7.2 Workshop Exercises 2

For more worked examples, see Gilbert and Jordan Guide to Mathematical Methods, pages 153–164.

True or False? Let $f(x,y)$ be a function of two variables.

i) If $f(x)\rightarrow 0$ as $x\rightarrow\infty$ then $\int_{1}^{\infty}f(x)\,dx$ converges.

ii) If $f(x)$ is well-defined and continuous for $0<x<1$ then $\int_{0}^{1}f(x)\,dx$ converges.

iii) $\int_{1}^{\infty}\frac{e^{-x}}{x^{2}}\,dx$ converges.

iv) If $\frac{\partial f}{\partial x}=0$ then $f$ is constant.

The two main themes this week are improper integrals (including Laplace transforms) and partial differentiation. For improper integrals, see questions W2.1-7, especially W2.2-2.5. For partial differentiation, see W2.8-13.

W2.1. Determine whether the following functions converge as $R\rightarrow\infty$, and find the limit where appropriate:

W2.2. Evaluate the improper integrals

In each case, use partial fractions to find $\int^{R}$ and then let $R\rightarrow\infty$.

W2.3. Evaluate $\int_{2}^{\infty}\frac{26}{f(x)}\,dx$, where $f(x)$ is as in W1.5.

W2.4. Show that each of the following integrals is improper, and evaluate:

W2.5. Find the Laplace transform $F(s)$ of $\cosh ax$ where $a>0$ is a constant, and $s>a$. Recall that $\cosh ax=(e^{ax}+e^{-ax})/2$.

W2.6. Let $0<a<1$. Find the improper integral $\int_{0}^{\infty}a^{x}\,dx.$

W2.7. (i) Show that

(ii) By integrating by parts, show that

W2.8. Find the first-order partial derivatives with respect to $x$ and $y$ (namely ${{\partial f}\over{\partial x}},{{\partial f}\over{\partial y}},{{\partial g}\over{\partial x}}$ and ${{\partial g}\over{\partial y}}$) of the functions

(i) $f(x,y)=\sin x\cosh y$ and    (ii) $g(x,y)=\cos x\sinh y$.

W2.9. Find the first-order partial derivatives with respect to $x$, $y$ and $z$ of the functions given by:

(i) $f(x,y,z)=z\sinh(yz^{3}+x^{2})$, and    (ii) $g(x,y,z)=e^{x+2y+3z}$.

W2.10. Let $u(x,t)=\sin(x^{2}-t).$ Show that $u$ satisfies the partial differential equation

W2.11. Find the third-order partial derivative $w_{xyz}$, when $w=(x+y^{5}+z^{7})^{6}$.

W2.12. (i) Two variables $x,y$ are related by the equation $x^{3}y+y^{2}=2$. Find $\frac{dy}{dx}$ in terms of $x$ and $y$.

(ii) Three variables $a$, $r$ and $S$ are related by the equation $e^{rS}=a^{2}S^{2}+r^{2}$. Find $\frac{\partial a}{\partial S}$, the rate of change of $a$ with respect to $S$, while keeping $r$ fixed.

(iii) Three variables $p$, $q$ and $m$ are related by the equation $(pm+pq+qm)=\cos(pqm)$. Find $\frac{\partial p}{\partial q}$, the rate of change of $p$ with respect to $q$ while keeping $m$ constant.

W2.13. (i) Find the partial derivatives $f_{x},f_{y},f_{xx},f_{yy}$ and $f_{xy}$ for $f(x,y)=y/x;$

(ii) Likewise , find $g_{x},g_{y},g_{xx},g_{yy}$ and $g_{xy}$ for $g(x,y)=\tan^{-1}(y/x),$ (where $\tan\theta=y/x$, and $(d/du)\tan^{-1}u=1/(1+u^{2})$.

(iii) Deduce that

W2.14. (i) Find the derivative of ${\hbox{sech}}\,s$.

(ii) Let $f(s)=-2^{-1}{\hbox{sech}}^{2}(s/2).$ Verify that $\left(\frac{df}{ds}\right)^{2}=f^{2}(2f+1).$