a_n=1/(n+3) (n=0.. infinity).

   [This sequence is bounded above by 1/3 and below by 0 and is decreasing].

7. Give an example of a sequence that converges even though it is not both bounded and monotonic. (Again keep it simple.)

   EL says:
   

   sin(n)/n (n=1..infinity) is bounded by 1, but is not monotonic. This sequence converges.

Due Friday 3/9 at 8:30am

Section 6.6: Improper Integrals

Reading questions:
Suppose that 0 < f(x) < g(x).

1. If int(g(x), x=1. .infty) diverges, what (if anything) can you conclude about int( f(x), x=1. . infty), and why?

   MS says:
   

   If ₁ ∫ ^∞ g(x)dx diverges, we cannot conclude anything about ₁ ∫ ^∞ f(x)dx.

   Because f(x) is less than g(x), ₁ ∫ ^∞ g(x)dx not converging does not force ₁ ∫ ^∞ f(x)dx to either be convergent or be divergent.

2. If int(f(x), x=1. .infty) converges, what (if anything) can you conclude about int( g(x), x=1. . infty), and why?

   KS says:
   

   Nothing can be said about the second integral because the Comparison Test only specifies relationships with the larger integral as the independent one when it comes to convergence. In this question, we are asked to make a conclusion based on something known about the smaller integral.

Due Wednesday 3/7 at 8:30am

Section 6.6: Improper Integrals

Reading questions:

1. What are the two ways in which an integral may be improper?

   MS says:
   

   The two ways in which an integral may be improper are:

   • if the integrand is discontinuous on the interval [a,b]
   • or if one or both of the limits of integration is infinite.

2. Rephrase the question "Does an improper integral with infinite limits of integration converge?" as a question about the graph of the integrand over that infinite interval.

   EC says:
   

   Does an improper integral with infinite limits have a finite signed area under the graph of the function?

Due Monday 3/5 at 8:30am

Guide to Writing Mathematics
Checklist
Section 6.6: Improper Integrals (Carried over from Friday 3/2)

Reading questions:

1. Rephrase the question "Does an improper integral with a discontinuous integrand converge?" as a question about the graph of the integrand.

   MC says:
   

   Rephrased:
   Does the area under a discontinuous curve approach a (finite) value L?

2. Explain why int( ¹⁄_x³, x=0..3) is improper.

   BL says:
   

   At the point 0, which is a in the integral, the function in discontinuous.

3. Does int( ¹⁄_x³, x=0..3) converge or diverge?

   FS says:
   

   The ₀∫ ^{3 1}⁄_x³ diverges because the limit does not exist:

   ₀∫ ^{3 1}⁄_x³ = lim_{R → 0+}   ( _R∫ ^{3 1}⁄_x³ )
               = lim_{R → 0+}   ( (-1/2)x^-2 from x=R to x=3)
               = lim_{R → 0+}   ( -1/18+1/2R^2)
               = infinity

4. Does int(¹⁄_x³, x=-2..3) converge or diverge?

   JC says:
   

   _-2∫ ^{3 1}⁄_x³ =_-2∫ ^{0 1}⁄_x³ + ₀∫ ^{3 1}⁄_x³

   Already know: ₀∫ ^{3 1}⁄_x³ diverges, so _-2∫ ^{3 1}⁄_x³ also diverges.

Due Monday 2/27 at 8:30am

Section 6.2: Integration by Parts
Section 6.5: Integration Tables and Computer Algebra Systems

Reading questions:

Each integral can be evaluated using u-substitution or integration by parts. Which technique would you use in each case? You do not need to evaluate the integral, but explain your choice.

1. ∫ xe^xdx

   KZ says:
   

   I would use integration by parts for this question. Set u=x and dv=e^xdx, then du=dx and v=e^x; and then we can solve the new integral ∫ e^xdx easily.

2. ∫ xe^x2dx

   CN says:
   

   Substitution, because with substitution one can keep it in one part (that is, substitution gets rid of the product).

Due Friday 2/24 at 8:30am

Section 6.2: Integration by Parts

Reading Questions

1. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?

   MC says:
   

   Integration by parts attempts to undo the product rule technique used in differentiation.

   The integration by parts formula can be easily reached by integrating the product rule.

2. Pick values for u and dv in the integral ∫ xcos(x)dx. Use integration by parts to find an antiderivative for xcos(x).

MB says:

Use u=x and dv=cos(x)dx.

With these, we can determine that du=dx and v = sin(x).

By following the steps for integrating by parts we can determine that the ∫ xcosxdx = xsin(x) - ∫ sin(x)dx= xsin(x)+cos(x) + C

Due Wednesday 2/15 at 8:30am

Section 5.3: Volumes By Cylindrical Shells

Reading questions:

1. Give an example where you would use cylindrical shells rather than washers to find volume.

   LH says:
   

   You would use shells rather than washers if you wanted x-integration about a vertical axis.

   Ex: Let R be the region bounded by the graphs of y=cos(x) and y=x². Compute the volume of the solid formed by revolving R about the line x=2.

2. Let S be a solid formed by rotating about the y-axis.

   What is the variable of integration, if you're finding the volume using shells? What is the variable of integration, if you're finding the volume using washers?

   SB says:
   

   Using shells, we would intrgrate with respect to x, but using washers we would integrate with respect to y.

Due Friday 2/10 at 8:30am

Section 5.2: Volume: Slicing, Disks and Washers

Reading questions:

1. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=3. What is the shape of the solid formed when R is rotated about the x-axis?

   RD says:
   

   The rotation forms a cylinder.

2. Let T be the triangle formed by the lines y=x, x=1 and the x-axis. What is the shape of the solid formed when T is rotated about the x-axis?

   KP says:
   

   The shape is a cone when T is rotated about the x axis.

3. Let R be the region bounded by the parabolas y=2(x-3)²-8 and y=(x-3)²-4.
   

   Note: As in your reading, this phrasing means that you look for the region that is completely enclosed by only these two curves.

   1. As best you can, describe the shape of the solid formed when R is rotated about the x-axis.

      EB says:
      

      The solid formed when R is rotated about the x-axis resembles a football with a smaller, thinner football missing from its center.

   2. Describe the solid formed when R is rotated about the y-axis.

      CW says:
      

      It would form a shape like a bowl, with the center raised

Due Wednesday 2/8 at 8:30am

Reading questions:
Let f(x)=10sin(πx) and g(x)=5sin(πx).

1. Find the area of the region bounded by y=f(x) and y=g(x) between x=0 and x=1.

   LW says:
   

   When x=0 or x=1, f(x)=g(x).

   Area = ₀∫  ¹ [10sin(πx)-5sin(πx)]dx = - ^5cos(π)⁄_π + ^5cos(0)⁄_π = ⁵⁄_π + ⁵⁄_π = ¹⁰⁄_π

2. Set up (but do not evaluate) the integral that gives the length of the curve y=g(x) from x=0 to x=1.

   BL says:
   

   Arclength = ₀∫  ¹ sqrt(1+[5πcos(π x)]²)dx

Due Monday 2/6 at 8:30am

Section 4.7: Numerical Integration

Reading questions:

1. Explain in words what K represents in Theorem 7.1.

   MS says:
   

   In Theorem 7.1, K is a constant that is representative of the concavity of a function. The more concave the function, the greater the value of K.

   It also determines how large the error is for the approximation of an integral for a specific function...

2. Consider $$_-3∫ ¹ x³dx. Is 5 a valid value for K in Theorem 7.1? Explain.

   KS says:
   

   What K=5 would mean is |f"(x)|=|6x| is less than or equal to 5 for all x between -3 and 1, which is just not true.

   For instance when x=-3, the value of the expression is 18 which is far greater than 5.

Due Friday 2/3 at 8:30am

Section 4.2: Sums and Sigma Notation
Section 4.3: Area
Section 4.4: The Definite Integral
Section 4.7: Numerical Integration

Reading questions:

1. Give an example of a partition of the interval [0,3].

   YZ says:
   

   With n=4, a partition may be {0,1,2,3}

2. Explain the idea of a Riemann sum in your own words.

   LW says:
   

   A Riemann Sum uses points in the intervals of a partition (whether they be the left, right, mid, or other) to estimate the area under a curve with rectangles.

3. Why might we want or need to approximate an integral?

   CZ says:
   

   To find integrals that can't be calculated exactly.

4. If a function f(x) is concave up, does the Trapezoidal Rule produce an over- or an under-estimate of the signed area between f(x) and the x-axis?

   JH says:
   

   An overestimate

Due Wednesday 2/1 at 8:30am

Section 4.6 Integration by Substitution

Reading questions:

1. Substitution attempts to undo one of the techniques of differentiation. Which one is it?

   CB says:
   

   The chain rule

2. Find one antiderivative of  ^ex⁄ _{(1-(e^x)²)^1/2}

   MB says:
   

   By using u = e^x and du = e^x, the general antiderivative is arcsin(e^x) + C.

