Write the equation of lines tangent and normal to the following function at (0, π). To find derivative, use implicit differentiation.
x^2cos^y - siny = 0
I do not know what cos^y means. I'll just use cosy and you can fix it as needed.
x^2 cosy - siny = 0
2x cosy - x^2 siny y' - cosy y' = 0
y' = (2x cosy)/(x^2 siny + cosy)
so, at (0,π), y' = 0
Since the tangent is horontal, the normal is vertical, and the lines are
tangent: y=π
normal: x=0
To begin, let's find the derivative of the given equation using implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x^2cos^y - siny) = d/dx(0)
Step 2: Apply the product and chain rules.
(d/dx(x^2cos^y) - d/dx(siny)) = 0
Step 3: Differentiate each term separately.
(d/dx(x^2) * cos^y + x^2 * d/dx(cos^y) - d/dx(siny)) = 0
Step 4: Simplify and solve for the derivative of y with respect to x.
2x * cos^y - x^2 * sin(y) * (d/dx(y)) - cos(y) * (d/dx(y)) = 0
Step 5: Notice that we have a common factor of (d/dx(y)). Factor it out.
(d/dx(y)) * (2x * cos^y - x^2 * sin(y) - cos(y)) = 0
Since we are interested in finding the equation of the tangent and normal lines at (0, π), we need to evaluate the derivative at this point.
Step 6: Substitute x = 0 and y = π into the derivative equation.
(d/dx(y))(2(0) * cos^π - (0)^2 * sin(π) - cos(π)) = 0
Step 7: Evaluate the values.
(d/dx(y))(-cos(π)) = 0
Step 8: Simplify and solve for (d/dx(y)).
(d/dx(y)) = 0
Thus, the derivative of y with respect to x is 0, indicating that the function is constant at the point (0, π). In other words, the slope of the tangent and normal lines at this point is 0.
To find the equation of a line, we need a point and the slope of the line. Since we already have the point (0, π), we can proceed to find the equations of both the tangent and normal lines.
1. Equation of the Tangent Line:
Since the slope of the tangent line is 0, we can write its equation as:
y - π = 0 * (x - 0)
y - π = 0
y = π
Therefore, the equation of the tangent line to the function at (0, π) is y = π.
2. Equation of the Normal Line:
Since the slope of the normal line is perpendicular to the tangent line (which means it is undefined or infinite), its equation can be written as:
x - 0 = undefined * (y - π)
The equation is of the form x = k, where k is a constant. Since the line is vertical, it intersects the y-axis at x = 0.
Therefore, the equation of the normal line to the function at (0, π) is x = 0.
In summary, the equation of the tangent line is y = π, and the equation of the normal line is x = 0.