Find the slope of the line tangent to the curve xsin(3y)−cosy=−π at the point (π,π2) .
To find the slope of the line tangent to the curve at the point (π,π²), we need to find the derivative of the curve with respect to x and then evaluate it at the given point.
First, let's rewrite the equation of the curve in terms of y:
xsin(3y) - cosy = -π
Subtract xsin(3y) from both sides:
-cosy = -π - xsin(3y)
Divide both sides by -1 to isolate cosy:
cosy = π + xsin(3y)
Now, differentiate both sides of the equation with respect to x using the chain rule:
-sin(y) * dy/dx = sin(3y) + xsin(3y)' = sin(3y) + 3xsin(3y)' = sin(3y) + 3x(3cos(3y)) = sin(3y) + 9xcos(3y)
Since we want to find the slope at the point (π,π²), substitute π for x and π² for y:
-sin(π²) * dy/dx|_(π,π²) = sin(3π²) + 9πcos(3π²)
To evaluate this expression, we need to know the values of sin(π²) and cos(3π²). Since π² is not a special angle, we will need to use a calculator or a mathematical software to find their values.
Therefore, the slope of the line tangent to the curve at the point (π,π²) is given by:
-m = sin(3π²) + 9πcos(3π²) / sin(π²)
To find the slope of the line tangent to the curve xsin(3y)−cos(y) = −π at the point (π,π^2), we can use the concept of implicit differentiation.
1. Start by differentiating both sides of the equation with respect to x.
d/dx(xsin(3y) - cos(y)) = d/dx(-π)
2. Apply the chain rule and product rule when differentiating.
[sin(3y) + x * (3cos(3y)*dy/dx)] - [cos(y) * dy/dx] = 0
3. Now, we need to find dy/dx. To do so, isolate dy/dx.
3xcos(3y) * dy/dx - cos(y) * dy/dx = -sin(3y)
Factor out dy/dx:
dy/dx(3xcos(3y) - cos(y)) = -sin(3y)
dy/dx = -sin(3y) / (3xcos(3y) - cos(y))
4. Substitute the coordinates of the given point (π, π^2) into the equation for dy/dx.
dy/dx = -sin(3*π^2) / (3*π*cos(3*π^2) - cos(π^2))
5. Simplify the expression if possible.
The final step may involve evaluating trigonometric terms and performing calculations.
Please note that the calculation could be quite complex due to the trigonometric expressions involved.
To find the slope of the line tangent to the curve at a given point, we can use the concept of derivatives. The slope of the tangent line at a particular point is equal to the derivative of the curve at that point.
Let's start by taking the derivative of the given curve with respect to x. We need to use the chain rule for differentiating the composite function xsin(3y):
First, differentiate the outer function: d/dx (xsin(3y)) = cos(3y) * d/dx(x)
Since d/dx(x) is simply 1, we can simplify the expression to: d/dx (xsin(3y)) = cos(3y)
Next, we need to differentiate the inner function 3y with respect to x:
d/dx (3y) = 3 * dy/dx
Therefore, the derivative of xsin(3y) with respect to x is: d/dx (xsin(3y)) = cos(3y) * 3 * dy/dx
Now, let's rewrite the original equation using the derivative we just calculated:
cos(3y) * 3 * dy/dx - sin(y) = 0
To find the value of dy/dx at the point (π, π^2), we need to substitute x = π and y = π^2 into the derivative expression.
cos(3(π^2)) * 3 * dy/dx - sin(π^2) = 0
Simplifying this equation will give us the value of dy/dx at the given point. From there, we can determine the slope of the tangent line.