Understanding Calculus is Largely Locally Linear

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Understanding Calculus is Largely Locally Linear
Dan KennedyPearson Author
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My Classs First Problem of the Day
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Point-Slope Equation of a Line
Linearization of f at x a
or
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This simple equation is the basis
for Instantaneous rate of change Differentials Li
near approximations Newtons Method for Finding
Roots Rules for Differentiation Slope
Fields Eulers Method LHopitals Rule
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What is Instantaneous Rate of Change?
If a rock falls 20 feet in 2 seconds, its average
velocity during that time period is easily
measured
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But what if we capture that rock at a moment in
time and ask for its instantaneous velocity at
that single moment? What is the logical reply?
Obviously, traditional algebra fails us when it
comes to instantaneous rate of change.
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But if we graph the rocks position as a function
of time, we see that velocity is the same as
slope. Instantaneous velocity is just the slope
of the zoomed-in picture which is linear!
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Thats what differentials are all about. Since
the derivative is the slope of the curve at a
point, there is no true ?y or ?xbut there is a
slope. So we write
If we zoom in on a differentiable curve, because
it is locally linear, ?y/ ?x is essentially
constant. That makes dy/dx defensible as
something other than 0/0.
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If we zoom in at a point (a, f(a))
So if (x, y) is close to (a, f(a)),
Theres that equation again!
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This is the basis for linear approximations. For
example, here is Problem of the Day 23
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The Good News Modern students can check how
close this approximation is by using a
calculator.
The Bad News Modern students do not appreciate
how much these approximation tricks meant to
their ancestors.
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Another such method is Newtons Method for
approximating roots. Here is Problem of the Day
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Then we name this x-intercept b. Using (b, f(b))
as the new point, I have them repeat the process.
Clever students quickly write
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Eventually this process will zoom in on an
x-intercept of the curve. This is Newtons Method
for approximating roots of equations.
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For example, let us find a root of the equation
sin x 0. Start with a guess of a 2.
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Local linearity even led to the discovery of the
differentiation rules. For example, heres the
product rule. Zoom in on a product function uv
until it looks linear. The slope will be the
derivative of uv.
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A slope field is all about local linearity.
At each point (x, y) the differential equation
determines a slope. This is the very essence of
point-slope!
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Here is Problem of the Day 37
This is Eulers Method. The picture at the right
shows how it works.
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You start with a point on the curve. Find the
slope, dy/dx. Now move horizontally by ?x. Move
vertically by ?y (dy/dx) ?x.This gives you a
new point (x ?x, y ?y).Repeat the process.
I like to use a table.
(x, y) dy/dx ?x ?y (x ?x, y ?y)


(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1)

(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2

(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1

(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2

(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 1.2)

(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 1.2)
(1.1, 1.2)
(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 1.2)
(1.1, 1.2) 2.3
(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 1.2)
(1.1, 2.2) 2.3 0.1
(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 2.2)
(1.1, 1.2) 2.3 0.1 0.23
(x, y) dy/dx ?x ?y (x ?x, y ?y)
(1, 1) 2 0.1 0.2 (1.1, 1.2)
(1.1, 1.2) 2.3 0.1 0.23 (1.2, 1.43)
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Problem of the Day 53
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And now for something completely different
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The Top Ten Student Errors on the AP Calculus
Examination according to Dan Kennedy
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Error 10 Mishandling the Chain Rule
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Error 9 Universal Logarithmic
Antidifferentiation
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Error 8 Not Showing the Pre-calculator Step
You can evaluate a definite integral on your
calculator, but you must show the integral
first. You can solve an equation on your
calculator, but you must show the equation first.
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Error 7 Using Ambiguous References
Avoid statements like "The graph of the
derivative is increasing, so the graph must be
concave up." Use statements like "The graph of
f ' is increasing, so the graph of f must be
concave up."
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Error 6 Omitting the Constant of Integration
This can be a very costly error if you are
solving a differential equation with an initial
condition, like
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Error 5 Not Separating the Variables
This can be a much more costly error if you are
solving a separable differential equation like
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Error 4 The Bad Washer Formula
Good
Bad
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Error 3 Messing up Average Rate of Change
Remember A derivative is the instantaneous rate
of change. It's a calculus concept. Average rate
of change is just . It's an algebra concept.
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Error 2 Justifying Maxima and Minima
Remember Showing that f ' 0 is only a
start. For local extrema, show how f ' changes
sign, or check concavity using the sign of f
''. For absolute extrema, give a global argument
on the entire domain, including endpoints if
relevant.
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Error 1 Thinking that Points of Inflection
occur whenever f '' 0.
Remember these two functions
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And what about the State Common Core Standards?
Prentice Hall textbooks either have the content
now or will make it available with website
supplements even if authors question whether it
is appropriate. Some of it is quite ambitious.
We must all become statistics teachers.
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The trickiest part for many teachers will be the
mathematical practices. We have been writing
textbooks with those in mind for about fifteen
years!
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• Make sense of problems and persevere in solving
  them
• Reason abstractly and quantitatively
• Construct viable arguments and critique the
  reasoning of others
• Model with mathematics
• Use appropriate tools strategically
• Attend to precision

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• Look for and make use of structure
• Look for and express regularity in repeated
  reasoning

If we can survive a few years of transition,
common state standards can be a great thing for
American mathematics education. After all, the
common curriculum is one of the main reason that
AP Calculus works so well!
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On the other hand, we may see fewer students
taking AP in the future. It will no longer be
possible to rush through the pre-calculus
curriculum. There will be a lot more to be
learned, and the expectation is that students
will learn it. And much of it will not
specifically relate to calculus preparation!
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dkennedy_at_baylorschool.org