Mathematical Tools for Physics - Department of Physics - University

1—Basic Stuff 17

OR, if you’re clever with partial fractions, you might realize that you can rearrangefas

x

a^2 −x^2

=

− 1 / 2

x−a

+

− 1 / 2

x+a

,

and then follow the ideas of techniques 8 and 9 to sketch the graph. It’s not obvious that this is any
easier; it’s just different.

Exercises

1 Expressexin terms of hyperbolic functions.

2 Ifsinhx= 4/ 3 , what iscoshx? What istanhx?

3 Iftanhx= 5/ 13 , what issinhx? What iscoshx?

4 Letnandmbe positive integers. Leta=n^2 −m^2 , b= 2nm,c=n^2 +m^2. Show thata-b-c

form the integer sides of a right triangle. What are the first three independent “Pythagorean triples?”
By that I mean ones that aren’t just a multiple of one of the others.

5 Evaluate the integral

∫a

0 dxx

(^2) cosx. Use parametric differentiation starting withcosαx.
6 Evaluate
∫a

0 dxxsinhxby parametric differentiation.

7 Differentiatexexsinxcoshxwith respect tox.

8 Differentiate

∫x 2

0 dtsin(xt)with respect tox.

9 Differentiate

∫+x

−xdte

−xt^4 with respect tox.

10 Differentiate

∫+x

−xdtsin(xt

(^3) )with respect tox.
11 Differentiate
∫√^3 sin(kx)

0 dte

−αt^3 J 0 (βt)with respect tox.J 0 is a Bessel function.

12 Sketch the functiony=v 0 t−gt^2 / 2. (First step: set all constants to one.v 0 =g= 2 = 1. Except

exponents)

13 Sketch the functionU=−mgy+ky^2 / 2. (Again: set the constant factors to one.)

14 SketchU=mg`(1−cosθ).

15 SketchV=−V 0 e−x

(^2) /a 2
.

16 Sketchx=x 0 e−αtsinωt.

17 Is it all right in Eq. (1.22) to replace “∆xk→ 0 ” with “N→∞?” [No.]

18 Draw a graph of the curve parametrized asx= cosθ,y= sinθ.

Draw a graph of the curve parametrized asx= coshθ,y= sinhθ.

19 What is the integral

∫b

adxe

−x^2?