Q. 15 - Calculus

Question

Let $$u_{1}$$ and $$u_{2}$$ be two nonparallel unit vectors in $$R^{2}$$. 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 $$u_{1}$$ and $$u_{2}$$ 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 $$u_{1}$$ and $$u_{2}$$ directions are sufficient to determine the tangent plane to the surface since they lie in the same plane.

Step 1. Given Information

• Let $$u_{1}$$ and $$u_{2}$$ be two nonparallel unit vectors in $$R^{2}$$.
• The function $$f(x, y)$$ is differentiable at a point $$(a, b)$$. 

Step 2. Explanation

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

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

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

Here, $$u_{1}$$ and $$u_{2}$$ be two nonparallel unit vectors in $$R^{2}$$ that lie in the same plane.

Hence, 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 $$u_{1}$$ and $$u_{2}$$ directions are sufficient to determine the tangent plane to the surface.