From part (b), we found that the tangent plane to the surface
at the point
was
![{\displaystyle \displaystyle -4x+8y-z=-1.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/244c93d0d8fdc2c062d44c7c6d87adca15e9ce84)
Since the point
is very close to
, the tangent plane approximation gives a good estimate for the true value of
. Plugging in
into the tangent plane equation we get
![{\displaystyle {\begin{aligned}-4(1.99)+8(1.01)-z&=-1\\-7.96+8.08-z&=-1\\0.12-z&=-1\quad \rightarrow \quad z=1.12\end{aligned}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/5f3a6904d7d92f62577eb12753035edec9b686be)
Therefore,
![{\displaystyle {\color {blue}f(1.99,1.01)\approx 1.12}.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/64f73c1d88b1d2c4a446f0330b792abdf190dc2e)