Here are the assigned readings and exercises (Reading 1, Reading 2, Reading 3, Reading 4, Reading 5, Reading 6, Reading 7, Reading 8, Reading 9, Reading 10, Reading 11, Reading 12, Reading 13, Reading 14, Reading 15, Reading 16, Reading 17, Reading 18, Reading 19, Reading 20, Reading 21, Reading 22, Reading 23, Reading 24, Reading 25, Reading 26, Reading 27, Reading 28, Reading 29, Reading 30, Reading 31, Reading 32, Reading 33, Reading 34, Reading 35, Reading 36, Reading 37, Reading 38); but anything whose assigned date is in the future is subject to change!

- Review:
- Date assigned: January 7 Monday.
- Date due: January 8 Tuesday.
- Reading from the textbook: Review at least Sections 5.4 and 5.5.
- Exercises due:
- If
*F*′(*x*) =*f*(*x*) for all*x*, then what is ∫*f*(*x*) d*x*, that is the indefinite integral of*f*(*x*) with respect to*x*? - If ∫
*f*(*x*) d*x*=*F*(*x*) +*C*, then what is ∫^{b}_{x=a}*f*(*x*) d*x*, that is the definite integral of*f*(*x*) with respect to*x*as*x*runs from*a*to*b*?

- If

- Work:
- Date assigned: January 8 Tuesday.
- Date due: January 9 Wednesday.
- Reading from the textbook: Pages 376–380 (§6.5).
- Exercises due:
- Suppose that a variable force is applied
on object moving from
*a*to*b*along the*x*-axis, such that the amount of the force in the direction of the positive*x*-axis is a function*F*of the object's position along the*x*-axis. Write down an integral for the work done on the object by that force. - Suppose that a spring with a spring constant
*k*is stretched or compressed a distance*x*from its equilibrium length. Write down a formula for the force of restoration on the spring.

- Suppose that a variable force is applied
on object moving from

- Moments:
- Date assigned: January 9 Wednesday.
- Date due: January 10 Thursday.
- Reading from the textbook: Pages 382–389 (§6.6).
- Exercises due:
- If you wish to find
the total mass of the plate
described in Exercise 6.6.15 on page 389 of the textbook,
will you do an integral with respect to
*x*or an integral with respect to*y*? - Give the formula for the centre of mass
(
*x̄*,*ȳ*) in terms of the total mass*M*and the moments*M*_{x}and*M*_{y}.

- If you wish to find
the total mass of the plate
described in Exercise 6.6.15 on page 389 of the textbook,
will you do an integral with respect to

- Differential equations:
- Date assigned: January 10 Thursday.
- Date due: January 11 Friday.
- Reading from my notes (first set):
- Section 6 (the middle of page 4);
- Sections 8&9 (most of pages 5–7);
- Optional: Sections 10&11 (the rest of page 7 and page 8).

- Reading from the textbook: Pages 403–409 (§7.2).
- Exercises due: Fill in the blanks with vocabulary words:
- An equation with differentials or derivatives in it is a(n) _____ equation.
- A quantity is growing _____ if it is growing in such a way that its rate of growth is proportional to its size.

- Integration by parts:
- Date assigned: January 14 Monday.
- Date due: January 15 Tuesday.
- Reading from my notes (first set): Section 7 (the bottom of page 4 and the top of page 5).
- Reading from the textbook: Pages 423–427 (§8.1).
- Exercises due:
Use the formula
∫
*u*d*v*=*u**v*− ∫*v*d*u*for integration by parts.- To find
∫
*x*e^{x}d*x*, what should be*u*and what should be d*v*? - To find ∫
*x*ln*x*d*x*, what should be*u*and what should be d*v*?

- To find
∫

- Trigonometric integration:
- Date assigned: January 15 Tuesday.
- Date due: January 16 Wednesday.
- Reading from the textbook: Pages 429–434 (§8.2).
- Exercises due:
- To find ∫ sin
^{2}*x*d*x*, what trigonometric identity would you use? That is, sin^{2}*x*= _____? - For which (if any) of the following other integrals
would you also need to use the identity from the previous exercise?
- ∫ sin
^{2}*x*cos^{2}*x*d*x*, - ∫ sin
^{2}*x*cos^{3}*x*d*x*, - ∫ sin
^{3}*x*cos^{2}*x*d*x*, - ∫ sin
^{3}*x*cos^{3}*x*d*x*.

