Question

What is the equation of the tangent line to the surface z=f(x,y) where f(x,y)=y²sinx?

Answer 1

The equation of the tangent line to the surface
`\( z = f(x, y) \)` where
`\( f(x, y) = y^2 \sin x \)` is
`\( z = y^2 \sin x \)`at the point of tangency.

Step-by-step explanation:

To find the equation of the tangent line to the surface
`\( z = f(x, y) \),` we use the partial derivatives with respect to
`\( x \)` and
`\( y \).` For the given function
`\( f(x, y) = y^2 \sin x \),` the partial derivatives are
`\( f_x = y^2 \cos x \)` and
`\( f_y = 2y \sin x \).`

The equation of the tangent plane is given by
`\( z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \),` where
`\( (x_0, y_0, z_0) \` ) is the point of tangency. In this case, let's assume the point of tangency is
`\( (a, b, f(a, b)) \), so \( z - f(a, b) = f_x(a, b)(x - a) + f_y(a, b)(y - b) \).` Substituting the given function and its partial derivatives, the equation becomes
`\( z - y^2 \sin x = y^2 \cos a (x - a) + 2by \sin a (y - b) \).` To simplify, we can rearrange the terms to get
`\( z = y^2 \sin x \),` which is the equation of the tangent line.

Understanding the concept of tangent lines to surfaces involves the use of partial derivatives and the tangent plane equation. The equation of the tangent line is derived by evaluating the partial derivatives at the point of tangency and using them in the equation of the tangent plane. The resulting equation represents a linear approximation to the surface at that specific point.