Advanced Calculus: Problem sheet 5 Please hand in any remaining solutions by by 1pm on 7 May 2001.

1. Obtain the Maclaurin series for the function f(x) = 1/(1 + x). Use this series to find the Maclaurin series for ln(1 + x). For what x does the resulting series converge?

2. Obtain the Maclaurin series for f(x) = 1/(1+x^{2}). Use this series to find the Maclaurin series for tan^{−1} x.

3. Obtain the Maclaurin series for sin x/x and hence express the integral

∫_{0}^{x} \frac{sin t}{t} dt

as a power series.

4. Use the binomial series and the fact that

\frac{d}{dx} sin^{−1} x = (1 − x^{2})^{−1/2}

to generate the first four nonzero terms of the Maclaurin series for sin^{−1} x. What is the radius of convergence of this series?

5. Recall that Taylor’s theorem states that

f(x) = P_{n}(x) + R_{n}(x)

where

P_{n}(x) = f(a) + f′(a)(x − a) + \frac{f′′(a)}{2!}(x − a)^{2} + ⋯ + \frac{f^{(n)}(a)}{n!}(x − a)^{n}

is the Taylor polynomial, and

R_{n}(x) = \frac{f^{(n+1)}(c)}{(n + 1)!}(x − a)^{n}

where c lies between a and x.

Take f(x) = e^{x} and a = 0 and write down the formula for P_{n}(x). Which value of n do you need to take to ensure that |e^{x} − P_{n}(x)| < 10^{−6} for (a) all x with 0 ≤ x ≤ 1?

(b) all x with 0 ≤ x ≤ 2? (c) all x with −2 ≤ x ≤ 0?

6. In each case find the equations of the tangent planes to the given surfaces at the point P.

(a) z = xy, P = (−2, 1, −2);

(b) z = x^{2} − y^{2}, P = (3, 2, 5);

(c) z = 1/(x + y), P = (1, −2, −1).

7. Find f_{x}, f_{y}, f_{xy}, f_{xx} and f_{yy} for the functions

(a) f(x, y) = y^{x};

(b) f(x, y) = e^{x^{2}y};

(c) f(x, y) = xy^{2} − x ln(xy);

(d) f(x, y) = y/(x^{2} + y^{2}).

8. Which order of differentiation will calculate f_{xy} faster: x first or y first? Try to answer without writing anything down.

(a) f(x, y) = x cos y + e^{−y};

(b) f(x, y) = 1/x^{2};

(c) f(x, y) = y^{2} + (x/y);

(d) f(x, y) = y ln xy.

9. Find the value of ∂z/∂x at the point (1, 1, 1) if the equation

2xy + z^{3}x − yz = 0

defines z as a function of the two independent variables x and y.

10. Show that the function f(x, y) = e^{−y} sin x satisfies the two-dimensional Laplace equa-

tion

\frac{∂^{2}f}{∂x^{2}} + \frac{∂^{2}f}{∂y^{2}} = 0

11. Show that the function w = ln(3x + 3ct) satisfies the two-dimensional wave equation

\frac{∂^{2}w}{∂t^{2}} = c^{2} \frac{∂^{2}w}{∂x^{2}}

12. Write the Chain Rule formula for \frac{∂w}{∂u} and \frac{∂w}{∂v} for w = h(x, y, z), x = f(u, v), y = g(u, v), z = k(u, v). You may find it useful to draw a diagram (as in Section 12.5 of Thomas and Finney to express the dependence of variables).

13. Show that, if x = e^{s} cos t, y = e^{s} sin t, then

u_{xx} + u_{yy} = e^{−2s} (u_{ss} + u_{tt}) .

RJC 12/3/2001