Consider the function given by f(x) = 3 sin(0.6x

Consider the function given by $ f(x) = 3 \sin\left(0.6x - 2\right) $. (a) Approximate the zero of the function in the interval $ \left[0,6\right] $. (b) A quadratic approximation agreeing with $ f $ at $ x = 5 $ is $ g(x) = -0.45x^2 + 5.52x - 13.70 $. Use a graphing utility to graph $ f $ and $ g $ in the same viewing window. Describe the result. (c) Use the Quadratic Formula to find the zeros of $ g $. Compare the zero in the interval $ \left[0,6\right] $ with the result of part (a).

Consider the function given by $ f(x) = 3 \sin\left(0.6x - 2\right) $.

(a) Approximate the zero of the function in the interval $ \left[0,6\right] $.

(b) A quadratic approximation agreeing with $ f $ at $ x = 5 $ is $ g(x) = -0.45x^2 + 5.52x - 13.70 $. Use a graphing utility to graph $ f $ and $ g $ in the same viewing window. Describe the result.

(c) Use the Quadratic Formula to find the zeros of $ g $. Compare the zero in the interval $ \left[0,6\right] $ with the result of part (a).

Transcript

00:01 In part a of this problem, we're asked to approximate the zero of this function on the interval from 0 to 6.

00:06 We can actually find the exact 0 using algebra, so i'll do that first.

00:11 So let's just set this equal to 0, divide by 3, and then i'm going to replace that quantity in parentheses with a theta for a moment.

00:26 So we're looking for the angles whose sine value is 0.

00:30 And we know that the sign of 0 is 0.

00:32 We know that the sign of pi is zero, and there are others, but remember we're going to be sticking in the interval from zero to six.

00:41 So let's keep that in mind.

00:43 So now let's say that 0 .6x minus 2 equals 0, or 0 .6x minus 2 equals pi, etc.

00:55 Okay, so we would add 2 to both sides, and then 0 .6 is the same as 3 5ths.

01:03 I think i'd rather work in fractions.

01:04 Multiply both sides by five -thirds.

01:08 And we get ten -thirds, and that is in the interval from zero to six.

01:11 And let's see what happens with the other one.

01:15 Pi minus two.

01:17 Oops, pi plus two add two to both sides.

01:19 So that's about 5 .14.

01:22 And then if we multiply that by five -thirds, we might have to grab a calculator for this.

01:32 Five -thirds times pi plus two is most likely greater than six.

01:36 So let's keep this one.

01:38 10 thirds is the exact intersection point.

01:41 Now if you do want to approximate that with your calculator, then make sure you set your radiance, go to your y equals menu, type in the function.

01:49 We're gonna have to be graphing it for part b anyway.

01:52 And then set your window to go from 0 to 6 on the x -axis, and mine is going from negative 4 to 4 on the y -axis.

01:58 Those numbers can vary...

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