My 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 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. (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) 2 (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) 20.1 (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 1.2) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 1.2) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 1.2) 2.3 (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 1.2) (1.1, 2.2) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 2.2) (1.1, 1.2) (x, y) dy/dx ΔxΔxΔxΔx ΔyΔyΔyΔy (x + Δx, y + Δy) (1, 1) (1.1, 1.2) (1.2, 1.43)

Since 2000, the AP Calculus Test Development Committee has been purposefully re-directing the philosophy of the AP courses away from mere computation and toward a better understanding of calculus concepts. A big component of this new philosophy has been the emphasis on multiple representations: Graphical Numerical (Tabular) Analytic Verbal

In differential calculus, this new emphasis has led to the emergence of some new kinds of AP problems: Linking graphically Derivatives from Tables Differential Equations Interpreting the Derivative Slope Fields Euler’s Method (BC) Logistic Growth (BC)

Problem of the Day #16:

2003 / AB-4 BC-4

Most tabular problems have derivative and integral parts. This is 2001 / AB-2 BC-2.

Differential Equations have changed emphases many times in the history of AP Calculus. At this time AB and BC students are responsible for solving two types: Exact, for example Separable, for example

A differential equation becomes an initial value problem when the solver knows a point on the solution curve, thus determining the value of the constant of antidifferentiation. This also implies a domain for the solution. Thus:

Notice that the point (0, 1) pins down the correct curve, but only on the interval containing that continuous piece of the graph.

To satisfy the differential equation and the initial condition, the graph could just as well look like this:

2006 / AB-5 combined slope fields with a differential equation…and a surprise.

2005 / BC-4 (with Euler’s Method)

2004 / BC-5 (Logistic Growth)

If students had been asked to solve the logistic differential equation, it would have required partial fractions. In fact, it can be done in the general case:

In the case of 2004 / BC-5:

Problem of the Day #53: