Question 3.33

Let \(f(x, z)=e^{x+y}\).

• (a) Find the first-order Taylor formula for \(f\) at (0, 0).
• (b) Find the second-order Taylor formula for \(f\) at (0, 0).

Question 3.34

Suppose \(L \colon \mathbb{R}^2 \to \mathbb{R}\) is linear, so that \(L\) has the form \(L(x, y)= ax+by\).

• (a) Find the first-order Taylor approximation for \(L\).
• (b) Find the second-order Taylor approximation for \(L\).
• (c) What will higher-order approximations look like?

Question 3.35

\(f(x,y) = (x+y)^2\), where \(x_0= 0, y_0 = 0\)

Question 3.36

\(f(x,y) = 1/(x^2 + y^2 +1)\), where \(x_0 =0, y_0 =0\)

Question 3.37

\(f(x,y) =e^{x+y}\), where \(x_0 = 0, y_0=0\)

Question 3.38

\(f(x,y) = e^{-x^2-y^2} \cos\,(xy)\), where \(x_0 = 0, y_0=0\)

Question 3.39

\(f(x,y) = \sin\,(xy) + \cos\, (xy)\), where \(x_0=0, y_0=0\)

Question 3.40

\(f(x,y) = e^{(x-1)^2} \cos y\), where \(x_0=1, y_0=0\)

Question 3.41

Calculate the second-order Taylor approximation to \(f(x, y)=\cos x \sin y\) at the point \((\pi, \pi/2)\).

Question 3.42

Let \(f(x, y)=x\cos (\pi y) - y\sin(\pi x)\). Find the second-order Taylor approximation for \(f\) at the point (1, 2).

Question 3.43

Let \(g(x, y)=\sin(xy) -3x^2 \log y +1\). Find the degree 2 polynomial which best approximates \(g\) near the point \((\pi/2, 1)\).

Question 3.44

For each of the functions in Exercises 3 to 7, use the second-order Taylor formula to approximate \(f(-1, -1)\). Compare your approximation to the exact value using a calculator.

Question 3.45

(Challenging) A function \(f{:}\,\, {\mathbb R}\to {\mathbb R}\) is called an analytic function provided \[ f(x+h) = f(x) + f'(x)h + \cdots + \frac{f^{(k)}(x)}{k!} h^k + \cdots \] [i.e., the series on the right-hand side converges and equals \(f(x+h)\)].

• (a) Suppose \(f\) satisfies the following condition: On any closed interval \([a,b]\), there is a constant \(M\) such that for all \(k =1,2,3,\ldots, |f^{(k)}(x)| \le M^k\) for all \(x \in [a,b]\). Prove that \(f\) is analytic.
• (b) Let \(f(x) = \Big\{\begin{array}{l@{\qquad}l} e^{-1/x} & x > 0\\[4pt] 0 & x \le 0. \end{array}\) Show that \(f\) is a \(C^\infty\) function, but \(f\) is not analytic.
• (c) Give a definition of analytic functions from \({\mathbb R}^n\) to \({\mathbb R}\). Generalize the proof of part (a) to this class of functions.
• (d) Develop \(f(x,y) = e^{x+y}\) in a power series about \(x_0 = 0\), \(y_0=0\).