Study guide and
2 practice problems
on:
First-order approximation of functions of several variables
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For a differentiable function $f(x,y)$ and small values of $\Delta x$ and $\Delta y$, we have the first-order approximation
$$f(x+\Delta x,y + \Delta y) \approx f(x, y) + \partial_x f(x, y) \Delta x + \partial_y f(x, y) \Delta y $$
Related topics
Functions of several variables
(36 problems)
Multivariable calculus
(147 problems)
Practice problems
Let $u(x,y) = x y $. Estimate the value of $u(0.8, 2.1)$ using a first-order approximation of $u$ at $(1,2)$.
Solution
Suppose that the temperature in a 2d region is given by $T(x,y) = e^{-x^2 -y^2}$. Suppose that you move along the curve $x(t) = t$ and $y(t) = t^2$.
At $t = 1$, approximately how much does $x$ change in time $\Delta t$?
How much does $y$ change in a time $\Delta t$?
Use a first order expansion to find how much $T(x(t), y(t))$ changes in time $\Delta t$ at $t=1$.
What is the rate of change of $T(x(t), y(t))$ with respect to $t$ at $t=1$?.
Solution