Meaning of y = c if f(x) = cg(x)

  • Thread starter seniorhs9
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In summary: I think you misunderstood my question. I was asking if the two graphs were tangent at the point of intersection. If they are not tangent, then how can the slope of \displaystyle y=\frac{\sin(x)}{x} be zero at the same value of x at which the graph of y = c x is tangent to the graph of y=\sin(x)\, near the second hump?
  • #1
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Homework Statement



Hi. I post part A too but I'm having trouble only with Part B.

Question.[PLAIN]http://img100.imageshack.us/img100/8297/20104ivwhytangenttoseco.png [Broken]

Homework Equations



I know...

[itex]y = sin x [/itex] and [itex] y = cx[/itex] intersects when ...

[itex]sin x = cx[/itex] so [itex] \frac{sin x}{x} = c[/itex]

The Attempt at a Solution



I just don't understand why [itex] \frac{sin x}{x} = c[/itex]
means that [itex] y = c [/itex] has to be tangent to
[itex] y = \frac{sin x}{x}[/itex] at the second hump?

I'm not seeing the connection...

Thank you.
 
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  • #2
seniorhs9 said:
...
I know...

[itex]y = \sin x [/itex] and [itex] y = cx[/itex] intersects when ...

[itex]\sin x = cx[/itex] so [itex]\displaystyle \frac{\sin x}{x} = c[/itex]

The Attempt at a Solution



I just don't understand why [itex]\displaystyle \frac{\sin x}{x} = c[/itex]
means that [itex] y = c [/itex] has to be tangent to
[itex]\displaystyle y = \frac{\sin x}{x}[/itex] at the second hump?

I'm not seeing the connection...

Thank you.
Look at the second graph. If y = c x is not tangent to the graph of y = sin(x) "near the second hump", then the graph of y = c x will intersect the graph of y = sin(x) in either 7 places, or in 3 places.
 
  • #3
Hi SammyS. Thanks for your answer.

But I don't understand your answer either

"If y = cx is not tangent to the graph of y = sin(x) "near the second hump", then the graph of y = cx will intersect the graph of y = sin(x) in either 7 places, or in 3 places "

Generally, I have trouble relating intersection of [itex] y = sin x [/itex] and [itex] y = cx [/itex] to graph of [itex] y = \frac{sin x}{x} [/itex] and [itex] y = c[/itex]. But I understand [itex] sin x = cx [/itex] means [itex] \frac{sin x}{x} = c [/itex].
 
  • #4
To solve y = c x , and y = sin(x) simultaneously, you have:

c x = sin(x) .

That's equivalent to [itex]\displaystyle c=\frac{\sin(x)}{x}\,.[/itex]

One way to solve this equation is graphically, i.e., where does the graph of y = c intercept the graph of [itex]\displaystyle y=\frac{\sin(x)}{x}\,?[/itex] I think the question to be asking is, "Should these two graphs be tangent at the point of intersection?"

If y = c is tangent to [itex]\displaystyle y=\frac{\sin(x)}{x}\,[/itex] near [itex]\displaystyle x=\frac{5\pi}{2}\,,[/itex] then it must be true that on either side of the point of tangency, [itex]\displaystyle c>\frac{\sin(x)}{x}\,.[/itex] It can be shown that this inequality holds.

Therefore, the slope of [itex]\displaystyle y=\frac{\sin(x)}{x}\,[/itex] is zero at the same value of x at which the the graph of y = c x is tangent to the graph of [itex]\displaystyle y=\sin(x)\,[/itex] near the second hump. Use this value of x to solve for c.
 
  • #5
Hi SammyS. Thanks.
 

1. What does the value of "c" represent in the equation y = c if f(x) = cg(x)?

The value of "c" represents the constant multiplier in the function. It is the coefficient that is applied to the input of the function to get the output value of "y".

2. How does changing the value of "c" affect the graph of the function f(x) = cg(x)?

Changing the value of "c" affects the slope and y-intercept of the graph of the function. A larger value of "c" will result in a steeper slope, while a smaller value of "c" will result in a flatter slope. The y-intercept will also change, as it is equal to the value of "c".

3. Can "c" be a negative value in the equation y = c if f(x) = cg(x)?

Yes, "c" can be a negative value. This will result in a reflection of the graph of the function across the x-axis, as the negative value will cause the output values to be reflected over the x-axis.

4. What is the significance of "c" in the equation y = c if f(x) = cg(x)?

"c" is a significant value in the equation as it helps to determine the behavior of the function. It affects the slope, y-intercept, and overall shape of the graph. It can also be used to shift the graph vertically or horizontally depending on its value.

5. Can the value of "c" be changed while keeping the function f(x) = cg(x) the same?

Yes, the value of "c" can be changed while keeping the function f(x) = cg(x) the same. This will result in a transformation of the graph, as the slope and y-intercept will be affected. However, the overall shape of the graph will remain the same.

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