Find the slope of the line tangent to the curve xsin(3y)−cosy=−π at the point (π,π2) . (1 point) Responses −3π − 3 π −1 − 1 0 0 1 1 3π
To find the slope of the line tangent to the curve, we need to first find the derivative of the curve and then evaluate it at the given point.
Taking the derivative of the curve xsin(3y) - cosy = -π with respect to x, we get:
sin(3y) + x(3cos(3y))(dy/dx) + sin(y)(dy/dx) = 0
Now we need to solve for dy/dx.
To do this, we can rearrange the equation and solve for dy/dx:
(3cos(3y))(dy/dx) + sin(y)(dy/dx) = -sin(3y)
(dy/dx)(3cos(3y) + sin(y)) = -sin(3y)
dy/dx = -sin(3y) / (3cos(3y) + sin(y))
Now we can evaluate this derivative at the point (π,π^2):
dy/dx = -sin(3π^2) / (3cos(3π^2) + sin(π^2))
Since sin(3π^2) = 0 and sin(π^2) = sin(π) = 0, we have:
dy/dx = 0 / (3cos(3π^2) + 0) = 0
Therefore, the slope of the line tangent to the curve at the point (π,π^2) is 0.
To find the slope of the line tangent to the curve at the point (π, π^2), we need to find the derivative of the curve function with respect to x, and then substitute the values of x and y from the given point.
The given curve equation is x*sin(3y) - cos(y) = -π.
Step 1: Find the derivative of the curve equation with respect to x.
Differentiating both sides of the equation implicitly with respect to x, we get:
d/dx[x*sin(3y)] - d/dx[cos(y)] = d/dx[-π]
The derivative of x*sin(3y) with respect to x can be found using the chain rule:
d/dx[x]*sin(3y) + x*d/dx[sin(3y)] - d/dx[cos(y)] = 0
The derivative of x with respect to x is 1, and the derivative of sin(3y) with respect to x can be found using the chain rule:
1*sin(3y) + x*(3*cos(3y)*d/dx[y]) - (-sin(y)*d/dx[y]) = 0
sin(3y) + 3x*cos(3y)*dy/dx + sin(y)*dy/dx = 0
Step 2: Substitute the values from the given point (π, π^2).
We substitute x = π and y = π^2 into the derivative equation:
sin(3*(π^2)) + 3π*cos(3*(π^2))*dy/dx + sin(π^2)*dy/dx = 0
Step 3: Solve for dy/dx (the slope).
Rearranging the equation to solve for dy/dx, we have:
dy/dx = -(sin(3*(π^2)) + sin(π^2)) / (3π*cos(3*(π^2)) + sin(π^2))
Now, simplify the expression:
dy/dx = -sin(3π^2) - sin(π^2) / (3π*cos(3π^2) + sin(π^2))
The slope of the line tangent to the curve at the point (π, π^2) is -sin(3π^2) - sin(π^2) / (3π*cos(3π^2) + sin(π^2)).
To find the slope of the line tangent to the curve at the point (π, π^2), we need to find the derivative of the curve with respect to x and evaluate it at that point.
First, let's rewrite the given equation:
xsin(3y) - cosy = -π
Now, we can differentiate both sides of the equation with respect to x using the chain rule:
d/dx (xsin(3y)) - d/dx (cosy) = d/dx (-π)
To differentiate xsin(3y), we use the product rule:
d/dx [x * sin(3y)] = x * d/dx[sin(3y)] + sin(3y) * d/dx[x]
The derivative of sin(3y) with respect to x is 3 * cos(3y) * (dy/dx).
Since we need to evaluate the derivative at the point (π, π^2), we substitute x = π and y = π^2 into the equation and differentiate:
π * 3 * cos(3*π^2) * (dy/dx) + sin(3*π^2) = 0
Now, we solve for (dy/dx):
3π * cos(3*π^2) * (dy/dx) = -sin(3*π^2)
(dy/dx) = -sin(3*π^2) / (3π * cos(3*π^2))
Thus, the slope of the line tangent to the curve at the point (π, π^2) is -sin(3*π^2) / (3π * cos(3*π^2)), which is approximately equal to -3π.