- ∫ sin

- To find ∫ sin

- Trigonometric substitution:
- Date assigned: January 16 Wednesday.
- Date due: January 17 Thursday.
- Reading from the textbook: Pages 435–438 (§8.3).
- Exercise due:
Give a trigonometric substitution
that will help to find
∫ √(
*x*^{2}+ 25) d*x*. (This one might take a little more work than most of these daily homework exercises need, but keep in mind that you just need to state the trigonometric substitution; you do*not*have to work out the entire integral.)

- Partial fractions:
- Date assigned: January 17 Thursday.
- Date due: January 18 Friday.
- Reading from the textbook: Pages 440–445 (§8.4).
- Exercise due:
Suppose that a rational expression has the denominator
(
*x*^{2}+ 3)(*x*+ 3)^{2}.- What are the denominators of its partial fractions?
- For each of these, indicate the maximum degree of the numerator.
(If you wish,
you may do this by writing a general form for each numerator,
such as
*A*or*A**x*+*B*.)

- Integration using computers and tables:
- Date assigned: January 23 Wednesday.
- Date due: January 24 Thursday.
- Reading from the textbook:
- Read pages 447–451 (§8.5);
- Skim pages T1–T6 (the table of integrals).

- Optional:
Look at these some or all of
these computer algebra systems that will do integrals (as well as much more):
- Sage (free to download, free for most uses online, $14.00 per month or more to make intensive computations online);
- Maxima (free to download);
- Maple ($99.00 to download);
- Wolfram Mathematica ($160.00 to download, $80.00 per year online, free 15-day trial);
- Wolfram Alpha (free for some uses online, $4.75 per month for advanced features, $2.99 for a smartphone app with intermediate features).

- Exercise due:
Which entry
in the table of integrals in the back of the textbook (pages T1–T6)
tells you how to integrate
∫ (
*x*^{2}+ 3)^{−1/2}d*x**without*using hyperbolic functions (sinh, cosh, etc) or their inverses? (Hints: That table doesn't use negative exponents; it uses fractions instead. And it doesn't use fractional exponents; it uses roots instead.)

- Numerical integration:
- Date assigned:
*Also*January 23 Wednesday. - Date due:
*Also*January 24 Thursday. - Reading from the textbook: Pages 452–459 (§8.6).
- Exercises due:
- Which numerical method of integration approximates a function with a piecewise-linear continuous function?
- Which numerical method of integration approximates a function with a piecewise-quadratic continuous function?

- Date assigned:
- Improper integrals:
- Date assigned: January 24 Thursday.
- Date due: January 25 Friday.
- Reading from the textbook: Pages 462–467 (all of §8.7 except for Tests for Convergence and Divergence).
- Exercise due:
Consider the integral
∫
^{∞}_{x=−∞}(*x*^{2}+ |*x*|^{1/2})^{−1}d*x*, that is the integral of (*x*^{2}+ |*x*|^{1/2})^{−1}d*x*as*x*runs from −∞ to ∞. List all of the reasons why this integral is improper.

- Infinite sequences:
- Date assigned: January 25 Friday.
- Date due: January 28 Monday.
- Reading from the textbook: Pages 478–487 (§9.1).
- Exercises due: Exercises 2 and 4 from §9.1 (on page 487) in the textbook.

- Infinite series:
- Date assigned: January 28 Monday.
- Date due: January 29 Tuesday.
- Reading from the textbook:
- Page 490 and through the top of page 492 (§9.2: introduction);
- Example 6 on page 494 and through page 497 (all of the rest of §9.2 except for Geometric Series).

- Reading from my notes (second set): Through Section 3 (through the top of page 4).
- Exercises due:
Fill in the vocabulary words:
- lim
_{n→∞}*a*_{n}, the limit of*a*_{n}as*n*goes to infinity, is the limit of a(n) ___. - Σ
^{∞}_{n=0}*a*_{n}, the sum of*a*_{n}as*n*runs from zero to infinity, is the sum of a(n) ___.

