This exercise will explore using the difference quotient for small values of h as well as using graphs to estimate the derivative. Execute the command defining the function f below.

> f:=x->3^x;

This definition will allow us to rapidly evaluate the difference quotient for smaller and smaller values of h .

>Q(1.);Q(.1);Q(0.01);Q(0.001);Q(0.0001);Q(0.00001);Q(0.000001);Q(0.0000001);Q(0.000000001);Q(0.0000000001);

From what we know about round off error, we don't believe the last answer at all, and we wonder about some of the previous ones too. Let us try to get a better estimate by changing the number of significant digits that Maple uses.

> Digits:=30;

> Q(0.000001);Q(0.0000001);Q(0.000000001);Q(0.0000000001);

It seems likely that the estimate 3.2958 is accurate to one part in 1/10000. Let us have Maple find the limit of the difference quotient, to give us an exact answer.

> limit(Q(h),h=0);

This answer is exact , but let us have Maple give us a floating point estimate of the answer.

> evalf(limit(Q(h),h=0));

Thus our estimate is confirmed by Maple.  We set Digits back down.

> Digits:=10;

Next, let us use a graph to estimate the slope by plotting the function and then zooming. Note that the scaling must be constrained to get a realistic idea of the slope in the picture.

> plot(f(x),x=0..2,scaling=constrained);

> plot(f(x),x=.9..1.1,scaling=constrained);

> plot(f(x),x=.99..1.01,scaling=constrained,color=red);

In the last graph, the curve appears to be a straight line (just like the earth appears flat). We see from the graph that the rise is about 0.066, so the slope estimate is

> .066/.02;

Submission worksheet: