Q. 14 - Calculus

Question

If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, explain why the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface.

Step-by-Step Solution

Answer

If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, then the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface because they all lie in the same plane.

Step 1. Given Information

The function $$f(x, y)$$ is differentiable at a point $$(a, b)$$ 

Step 2. Explanation

We know that the function, $$f(x, y)$$ is differentiable at a point $$(a, b)$$.

$$\implies$$ All the lines tangent to the graph of the function, $$f$$ at the point $$(a,b)$$ lie in the same plane.

Here, we can use any two distinct lines in that plane to determine the equation of the plane. 

Therefore, If the function $$f(x, y)$$ is differentiable at a point $$(a, b)$$, then the tangent lines to the graph of $$f$$ at $$(a, b)$$ in the $$x$$ and $$y$$ directions are sufficient to determine the tangent plane to the surface.