- lim

- Evaluating special series:
- Date assigned: January 30 Wednesday.
- Date due: January 31 Thursday.
- Reading from my notes (second set): Section 4 (the rest of page 4 and the first half of page 5).
- Reading from the textbook: The rest of page 492 and through the end of Example 5 on page 494 (§9.2: Geometric Series).
- Exercises due:
Finish these formulas and attach any conditions necessary for them to be true:
- The sum of
*b*_{n+1}−*b*_{n}as*n*runs from*i*to infinity (where*i*is a natural number and*b*is an infinite sequence of real numbers): Σ^{∞}_{n=i}(*b*_{n+1}−*b*_{n}) = _____. - The sum of
*r*^{n}as*n*runs from*i*to infinity (where*i*is a natural number and*r*is a real number): Σ^{∞}_{n=i}*r*^{n}= _____.

- The sum of

- The Integral Test:
- Date assigned: January 31 Thursday.
- Date due: February 1 Friday.
- Reading from the textbook: Pages 499–504 (§9.3).
- Exercises due:
- Does the Integral Test apply
to the function
*f*(*x*) = sin^{2}(π*x*)? Why or why not? - For which values of
*p*does Σ^{∞}_{n=1}(1/*n*^{p}), the sum of 1/*n*^{p}as*n*runs from 1 to infinity, converge?

- Does the Integral Test apply
to the function

- Comparison tests for integrals:
- Date assigned: February 1 Friday.
- Date due: February 4 Monday.
- Reading from the textbook: The rest of page 468 and through page 470 (§8.7: Tests for Convergence and Divergence).
- Exercises due:
Suppose that you want to know whether
∫
^{∞}_{x=1}(*x*+ sin*x*)/*x*^{2}d*x*, the infinite integral of (*x*+ sin*x*)/*x*^{2}, converges.- Knowing that the infinite integral of 1/
*x*^{2}converges, can you use the Direct Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*^{2}converges, can you use the Limit Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*diverges, can you use the Direct Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*diverges, can you use the Limit Comparison Test to decide?

- Knowing that the infinite integral of 1/

- Comparison tests for series:
- Date assigned: February 4 Monday.
- Date due: February 5 Tuesday.
- Reading from the textbook: Pages 506–509 (§9.4).
- Exercises due:
Suppose that you want to know whether
Σ
^{∞}_{n=1}(*n*+ (−1)^{n})/*n*^{2}, the infinite series of (*n*+ (−1)^{n}))/*n*^{2}, converges.- Knowing that the infinite series of 1/
*n*^{2}converges, can you use the Direct Comparison Test to decide? - Knowing that the infinite series of 1/
*n*^{2}converges, can you use the Limit Comparison Test to decide? - Knowing that the infinite series of 1/
*n*diverges, can you use the Direct Comparison Test to decide? - Knowing that the infinite series of 1/
*n*diverges, can you use the Limit Comparison Test to decide?

- Knowing that the infinite series of 1/

- Alternating series:
- Date assigned: February 6 Wednesday.
- Date due: February 7 Thursday.
- Reading from the textbook: Pages 516–518 (§9.6: introduction).
- Exercises due:
Identify which of these series are alternating:
- The sum of (−1)
^{n}/(2 +*n*); - The sum of (2 + (−1)
^{n})/*n*; - The sum of (cos
*n*)/*n*.

- The sum of (−1)

- Absolute convergence:
- Date assigned: February 7 Thursday.
- Date due: February 8 Friday.
- Reading from the textbook:
- Pages 510&511 (§9.5: introduction);
- Page 519 and through the paragraph following Example 5 on page 520 (§9.6: Conditional Convergence; Rearranging Series).

- Exercises due:
- If a series converges, is it necessarily true that its series of absolute values also converges?
- If the series of absolute values converges, is it necessarily true that the original series converges?

- The Ratio and Root Tests:
- Date assigned: February 8 Friday.
- Date due: February 11 Monday.
- Reading from the textbook:
- Pages 512–515 (§9.5: The Ratio Test; The Root Test);
- The rest of page 520 (§9.6: Summary of Tests).

- Reading from my notes (second set): Skim Section 5 (the rest of page 5 and through page 8).
- Exercises due:
- Under what circumstances
does the Ratio Test
*not*tell you whether a series converges? - Under what circumstances
does the Root Test
*not*tell you whether a series converges? - If the Ratio Test doesn't tell you, is it possible that the Root Test will?

- Under what circumstances
does the Ratio Test

- Power series:
- Date assigned: February 11 Monday.
- Date due: February 12 Tuesday.
- Reading from the textbook: Pages 522–530 (§9.7).
- Exercises due:
Which of the following are (or are equivalent to)
power series (in the variable
*x*)?- Σ
^{∞}_{n=0}*n*^{2}(*x*− 3)^{n} - Σ
^{∞}_{n=5}(2*x*− 3)^{n} - Σ
^{∞}_{n=0}(√*x*− 3)^{n} - 5 + 7
*x*− 3*x*^{3}

- Σ

- Taylor polynomials:
- Date assigned: February 13 Wednesday.
- Date due: Feburary 14 Thursday.
- Reading from my notes (second set): Section 6 (page 9 and through the first half of page 11).
- Reading from the textbook:
- Pages 534–536 (§9.8: Taylor Polynomials), but don't worry yet about when they talk about infinite series and convergence;
- Page 537 and through Theorem 24 on page 539 (§9.9: introduction; the very beginning of Estimating the Remainder).

- Exercises due:
- Suppose that
*f*is a function,*a*is a number, and*f*is infinitely differentiable at*a*. Given a natural number*k*, let*P*_{k}be the Taylor polynomial of order*k*generated by*f*at*a*(so*P*_{k}is also a function). Then the derivatives of*f*at*a*and the derivatives of*P*_{k}at*a*must be equal up to what order? (Up to their first derivatives, up to their second derivatives, up to their*k*th derivatives, or what?) - If a function
*f*is to have a good approximation on an interval by a polynomial of degree at most*k*, then it's best if its derivative of what order is close to zero on that interval? (Its first derivative, its second derivative, its*k*th derivative, or what?)

- Suppose that

- Taylor series:
- Date assigned: Feburary 14 Thursday.
- Date due: February 15 Friday.
- Reading from my notes (second set): The beginning of Section 7 (the rest of page 11 and the first two lines of page 12).
- Reading from the textbook: Pages 532–534 (the rest of §9.8).
- Exercises due:
- Fill in the blank:
If
*f*is a function whose derivatives of all orders exist everywhere, then the Maclaurin series generated by*f*is the Taylor series generated by*f*at ___. - True or false:
Whenever
*a*is a constant and*f*is a function whose derivatives of all orders exist at*a*, if the Taylor series generated by*f*at*a*converges anywhere, then it must converge to*f*there.

- Fill in the blank:
If

- The Binomial Theorem:
- Date assigned: February 15 Friday.
- Date due: February 18 Monday.
- Reading from the textbook: Page 543 and through Example 2 on page 545 (§9.10: introduction; The Binomial Series for Powers and Roots).
- Reading from my notes (second set): The parts of Section 7 about the Binomial Theorem (the first line in the list on page 12 and the paragraph in the middle of page 12).
- Exercises due:
- Using the Binomial Theorem,
expand (
*x*+ 1)^{6}. - Using the Binomial Theorem,
write (1 +
*x*^{2})^{−1}as an infinite series (assuming that*x*^{2}< 1 so that the series converges).

- Using the Binomial Theorem,
expand (

- More common Taylor series:
- Date assigned: February 18 Monday.
- Date due: February 19 Tuesday.
- Reading from the textbook:
- The rest of page 539 and through page 541 (the rest of §9.9);
- The rest of page 545 and through page 548 (the rest of §9.10).

- Reading from my notes (second set): The rest of Section 7 (the rest of page 12 and page 13).
- Exercises due:
- Assuming that
*a*and*b*are real numbers, fill in the blank using trigonometric operations applied to real numbers (and other operations as appropriate): exp (*a*+ i*b*) = e^{a+ib}= ___. - Can any limit using L'Hôpital's Rule be done using Taylor polynomials or series instead? Explain why or why not.

- Assuming that

- Graphs in three dimensions:
- Date assigned: February 21 Thursday.
- Date due: February 22 Friday.
- Reading from the textbook: Pages 596–599 (§11.1).
- Reading from my notes (last set): through Section 1 (the first half of page 1).
- Exercises due:
- In a right-handed rectangular coordinate system
using the variables
*x*,*y*, and*z*, if you curl the fingers of your right hand from the direction of the positive*x*-axis to the direction of the positive*y*-axis and stick out your thumb, then in what direction approximately should your thumb point? - What is the name of a shape
whose equation in a three-dimensional rectangular coordinate system
is linear and contingent?
(For example, 2
*x*+ 3*y*+ 5*z*= 8.) - What is the equation
in the rectangular
(
*x*,*y*,*z*)-coordinate system of a sphere whose radius is*r*and whose centre is (*h*,*k*,*l*)?

- In a right-handed rectangular coordinate system
using the variables

- Vectors:
- Date assigned: February 22 Friday.
- Date due: February 25 Monday.
- Reading from my notes (last set): Sections 2–4 (the rest of page 1 and through the first half of page 5).
- Reading from the textbook:
- From page 601 to the very top of page 605 (§11.2: introduction; Component Form; Vector Algebra Operations);
- The bottom half of page 606 (§11.2: Midpoint of a Line Segment).

- Exercises due:
- Give a formula
for the vector
from the point (
*x*_{1},*y*_{1}) to the point (*x*_{2},*y*_{2}). - If
**u**,**v**, and**w**are vectors, simplify the expression 2(**u**+ 3**v**) − 6(**v**− 3**w**) − 18(**w**+**u**/9).

- Give a formula
for the vector
from the point (

- Length and angle:
- Date assigned:
*Also*February 22 Friday. - Date due:
*Also*February 25 Monday. - Reading from my notes (last set): Section 5 (the rest of page 5 and through the first half of page 7).
- Reading from the textbook:
- Pages 605&606 (§11.2: Unit Vectors);
- Page 607&608 (§11.2: Applications).

- Exercises due:
- Give a formula for the magnitude (or norm, or length)
of the vector ⟨
*a*,*b*,*c*⟩. - If |
**u**+**v**|^{2}= |**u**|^{2}+ |**v**|^{2}, then what is the angle between**u**and**v**? - If |
**u**+**v**| = |**u**| + |**v**|, then what is the angle between**u**and**v**?

- Give a formula for the magnitude (or norm, or length)
of the vector ⟨

- Date assigned:
- The dot product:
- Date assigned: February 25 Monday.
- Date due: February 26 Tuesday.
- Reading from my notes (last set): Sections 6–8 (the rest of page 7 and through most of page 10).
- Reading from the textbook: Pages 610–615 (§11.3).
- Exercises due:
- State a formula for
the dot product
**u**⋅**v**of two vectors using only their lengths |**u**| and |**v**|, the angle ∠(**u**,**v**) between them, and real-number operations. - State a formula for the vector projection of
**u**onto**v**using only dot products and real-number operations (so in particular, no lengths or angles unless expressed using dot products).

- State a formula for
the dot product

- The cross product:
- Date assigned: February 26 Tuesday.
- Date due: February 27 Wednesday.
- Reading from my notes (last set): Sections 9–12 (the rest of page 10 and through the very top of page 15).
- Reading from the textbook: Pages 618–622 (§11.4).
- Exercises due:
- State a formula for
the magnitude |
**u**×**v**| of the cross product of two vectors**u**and**v**, using only their lengths |**u**| and |**v**|, the angle ∠(**u**,**v**) between them, and real-number operations. - If
**u**and**v**are vectors in 2 dimensions, then is**u**×**v**a scalar or a vector? - If
**u**and**v**are vectors in 3 dimensions, then is**u**×**v**a scalar or a vector?

- State a formula for
the magnitude |

- Parametrized curves:
- Date assigned: February 28 Thursday.
- Date due: March 1 Friday.
- Reading from the textbook: Pages 557–562 (§10.1).
- Reading from my notes (last set): The first half of Section 13 (the rest of page 15 and the first two lines of page 16).
- Exercises due:
Define a parametrized curve
by (
*x*,*y*) = (2*t*^{2}, 3*t*^{3}) for 0 ≤*t*≤ 2.- Which variable(s) is/are the parameter(s)?
- What are the beginning/initial point and the ending/final/terminal point of the curve?

- Geometry with vectors:
- Date assigned: March 1 Friday.
- Date due: March 4 Monday.
- Reading from the textbook: Pages 624–630 (§11.5).
- Reading from my notes (last set): Section 14 (most of page 17).
- Exercises due:
- Give a parametrization for
the line through the point
(
*x*_{0},*y*_{0},*z*_{0}) and parallel to the vector ⟨*a*,*b*,*c*⟩. - Give an equation for
the plane through the point
(
*x*_{0},*y*_{0},*z*_{0}) and perpendicular to the vector ⟨*a*,*b*,*c*⟩.