   One antiderivative would be arcsin(e^x) + 5.

Due Monday 1/30 at 8:30am

A Description of Calculus 2 Homework Assignments
Problem Set Guidelines
Section 0.4 Trigonometric and Inverse Trigonometric Functions
Section 2.8: Implicit Differentiation and Inverse Trigonometric Functions

Reading questions:

1. Why do you think mathematicians often prefer to use arcsin(x) (or arccos(x), etc) rather than sin^-1(x)?

   EC says:
   

   Mathematicians most likely prefer to use arcsin(x), arccos(x), etc. because sin^-1(x) can easily get confused with 1/sin(x), which is csc(x).

2. What is the domain of the function arccos(x)? Why is this the domain?

   NI says:
   

   TheÊdomain of the function arccos(x) is [-1,1], because the range of cos(x) is [-1,1]: the inverse of cos(x) has a domain equivalent to the range of cos(x).

3. Why do you think we are studying the inverse trig functions now?

   FS says:
   

   The inverse trig functions are being studied now in order to determine their derivatives. Once the derivatives of the inverse trig functions are mastered, they can be used in integration.

4. Find one antiderivative of 1 / (1+x²).

   ZJ says:
   

   arctan(x), because d/dx arctan(x)=1/(1+x²)

Due Friday 1/27 at 8:30am

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Section 4.1: Antidifferentiation
Section 4.2: Sums and Sigma Notation
Section 4.3: Area
Section 4.4: The Definite Integral
Section 4.5: The Fundamental Theorem of Calculus

Reading questions:

1. In Section 4.1, the text states that we do not yet have formulae for the indefinite integrals of several elementary functions, including ln(x), tan(x), and sec(x). Why not? For instance, why do we know ∫ cos(x) dx but not ∫ tan(x)?

   HA says:
   

   This is because we know that the derivative of sinx is cosx but we do not yet know what tanx is derived from. Also in this sense tanx is a more complex function because it is composed on sinx and cosx (tanx=sinx/cosx). The same applies to secx.

2. The following questions relate to the antiderivatives of the two products xe^x2 and of xe^x.

   (a) Which differentiation rule would you use to verify that an antiderivative of xe^x2 is ½ e^x2?
   

   SB says:
   

   I would use the chain rule:
   (1/2) e^x2 would give (1/2) e^x2* 2x = xe^x2.

   (b) Which differentiation rule would you use to verify that an antiderivative of xe^x is e^x(x-1)?
   

   JC says:
   

   f(x) = e^x(x-1).
   Using the product rule, f '(x) = e^x + e^x(x-1) = x(e^x)

   (c) Why do your answers to (a) and (b) make it unlikely that we will find a general product rule for antidifferentiation?

   MC says:
   

   It is unlikely because although these are both products, to find what differentiates to them we must use two different rules.

3. Find the signed area between 3x⁵+sin(x) and the x-axis from x=1 to x=2.

   EL says:
   

   ₁∫ ² 3x⁵+sin(x) dx = ₁∫ ² 3x⁵ dx + ₁∫ ² sin(x) dx = (1/2)x⁶ - cos(x)
                                 =[(1/2)(2)⁶ - cos (2)] - [1/2 (1)⁶ -cos(1)]=63/2+cos(1)-cos(2) = 32.456

4. If f(x) is continuous, must it have an antiderivative? If your answer is yes, does that mean there must be a nice formula (or any formula at all) for the antiderivative?

   LH and AS say:
   

   If f(x) is continuous then yes, it must have an antiderivative -- this is part of the Fundamental Theorem of Calculus. This doesn't mean that there must be a nice formula (or any formula at all) for the antiderivative because the function may not be simple like a second degree polynomial, for example.

5. Explain the fundamental difference between a definite integral and an indefinite integral. Please go deeper than saying one has limits of integration and one doesn't, or even that one is a real number and the other a family of functions. Why is a definite integral a real number (what does it represent)? Why is an indefinite integral a family of functions (again, what does it represent)?

   EB says:

   A definite integral represents the area between the curve of f(x) and the x-axis from x=a to x=b.Ê The outcome is the limit of the Riemann Sum, which is used to estimate the area under a curve.

   An indefinite integral represents the inverse of differentiation.Ê Included is a constant of integration.Ê This is a family of functions because there are multiple functions that would complete an antiderivative (infinite possible constants).

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 1306
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu