Q. 17 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

Step 2. Explanation

It is given that the function, $$f(x, y,z)$$ is differentiable at a point $$(a, b,c)$$.

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

So, we can use any three non-coplanar lines in that same hyperplane to determine the equation of the hyperplane.

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