Reflections on Good Calculus Questions

Reflections on Good Calculus Questions from Students and Colleagues

AMATYC 2015 New Orleans, LA Keith Nabb ([email protected]) Daniel Nghiem ([email protected]) Moraine Valley Community College Palos Hills, Illinois

What is a “good” question? • Based on a hunch/curiosity/coincidence • Students get a taste of the nature of math • Should have a reasonable degree of accessibility (e.g., Vygotsky’s ZPD) • Investigation has the potential to… – Extend/generalize mathematical content – Lead to something new (or something already known but in a new light)

Counterexample sin x sin x   six  6 n n Again, why do the n's cancel?

Table of Contents The questions and potential “answers” – Student-generated solutions – Teacher-lead discussion – Ideas for projects

THE QUESTIONS & SOME EXPLORATIONS

Question 1

Why is  f  x   g  x    f   x   g   x  ?

Dunkels & Persson (1980); Gay, Tefera, & Zeleke (2008)

Student Work

Student Work (continued)

Student Work (actual work submitted to teacher)

Question 2 d d 2  r   2 r or  Acircle   Ccircle .  dr dr Is this a neat coincidence or something deeper?

Perrin & Quinn (2008); Zazkis, Sinitsky, & Leikin (2013)

Analytical Approach

A  r  r   A  r  A  r   lim r 0 r

  r  r    r 2

 lim

r 0

r

 2 r  Circumference

2

Geometric Approach

Area of band  2 r  r A  2 r  r

Radius increases by r A 2 r  r   2 r r r

Generalizing

d 4 3 2   r   4 r dr  3 

Generalizing 1. Why do these results not extend to squares or cubes? Or rather, what needs to be done to obtain consistency (Zazkis, Sinitsky & Leikin, 2013)? 2. How about polygons?

3. Platonic solids?

Question 3 What is the point of writing "dx " when integrating? I keep forgetting to write it down!

Application Estimate the total force exerted on a window fully submerged under water. Pressure varies with height so consider horizontal strips. Force = (Pressure)(Area) n

Force   Pk A  Force  k 1

Jones (2013)

bottom strip



top strip

PdA

Student Work Traditional: 6

6

u tan x  tan x sec x dx   u du  6  C  6  C 5

2

5

Better? 6

tan x  tan x sec x dx   tan x d (tan x)  6  C 5

2

5

Challenging Exercises d 4 x   ?? 2  d x  The symbols dx, d(tan x), dr, or d(anything) are more than placeholders telling us the “variable of interest.” Differentials have a tangible meaning through the operation of multiplication.

Question 4

When does



f  x  dx 



f  x  dx ?

Context

Students were asked to evaluate



tan x dx (semester project).

1. Can I do 2. Is

 tan x dx

instead?

tan x dx related in any 

way to



tan x dx ?

Direct attempt

An indirect attempt

Question 5 While using the Second Derivative Test, we found that all assumptions were met and f   0   2  0. Hence, we concluded there was a relative minimum at

 0, f  0   .

What is the meaning of the number 2?

Concavity vs. Curvature

What’s the difference?

"We can make another observation related to the degree of concavity (also called curvature). A large value of | f ( a) | (large curvature) means ... the slope of the curve changes rapidly and the graph of f separates quickly from the tangent line." (Briggs, Cochran, Gillett, 2015)

"...absolute errors in linear approximation are larger when | f (a) | is large."

Mostly true, but not always…

A "Turning Angle" θ

dy y(t ) tan    dx x(t )

What is dθ/dt? tan  d sec   dt d dt 2

dy y(t )   dx x(t ) xy  xy  2  (x ) xy  xy  ( x) 2  ( y) 2

Implicit Differentiation + Quotient Rule d𝜃

Solve for 𝑑𝑡 and use sec 2 𝜃 = 1 + tan2 𝜃

d f (t )  2 dt 1  ( f (t ))

d f (t )  dt 1  ( f (t ))2 At t  0 : f   0, f   2 d   2 radians/unit ! dt At t  2 : f   4, f   2 d 2   radians/unit ! dt 17

Question 6 Can we find the volumes of solids of revolution where the axis of revolution is a skew line such as y  x ?

Idea

Challenge for the Student: Find a "change of variables" from  x to u.

Area and Volume Elements

r ( x)  u 

f ( x)  mx

1  m2 1  mf ( x) 1 m

2

x

Area Element: A  r u ( f ( x)  mx)(1  mf ( x))  x 2 1 m Volume Element: V   r 2 u ( f ( x )  mx )2 (1  mf ( x ))  x 2 3/2 (1  m )

Question 7

Is there an "integration by parts" for quotients?

Deveau & Hennigar (2012); Switkes (2005)

Why yes!  u   vu   uv Begin with    . Integration 2 v v u  vu  uv  wrt x gives   2 dx   2 dx or  v v  v u  vdu  u dv   2   2 . Simplify and  v v  v u dv  du u  rearrange to get  2    .  v  v v

Features

 u dv   du  u  2   v  v v • Choose v (or v²), then find dv, u must correspond to the remainder of the integrand. • All we need on the right hand side is du!

