Question

Show a graphical method of approximating the instantaneous rate of change at x = 3 for the function ƒ(x) = -x2 + 4x + 1 using secants. Show two graphical approximations.

I had the same question but instructed to find it using a numerical method.
No problem with that, but Im not sure how to calculate graphically.

Answer 1

just draw several position vectors <b>v</b>_{_k}, for i=1..n, getting closer to the vector at the given point, <b>r</b>. For each <b>v</b>_{_k}, draw <b>v</b>_{_k}-<b>r</b><sub>. These serve the same purpose as secants in the usual x-y plots illustrating the approach to a tangent.

Pick up any good book on vector analysis and you will find such illustrations. Also, as always, google is your friend.

Answer 2

Shoot. I see that you were in fact just talking about an x-y plot. Pick any point P on the curve, and several nearby points, q_{_k}. Draw the secants to the various points. You can see that as q_{_k} approaches P, the secant is ever more nearly a tangent.

I'm sure your textbook has this very illustration, maybe for some other f(x).