Linear approximation in two variables practice problems
The tangent-plane approximation L(x, y) = f(a, b) + f_x(x − a) + f_y(y − b), and using it to estimate. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{- x^{2} - 7 y^{2} + 20} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{19}{10}, \frac{21}{20}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = \frac{x}{x + y} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{41}{20}, \frac{11}{10}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{- x^{2} - 7 y^{2} + 20} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{41}{20}, \frac{21}{20}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = e^{x} \cos{\left(y \right)} \) at \( \displaystyle (0, 0) \), and use it to estimate \( \displaystyle f\left(\frac{1}{20}, \frac{1}{10}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{- x^{2} - 7 y^{2} + 20} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{19}{10}, \frac{51}{50}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{x^{2} + y^{2}} \) at \( \displaystyle (3, 4) \), and use it to estimate \( \displaystyle f\left(\frac{61}{20}, \frac{201}{50}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = x \sqrt{y} \) at \( \displaystyle (1, 4) \), and use it to estimate \( \displaystyle f\left(\frac{11}{10}, \frac{39}{10}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = x \sqrt{y} \) at \( \displaystyle (1, 4) \), and use it to estimate \( \displaystyle f\left(\frac{9}{10}, \frac{81}{20}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = x^{2} y^{3} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{41}{20}, \frac{11}{10}\right) \).
Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{- x^{2} - 7 y^{2} + 20} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{99}{50}, \frac{51}{50}\right) \).