Which Tangent Line to y=sin(x) Has the Highest Y-Intercept Between 0 and 2π?

[Absolut10]

Homework Statement

Everyline tanjent to the function y=sin x has a y-intercept. Among all these tanjent lines, somewere between 0<x<2pi, find the equation of the line with the highest y-intercept.

Homework Equations

derivative of sinx=cosx
Second derivative is -sinx

The Attempt at a Solution

I got both of the derivatives and know that x = 0 when it equals pi and 2pi, I know the slope when x=pi is -1 so that's probabley the slope. But how would i figure out the y-intercept and put it in an equation?

 

[vela]

For x=x₀, find the slope of the line and a point (x₀, y₀) it passes through. Then write down the equation of the line using point-slope form and then solve for the y-intercept of the line.

 

[Mark44]

Homework Statement

Everyline tanjent to the function y=sin x has a y-intercept. Among all these tanjent lines, somewere between 0<x<2pi, find the equation of the line with the highest y-intercept.

Homework Equations

derivative of sinx=cosx
Second derivative is -sinx

The Attempt at a Solution

I got both of the derivatives and know that x = 0 when it equals pi and 2pi

Who is "it"? I think what you're trying to say is that sin(pi) = 0 and sin(2pi) = 0.

, I know the slope when x=pi is -1 so that's probabley the slope.

But it might not be. It seems reasonable to me to look for a number in the interval [pi/2, pi]. What you're looking for is the number for which the y-intercept of the tangent line is largest. You have a point on a curve (x, sin(x)) and the slope of the tangent line at that point is cos(x). If you know a point on a line (the tangent line) and its slope, you should be able to find an equation of that line. In particular, you should be able to find an expression for the y-intercept. That's what you want to maximize.

But how would i figure out the y-intercept and put it in an equation?

 

[berkeman]

Homework Statement

Everyline tanjent to the function y=sin x has a y-intercept. Among all these tanjent lines, somewere between 0<x<2pi, find the equation of the line with the highest y-intercept.

Homework Equations

derivative of sinx=cosx
Second derivative is -sinx

The Attempt at a Solution

I got both of the derivatives and know that x = 0 when it equals pi and 2pi, I know the slope when x=pi is -1 so that's probabley the slope. But how would i figure out the y-intercept and put it in an equation?

At each point along the curve y(x)=sin(x), you can calculate the derivative (as you have done), and that number at that point is the slope of the tangent line. So you have the point (x,y) and the slope of the tangent line that goes through that point. That should be enough for you to calculate the y-intercept of that line.

Once you have the equation for the y-intercept as a function of x, you can use differentiation to figure out the max and min values of the function over the interval specified.