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Understanding Calculus is Largely Locally Linear. Dan Kennedy Pearson Author. My Class’s First Problem of the Day:. Point-Slope Equation of a Line:. Linearization of f at x = a:. or. This simple equation is the basis for: Instantaneous rate of change Differentials
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Understanding Calculus is Largely Locally Linear Dan KennedyPearson Author
Point-Slope Equation of a Line: Linearization of f at x = a: or
This simple equation is the basis for: Instantaneous rate of change Differentials Linear approximations Newton’s Method for Finding Roots Rules for Differentiation Slope Fields Euler’s Method L’Hopital’s Rule
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:
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.
But if we graph the rock’s 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!
That’s what differentials are all about. Since the derivative is the slope of the curve at a point, there is no true Δy or Δx…but there is a slope. So we write: If we zoom in on a differentiable curve, because it is locally linear, Δy/ Δx is essentially constant. That makes dy/dx defensible as something other than 0/0.
If we zoom in at a point (a, f(a)): So… if (x, y) is close to (a, f(a)), There’s that equation again!
This is the basis for linear approximations. For example, here is Problem of the Day #23:
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.
Another such method is Newton’s Method for approximating roots. Here is Problem of the Day #22:
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:
Eventually this process will zoom in on an x-intercept of the curve. This is Newton’s Method for approximating roots of equations.
For example, let us find a root of the equation sin x = 0. Start with a guess of a = 2.
Local linearity even led to the discovery of the differentiation rules. For example, here’s the product rule. Zoom in on a product function uv until it looks linear. The slope will be the derivative of uv.
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!
Here is Problem of the Day #37: This is Euler’s Method. The picture at the right shows how it works.
You start with a point on the curve. Find the slope, dy/dx. Now move horizontally by Δx. Move vertically by Δy = (dy/dx) Δx.This gives you a new point (x + Δx, y + Δy).Repeat the process. I like to use a table.
The Top Ten Student Errors on the AP Calculus Examination …according to Dan Kennedy
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.
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."
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:
Error #5: Not Separating the Variables This can be a much more costly error if you are solving a separable differential equation like:
R Error #4: The Bad Washer Formula Good: Bad:
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.
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.
Error #1: Thinking that Points of Inflection occur whenever f'' = 0. Remember these two functions:
While you’re here, be sure to check out: FDWK4 “…the only book that could possibly follow FDWK3…”
New Features • All answers in back of the TE • Improved slope fields, series, etc. • Guidance on calculator usage • More AP® Quick Quizzes
New Features • New “More Derivatives” chapter • Updated data problems • Better answers and solutions • Better Teacher Edition stuff
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.
The trickiest part for many teachers will be the mathematical practices. We have been writing textbooks with those in mind for about fifteen years!
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
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!
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!