Example (Standard)    x2  1 1  2 x2 1 dx  cos  d  dx  dx  dx 2     2 2 2 2 2  2  2   x  1   x  1   x  1   x  1

 1 1   2 dx   dx 2 2  x 1   x  1 Let x  tan 

1  cos 2  d   2 1 1    sin 2  C 2 4  (apply sin 2  2sin  cos  )  12 tan 1 x 

x

2  x  1 2

C

Example (Quotient Rule IBP)  u dv   du  u  2   v  v v 1   12 x  2 xdx  12 dx x2 2 x dx    2  2   2 2  2 2  x 1 x 1   x  1   x  1

1 2

1

tan x 

x

2  x  1

Integration by Parts

Discuss pros/cons of this formula with students!

2

C

Question 8

Which quantity is larger, 

 1

f  x  dx or



 f n ? n 1

Hasty thought 

 1



f  x  dx   f  n  n 1

Misconception: The “continuous” version is greater than the “discrete” version.

Hmmmmm… 

1  (i)  2 dx  1 whereas x 1 

1 2   1.65  2 6 n 1 n 

1 1 (ii)    dx   0.72 whereas 2 ln 2 2 x



1    1 n 1  2 

1 

(iii)

 xe 1

x

2

1 dx   0.18 while 2e



n

e n 1

n2

n

 0.40



More support for

 1



f  x  dx   f  n  n 1

Re-examine the assumptions of the Integral Test!

Pedagogical Considerations • The diagrams outline a proof of the Integral Test • The proof re-ignites older material (Riemann Sums) • Students need little convincing of the fact that if one converges, so does the other.

Challenge: Produce an example in 

which

 1



f  x  dx   f  n . Explain n 1

how you arrived at your result.

Question 9 

Can an alternating series

  1 a n 1

n

n

converge without the condition an 1  an ?

Nabb (2010)

Student 1 Solution sin  n   2  1  2 4n n 1 

n

A plot of y  an for large n :

A list of a1 through a20 : 0.7103677462 , 0.1818310892 , 0.05947555578 , 0.01942496101 , 0.01041075725 , 0.01194850349 , 0.01355605407 , 0.01167718065 , 0.007444810139 , 0.003639947223 , 0.002066135937 , 0.002540672017 , 0.003580128753 , 0.003814550198 , 0.002944764267 , 0.001671969417 , 0.0008984450760 , 0.0009637444088 , 0.001488834632 , 0.001820590782

Student 2 Solution 1 1 1 1 1 1       2 3 4 9 8 27  n 1 / 2  , n  1,3,5, 7, 1/ 2 an   n/2 n  2, 4, 6,8,   1/ 3 ,

Student 3 Solution 1 2 1 11 3 a3      4 24 8 a1 

1 11 3 a5      8 2  8  16 1 1 1  3 a7      16 2  16  32 a9 

1 1 1  3    32 2  32  64

1 4 1 a4  8 1 a6  16 1 a8  32 1 a10  64 a2 

Student 3 Solution…continued 

  1 n 1

n 1

1  1 1 3 1  3 1   3 1   3 an                       2 4   8 8   16 16   32 32   64 64  1 1 1 1 1       4 4 8 16 32 geometric

1  1 n    2  4 n2 

Student 4 (Beginning Idea)

Unexplored Questions

Question 10

Is Calculus much the same when working in degrees instead of radians?

Question 11

Are there coordinate systems that make the "business" of Calculus easier?

Question 12  x  a   sin   Given   y  a 1  cos   (parametric form for a cycloid), d why is  x  y ? d

Question 13 (asked Fall 2015) 

Given a convergent series

a , n 1



will

  1 n 1

n

n

an always converge?

Question 14 (asked Fall 2015)  n 1 We know lim   1 but why  n   n  3

 n 1 is lim   1? Can one apply  n   n  the "standard techniques" from the former problem to solve the latter? n

WHAT IS THE POINT?

• Students are inherently curious (Harel’s intellectual need) • Questions often connect to other areas of Calculus • “New” mathematics may be just around the corner

ADVICE

• Don’t ignore these questions! • Look for opportunities to use them in instruction • Class projects • Supplemental work • Let a student “show off”

Takeaway All subjects are inherently inquiry-based. New knowledge is developed by asking questions and attempting to answer them!

References Briggs, W., Cochran, L., & Gillett, B. (2015). Calculus: Early transcendentals. Boston, MA: Pearson. Deveau, M., & Hennigar, R. (2012). Quotient-rule-integration-by-parts. The College Mathematics Journal, 43 (3), 254-256. Dunkels, A., & Persson, L.-E. (1980). Creative teaching by mistakes. The Two-Year College Mathematics Journal, 11, 296-300. Gay, C.C., Tefera, A., & Zeleke, A. (2008). The naïve product rule for derivatives. The College Mathematics Journal, 39 (2), 145-148. Jones, S.R. (2013). Adding it all up: Reconceiving the introduction of the integral. Mathematics Teacher, 107 (5), 372-377. Nabb, K.A. (2010). A close encounter with infinity: Inventing new mathematics. Mathematics Teacher, 104 (5), 373-378. Perrin, J.R., & Quinn, R.J. (2008). The power of investigative calculus projects. Mathematics Teacher, 101 (9), 640-646. Switkes, J. (2005). A quotient rule integration by parts formula. The College Mathematics Journal, 36 (1), 58-60. Zazkis, R, Sinitsky, I., & Leikin, R. (2013). Derivative of area equals perimeter— Coincidence or rule? Mathematics Teacher, 106 (9), 686-692.

Thank You!

Questions? Keith:

[email protected]

Daniel: [email protected]