- Give a parametrization for
the line through the point
(

- Calculus with parametrized curves:
- Date assigned: March 4 Monday.
- Date due: March 5 Tuesday.
- Reading from the textbook: Page 564 and through the end of Example 3 on page 566 (§10.2: introduction; Tangents and Areas).
- Reading from my notes (last set): Section 15 (the rest of page 17 and through most of page 19).
- Exercises due:
If
*x*and*y*are each functions of*t*:- Give a formula for the derivative of
*y*with respect to*x*in terms of the derivatives of*x*and*y*with respect to*t*. (There is basically only one possible correct answer to this.) - Give a formula for
the second derivative of
*y*with respect to*x*in terms of derivatives of*x*and*y*with respect to*t*. (There is more than one possible correct answer to this, and you only need to give one of them, but make sure that all of the derivatives appearing are with respect to*t*as required!)

- Give a formula for the derivative of

- Arclength of parametrized curves:
- Date assigned: March 5 Tuesday.
- Date due: March 6 Wednesday.
- Reading from the textbook: Pages 566 to 572 (the rest of §10.2).
- Reading from my notes (last set): Section 16 (the rest of page 19 and page 20).
- Exercises due:
- If a curve is parametrized by
*x*=*f*(*t*) and*y*=*g*(*t*) for*a*≤*t*≤*b*(and assuming, as usual, that the parametrization is continuously differentiable and one-to-one), then what integral in the variable*t*gives the length of this curve? - How does changing the parametrization of a curve affect its arclength?

- If a curve is parametrized by

- Polar coordinates:
- Date assigned: March 8 Friday.
- Date due: March 11 Monday.
- Reading from the textbook: Pages 574–577 (§10.3).
- Exercises due: True or false:
- For every point
*P*in the coordinate plane, for every pair (*r*,*θ*) of real numbers that gives*P*in polar coordinates,*r*≥ 0 and 0 ≤*θ*< 2π. - For every point
*P*in the coordinate plane, for at least one pair (*r*,*θ*) of real numbers that gives*P*in polar coordinates,*r*≥ 0 and 0 ≤*θ*< 2π.

- For every point

- Graphs in polar coordinates:
- Date assigned: March 11 Monday.
- Date due: March 12 Tuesday.
- Reading from the textbook: Pages 578–580 (§10.4).
- Exercises due:
Suppose that a curve is parametrized in polar coordinates
by
*r*=*f*(*θ*) for some differentiable function*f*. (In the following answers, refer directly to only*f*, its derivatives, and*θ*.)- What is the slope of the curve at a given value of
*θ*? - Under what circumstances is this slope undefined?

- What is the slope of the curve at a given value of

- Area in polar coordinates:
- Date assigned: March 12 Tuesday.
- Date due: March 13 Wednesday.
- Reading from the textbook: Pages 581&582 and most of page 583 (§10.5: introduction; Area in the Plane).
- Exercises due:
- Can the area of a region in the plane ever be negative?
- What is the formula for the area of the region
satisfying
*f*(*θ*) ≤*r*≤*g*(*θ*) and*α*≤*θ*≤*β*in polar coordinates? (Assume that*α*and*β*are real numbers with*α*≤*β*and*β*−*α*≤ 2π, and that*f*and*g*are continuous functions defined at least on [*α*,*β*] with 0 ≤*f*(*θ*) ≤*g*(*θ*) whenever*α*≤*θ*≤*β*.)

- Length in polar coordinates:
- Date assigned:
*Also*March 12 Tuesday. - Date due:
*Also*March 13 Wednesday. - Reading from the textbook: The rest of page 583 and page 584 (§10.5: Length of a Polar Curve).
- Exercises due:
- Can the length of a curve ever be negative?
- What is the formula for the length of the curve
given by
*r*=*f*(*θ*) and*α*≤*θ*≤*β*in polar coordinates? (Assume that*α*and*β*are real numbers with*α*≤*β*, and that*f*is a continuously differentiable function defined at least on [*α*,*β*] with (*f*(*θ*_{1}),*θ*_{1}) always defining a different point in polar coordinates than (*f*(*θ*_{2}),*θ*_{2}).)

- Date assigned:

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