Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180.
This will also lead y(1) from the tangent line approximation to be equal to approximately .76323248, NOT .81
Just an edit to the previous response, 3(3√3-π) / 4√2(π-4) is NOT equal to 1.27, but rather about -1.27
Using implicit derivatives,
y' =
y^2(sinx - x cosx)
------------------------------
x^2(y cosy - siny)
At (π/3,π/4), that is y' = 3(3√3-π) / 4√2(π-4) ≈ 1.27
Now, using the approximation ∆y/∆x ≈ dy/dx we get
y1 = y0 + y' (x1-x0)
= π/4 + 1.27(π/180) ≈ 0.81
Ah, the world of math, where even clowns have to put on their serious faces! Well, fear not, my friend, for I shall guide you through this mathematical maze.
First, let's differentiate the equation:
(d/dx)(sinx/x) + (d/dy)(siny/y) = (d/dx)C
Now, using some nifty calculus, we can find the derivatives of sinx/x and siny/y:
(sin(x)/x)' = (x*cos(x) - sin(x))/x^2
(siny/y)' = (cos(y))/y - (siny)/(y^2)
Plugging these derivatives back into the equation, we get:
(x*cos(x) - sin(x))/x^2 + (cos(y))/y - (siny)/(y^2) = 0
Now, let's find the equation of the tangent line to the curve at the point (pi/3, pi/4). We can use the point-slope form of a line:
(y - pi/4) = m(x - pi/3)
To find the slope m, we'll differentiate the equation of the curve with respect to x and substitute x = pi/3 and y = pi/4:
(cos(pi/3) - sin(pi/3))/(pi/3)^2 + (cos(pi/4))/(pi/4) - (sin(pi/4))/(pi/4)^2 = m
Simplifying and calculating, we get:
(1/2 - sqrt(3)/2)/(pi/9) + (sqrt(2))/2/(pi/4) - (1/2)/(pi^2/16) = m
After evaluating this expression, you can substitute m into the equation of the tangent line. Finally, plug in x = pi/3 + pi/180 into the equation of the tangent line to find the y-coordinate of the point on the curve.
And just like that, my friend, you'll have your answer. Good luck, and keep those math clowns running in circles!
To find the y-coordinate of the point on the curve with the given x-coordinate using the tangent line approximation, we can follow these steps:
Step 1: Start with the equation sin(x)/x + sin(y)/y = C.
Step 2: Differentiate the equation with respect to x to find the derivative of y with respect to x. This allows us to find the slope of the tangent line at the point (pi/3, pi/4).
d/dx [sin(x)/x + sin(y)/y] = d/dx [C]
Using the product rule and chain rule, we can differentiate both terms:
[cos(x)/x - sin(x)/x^2] + [sin(y)/y^2 * dy/dx] = 0
Step 3: Substitute the x-coordinate of the given point, pi/3, into the equation:
[cos(pi/3)/(pi/3) - sin(pi/3)/(pi/3)^2] + [sin(y)/(y^2) * dy/dx] = 0
Simplifying further:
[sqrt(3)/(pi/3) - 1/(pi^2/9)] + [sin(y)/(y^2) * dy/dx] = 0
Step 4: Substitute the y-coordinate of the given point, pi/4, into the equation:
[sqrt(3)/(pi/3) - 1/(pi^2/9)] + [sin(pi/4)/(pi/4)^2 * dy/dx] = 0
Simplifying further:
[sqrt(3)/(pi/3) - 9/pi^2] + [1/(pi^2/16) * dy/dx] = 0
Step 5: Solve for dy/dx, which represents the slope of the tangent line at the point (pi/3, pi/4):
[1/(pi^2/16) * dy/dx] = [9/pi^2 - sqrt(3)/(pi/3)]
Multiplying both sides by (pi^2/16), we get:
dy/dx = (pi^2/16) * [9/pi^2 - sqrt(3)/(pi/3)]
Simplifying:
dy/dx = (9/16) - (4sqrt(3)/3pi)
Step 6: Finally, using the tangent line approximation, we can find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180. We'll use the point-slope form of a line:
y - y0 = m (x - x0)
Plugging in the values, we have:
y - (pi/4) = [(9/16) - (4sqrt(3)/3pi)] * [pi/3 + pi/180 - (pi/3)]
Simplifying:
y - (pi/4) = [(9/16) - (4sqrt(3)/3pi)] * [pi/180]
Now, solving for y:
y = (9/16pi - 4sqrt(3)/180) * (pi/180) + (pi/4)
After simplifying the expression, you would get the approximate value of y.