Problems

01

Linear approximation

Use linear approximation to estimate the value of $\frac{8.01}{\sqrt{(1.99)(2.01)}}$ without a calculator, using the notation of differentials. Then compute the exact value on a calculator and find the percentage error.

02

Tangent planes

• (a) Find the tangent plane to the graph of $f(x,y)=x{y}^3+x^2$ at the point $(2,-2,-12)$.
• (b) Suppose $f(1,2)=10$, $f(1.1,2.01)=10.3$ and $f(1.04,2.1)=9.7$. Find an approximate equation of the tangent plane to the graph $z=f(x,y)$ at the point $(1,2,10)$.
• (c) Find out where the tangent plane to $z=3x^2-4y^2$ has normal vector $(3,2,2)$. (Hint: by inspection of the equation $z=L(x,y)$, you can write a general formula for the normal vector to a tangent plane. Insert an arbitrary point $(a,b)$ and solve for $a$ and $b$ to be where this normal vector aligns with the given one.)