1. The Tangent problem.

Consider the function given by the equation . We can define this function with Maple.

> f:=x->sqrt(x);

We can now evaluate the function at 4 and 5.

> f(4);

We hope that everyone knew this one already!

> f(5);

Well that's good enough (in fact it's perfect) and if we want a decimal approximation we can use evalf later.  We can compute the slope of the secant line through these two points, ( ) and ( ) .

> (f(5)-f(4))/(5-4);

Now we can write the equation of the secant line through these two points.

> S:=x->(sqrt(5)-2)/(5-4)/(5-4)*(x-4)+2;

We can now plot both and on the same axis.

> plot([f,S],0..10);

Let's plot from 4 to 5.

> plot([f,S],4..5);

It's hard to tell these two graphs apart. That is the secant line approximates the function closely between 4 and 5. So the slope of the secant line (which is the rate of change of the secant line)

> evalf(sqrt(5)-2);

approximates the rate of change of our function at . If we want a better approximation of the rate of change at we can pick the second value closer to 4.

Submission:

(a) Compute the equation for the secant line through the points ( ) and ( ) . Also compute the equation for the secant line through the points ( ) and ( ) . Now graph the two secant lines and the function on the same set of axes.

(b) Repeat the above by changing 4.1 to 4.01 and 3.9 to 3.99 .

(c) Redo the above by changing 4.01 to 4.001 and 3.99 to 3.999.

(d) Make your best 'educated' guess at the slope of the tangent line to at ( ) . Then compute the equation of the tangent line and plot the tangent line together with the function on the same set of axes.