# Finding point where slope of line equals curve

• fran1942
In summary: So your solution points are (##\pi/6 + 2n*##π, ##\pi/3 + 4n*##π - √3), where n is any integer.In summary, to find the points on the curve y=2(x-cosx) where the tangent is parallel to the line 3x-y=5, you must use the equation y=2(x-cosx) and set x=pi/6 + 2n*pi, where n is any integer. This will give you the x values of the points, and then you can plug them into the equation to
fran1942

## Homework Statement

At what point on the curve y=2(x-cosx) is the tangent parallel to the line 3x-y=5.

## The Attempt at a Solution

1. rewrite 3x-y=5 as y-3x-5
2. equate 2(x-cosx) = y-3x-5
3. differentiate: 2+2sinx = 3
4. solve for x: sin^-1(,5) = 0.524
5. plug into y=2(x-cosx) to get y value: y = -0.684

I am not sure I have this correct though. At stage 4 above, I have a trig equation. So doesn't that give me an infinite number of solutions. (I was not given a range in this question). Or do I just look at the graph of 2(x-cosx) and y-3x-5 to see logically where the point of equal slope is ?

Thanks for any help.

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fran1942 said:

## Homework Statement

At what point on the curve y=2(x-cosx) is the tangent parallel to the line 3x-y=5.

## The Attempt at a Solution

1. rewrite 3x-y=5 as y-3x-5
2. equate 2(x-cosx) = y-3x-5
No. The problem isn't asking where the two curves intersect, which is what you might be thinking you're doing in the step above.

What you want is to find any point on the graph of y=2(x-cosx) whose slope is 3, the slope ope of the line.

fran1942 said:
3. differentiate: 2+2sinx = 3
4. solve for x: sin^-1(,5) = 0.524
This looks OK, but I suspect there are a whole lot of points that are solutions. The inverse sine gives you only one.
fran1942 said:
5. plug into y=2(x-cosx) to get y value: y = -0.684

I am not sure I have this correct though. At stage 4 above, I have a trig equation. So doesn't that give me an infinite number of solutions. (I was not given a range in this question). Or do I just look at the graph of 2(x-cosx) and y-3x-5 to see logically where the point of equal slope is ?

Thanks for any help.

You are right, there is an infinite number of solutions, solving for x gives you your answer or pi/6, and so the tangent of y=2(x-cosx) is parallel to 3x-y=5 whenever x=pi/6 +2pi*n where n is an integer. As said above, instead of considering the intersection of the two functions, you are looking for points where the tangent line (whose slope is the derivative of y=2(x-cosx)) is parallel to 3x-y=5.

thanks guys.
I understand what you are saying.

So to clarify, when the question asks: at what point(s) on the curve is the tangent parallel to the line, then the answer would just be:
"the tangent of y=2(x-cosx) is parallel to 3x-y=5 whenever x=pi/6 +2pi*n where n is an integer" ?
i.e. I would not need to include y value reference in the answer ?

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To identify the points they're asking for, you need to supply a y value as well.

sorry to persist with this, but I just don't know how to express the recurring y values (due to the infinite amount of possible answers).

Can someone please show me to express the y values ?

Perhaps I would just express the points as:
x=pi/6 +2pi*n, y = f(pi/6 +2pi*n)
x=3pi/6 +2pi*n, y = f(3pi/6 +2pi*n)

Last edited:
Use the equation you're given: y = 2(x - cos(x))

If x = ##\pi/6## + 2n*##\pi## , then y = 2(##\pi/6## + 2n*##\pi## -√3/2)

## 1. What is the slope of a line at a given point?

The slope of a line at a given point is the measure of its steepness and direction at that specific point. It is calculated by finding the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

## 2. How do I find the point where the slope of a line equals a curve?

To find the point where the slope of a line equals a curve, you can use the derivative function. The derivative function will give you the slope of a curve at a specific point. Set the derivative function equal to the slope of the line and solve for the value of x to find the point of intersection.

## 3. Can the slope of a line ever equal the slope of a curve?

No, the slope of a line and the slope of a curve are two different concepts. The slope of a line is a constant value, while the slope of a curve changes at every point along the curve.

## 4. Do all lines have a slope at every point?

Yes, all lines have a slope at every point. This is because a line is a straight path with a constant slope, meaning the ratio of the rise to the run is the same at every point.

## 5. How can I visualize the point where the slope of a line equals a curve?

You can visualize the point where the slope of a line equals a curve by graphing both the line and the curve on the same coordinate plane. The point of intersection between the two will be the point where their slopes are equal. You can also use online graphing tools or software to plot the functions and find the point of intersection.

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