Exercises

01

Tangent plane to a graph

Let $f(x,y)=x{y}^3+x^2$. Find the tangent plane to the graph of $f(x,y)$ at the point $(x,y)=(2,-2)$.

02

Linear approximation

Use a linear approximation to estimate the value of $(3.99{)}^3(1.01{)}^4(1.98{)}^{-1}$.

• (a) Write this using a function $L(x,y,z)$.
• (b) Write this using the notation of differentials.

What is the percentage error of your estimate from the true value?

03

When partials aren’t continuous

Compute the partial derivatives of $f(x,y)=\frac{2xy(x+y)}{x^2+y^2}$ and explain carefully why this non-differentiable function does not violate the theorem.

04

Differentiability of the radius function

Let $r(x,y)=\sqrt{x^2+y^2}$. Is $r(x,y)$ differentiable? Where?