**A tour of calculus features of GeoGebra 4**

We start with an interesting function, f(x) = x^2/10 + sin(2 x) and look at its graph. We also add two points, A and B, on the curve and one point, C, off the curve for later use.

I can use the commands

TanA=Tangent[A,f] and TanC=Tangent[C,f] to draw lines tangent to the graph of f(x) at A, and at the point on the graph with the same x value as C.

With the Slope command I can highlight the slopes of the tangent lines.

I can create a point DfA (Derivative of f at A) with DfA=(x(A),Slope[TanA]). This point has the form (x, f ' (x)) with x equal to the x coordinate of A.

With DfA, I can turn on trace to drag A around and see the derivative at a lot of points. I can also take the locus of DfA depending on letting A vary as a point on the graph of f(x).

Switching to a symbolic viewpoint, I can define the derivative either with

g(x) = f '(x)

or with

h(x) = Derivative[f]

Notice that the algebra panel is showing the symbolic formula for the derivative. (In the graphics view we can show name and value as a label. The formula is then formatted nicely.)

I can look at higher order derivatives. If we wanted a 4th derivative, we can either define it like k(x) in the example below, with 4 primes, or like K(x) with the command K(x)=Derivative[f,4].

We see that we can use the Min, Max, Extremum, and Root commands to find the appropriate points.

We now turn to the function inspector tool. This lets us look at information for a chart of points. This gives a good presentation for difference quotients.

Choosing the other tab of the function inspector, we can also look at information related to an interval.

We turn now to Reimann sums and integrals. I can use the LeftSum command to compute a simple Riemann sum for my curve, with a slider so that I can vary the number of intervals.

Similarly, I can compute other versions of the Riemann sum. LowerSum and UpperSum give lower and upper bounds on the integral. RectangularSum requires a position number on how far across each interval to evaluate the function. TrapezoidalSum computes the Trapezoidal rule.

This lets us explore the rate of convergence for the various methods.

We can also use "integral" to add in the definite integral.

The integral command without an interval produces an indefinite integral.

Finally, I can compute Taylor polynomial using the "TaylorPolynomial[f(x),a,n]" command.

Note that the polynomials are expressed in terms of powers of (x-a).

© 2011, Mike May, S.J.

**Except where otherwise noted, content on this site is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 license, Mike May, S.J. maymk@slu.edu**

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