Approximate range of sum of two trig functions.

Question

Find the approximate range of the function

y = 2 sin (6x) + sin (4x).

My initial reasoning is that sin of anything maxes out at 1, so this function can be rewritten as

y = 2 (1) + 1

The maximum of y is 3.

Then I tried to refine my answer.

If sin(6x) is 1, then 6x must be π/2. It follows that x = π/12.

So, if 6x is π/2, then 4x is π/3. Correspondingly, when sin(6x) is 1, sin(4x) is ½√3 or ~0.866.

Hence, the max should be 2(1) + 0.866 = 2.866. The minimum should be the opposite -2.866.

But the answer keys shows the answer as [-2.889,2.889]
http://www.sosmath.com/cyberexam/precalc/TA4001/TA4001.html
Question 9.

My solution is off by 0.023 and I can't figure out why.

Anyone see it?

Answer 1

You were in the good way at the beginning but you forgot that the maximum value of the function will correspond to an "x" such that y'(x) will be zero. Your derivative is given by
f(x) = y'(x) = 4 Cos[4 x] + 12 Cos[6 x]
and you want that to be zero. So, use a Newton method for locating more accurately where f(x) is zero. Let us start at a guess which we shll note x_old and write, according to Newton, the following iteration scheme

x_new = x_old - f(x_old) / f'(x_old)

Starting at x_old = 0.261799 (this is Pi/12, your first guess), the successive iterates will then be 0.285094, 0.285062. For this last value of "x", the value of f(x) is 1.47708*10^-6 which means that you are at an extremum (or very close to it). Now, for this value of "x", y(x) is 2.88929. I bet that the first iterate would be sufficient.

Is this clearer to you ? Merry Xmas.

Answer 2

For your specific problem, you properly identified that the solution is close to Pi / 12. So, what you could have done is to expand your function y(x) as a Taylor series around x = Pi / 12 up to the second term. This leads to the following local approximation

(2 + Sqrt[3]/2) + 2 (x - Pi/12) - (36 + 4 Sqrt[3]) (x - Pi/12)^2

Since you look for an extremum, take the derivative of the above expression with respect to variable x and set the result to zero; you then get

2 - 2 (36 + 4 Sqrt[3]) (x - Pi/12) = 0

from which x = Pi/12 + 1/(36 + 4 Sqrt[3]) = 0.285094 which is identical to the first iterate in my first answer.

Plugging this value of x in the definition of y gives a maximum value equal to 2.88929.