use newton's method to find the smallest positive value of a so that the graph of f(x)=xsin(3x) has a horiz...

Question

use newton's method to find the smallest positive value of a so that the graph of f(x)=xsin(3x) has a horizontal tangent line at x=a. Answers correct to three decimal places.

Answer 1

A function F(x) has a horizontal tangent line wherever it's derivative is 0 (the derivative is the slope of the tangent line), so we are looking for the smallest x such the derivative of F(x) = 0

Newton's method is an iterative algorithm for finding a root (a value where the function is 0). Starting with a guess, x, you compute the next guess with:

x' = x - f(x)/f'(x)

And keep iterating until the answer is good enough.

In our case, we want the derivative of F(x) to be zero so we take f(x) to be F'(x). This makes f'(x) the second derivative of F(x).

Newton's method works much better when you are close to the root, so we want to find a good starting value.

x sin(3x) starts off at 0 because both factors are 0. It rises rapidly (as x^2) because both factors rise until 3x = pi/4, or x = pi/12

At pi/4, the sin(3x) term first goes flat and then starts decreasing. So it seems reasonable to start the iteration at pi/12.

With a scientific calculator (or a calculator app on a computer), computing sines and cosines is easy so you can just start in with Newton's method.

If you had to do the computations by hand or with a 4-function calculator, you'd go one step further.

sin(x) = x - (1/3!)x^3 + (1/5!)x^5 + ...
cos(x) = 1 - (1/2!)x^2 + (1/4!)x^4 + ...

With series such as this, the error is less than that of the first term neglected.

So for three decimal places, you only need the first three terms for the sine and probably the cosine as well.

If you were to subsitute these, everything in sight becomes a polynomial, which is easier to compute than a transcendental function.

But as I noted, this is not needed with a good calculator.