Sketch the graph of y = x + sin(x)

[bengalibabu]

Hey guys, just needed help with two questions, any help would be appreciated

1. a radar antenna on a ship tha tis 4km from a straight shoreline revolves at 32 rev/min. How fast is the radar beam moving along the shore line when the beam makes an angle of 30° with the shore? Give an exact value answer.

http://illusionpixels.com/dt/diagram.jpg

tan 30 = x/4
4tan30 = x
4(sec^2 30)dy/dt = dx/dt
4(sec^2 30)(64pi) = dx/dt
= 1072.330292m/min

2. Sketch the graph of y = x + sin(x), showing derivative, intervals of increase and decrease, points of inflection, concavity, etc. Use exact values

i kno its stated in the rules that ur supposed to show what u did and all that stuff but for this questions I am not even sure where or how to start


any help would be appreciated...thx in advanced

 

[Knavish]

1) Instead of setting in 30 degrees, give the angle the arbitrary value of, say, y. Thus tan(y)=x/4; dx/dt=128sec^2(y). Then subsitute y=30.

2) Can you be more specific? It's mostly just knowing the definitions of the terms. For example, y'=1+cos(x); the graph increases when y is greater than 0, and decreases when less than; the point of inflection is y''=0; and concavity exists when y'' is greater or less than 0.

 

[bengalibabu]

1) wud it really matter if i set the angle as y instead of 30, should i not get the same answer
2) that's all the questions asks, can't get any more specific, but that's pretty much what i thought the answer would be

 

[Jelfish]

I think you have the right answer for the first part.

For the second, just think about each term separately - what does y=x and y=sin(x) look like? Now, if you add them together, that means that the y component of the graph at each x will be the sum of those two individual graphs. Sin(x) hovers around y=0 up to +/- 1. Therefore, sinx+x will hover around y=x up to x +/- 1.

Here's a way to help you sketch the graph. The first thing to do is draw y=x. Then consider the points where sin(x)=0, i.e. x=0, pi, 2pi, etc. so you can draw those points onto your line because you know that at those points, they will be the same as y=x. Then consider where sin(x)=1, i.e. x=pi/2, 5pi/2, 9pi/2 etc. and draw those a height of 1 above the y=x line at those points. Do the same for when sin(x)=-1. After that, you should have a y=x line with points that go above, below, and lie on that line. Knowing the shape of a sinusoidal graph, you should be able to connect them together and that will be your graph.

 

[OlderDan]

1) wud it really matter if i set the angle as y instead of 30, should i not get the same answer
2) that's all the questions asks, can't get any more specific, but that's pretty much what i thought the answer would be

You in fact did not treat the angle as a constant the way you wrote it, so you did not "set the angle". Your notation is sloppy. Since there was a y coordinate in your diagram, you should have used some other letter for the angle. I'll call it A. Your equation should have been

tan A = x/4km
x = 4km tan A
dx/dt = 4km (sec^2 A)dA/dt

When you evaluate at A = 30 degrees and dA/dt = 32 rev/min = 64pi rad/min you get your numerical answer, but not your units. There is a good chance a given answer will be expressed in km/sec or m/sec rather than distance per minute.

Your second problem is a typical calculus curve sketching problem. You need to find the first and second derivatives of y wrt x and use what you know about maxima, minima and concavity as related to those derivatives. The list of things you have been asked to consider are all related to those derivatives. I expect your text has an example or two with everything worked out that you can use as a model.