Question

Let
\[
f(x)= 5 x^3 + 3 x^2 + x - 5\\
x_0=-3\\
\text{ The tangent to the graph of f at } x_0 \\
\text { cut the graph of f at } x_1\\
\text { Let A be the area between f and this tangent from } x_0 \text { to } x_1\\

\text { Draw the tangent to graph of f at } x_1\\
\text { This new tangent cut the graph of f at } x_2 \\
\text { Let B be the area between f and this tangent from } x_2 \text { to } x_1\\

\text{ Compute } \frac B A

\]

Answer 1

See the attached graph.

Answer 2

u should find the integral of function - tangent to find the space between and then compare them .

Answer 3

can mortals do this problem doc?

Answer 4

@dpalnc yes you can.
You know how to find the equation of a tangent at f at a given point and you can
find the intersections of a line and f.
You can find the area between two curves,
You can do that.

Answer 5

yep.. i am mortal. this looks doable... :)

Answer 6

You have to deal with messy numbers.

Answer 7

no problem.... maybe... :)

Answer 8

Actually, you would be amazed to
1) find the ratio B/A
2) you can also test that this ratio is the same if you start with a different point x_0 not equal to the infection point of f.
3) This ratio is the same for any polynomial of degree 3 and x0 different from the inflection point of f.

Answer 9

inflection not infection.

Answer 10

16...:)

Answer 11

i'm afraid to test out starting at a different x_0... those were ugly numbers!

Answer 12

@dpalnc Try another 3 rd degree polynomial.

Answer 13

You also get 16 if you choose x_0 not where the polynomial has an inflection point.