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Part I Review of Fundamentals

Module 2 Basic Physics and MathematicsUsed in Radiation Protection

Session 10 Basic Mathematics

Session I.2.10

IAEA Post Graduate Educational CourseRadiation Protection and Safety of Radiation Sources

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Introduction

Basic mathematics needed to perform health physics calculations will be reviewed

Students will learn about differentiation and integration; exponential and natural logarithmic functions; properties of logs and exponents; properties of differentials and integrals; and work example health-physics related problems

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Content

Concepts of differentiation and integration Exponential and natural logarithmic

functions Properties of logs and exponents Properties of differentials and integrals Solve sample health-physics related

problems

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Overview

Basic health-physics related mathematics will be discussed

Health physics-related sample problems will be worked to illustrate use of the mathematical principles discussed

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Definition of the Derivative

y = f(x)

f (x) is the derivative of f(x)

f (x) is also called the differential of y with respect to x

f (x) is defined as:

x 0 f (x) = = lim

dydx x

f(x + x) – f(x)

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Constant Rule for Differentiation

ddx (c) = 0 where c is a constant

ddx (cu) = c

dudx

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Sum and Difference Rule and Power Rule for Differentiation

ddx (u v) = du

dxdvdx

ddx (xn) = nxn-1

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Product and Quotient Rulesfor Differentiation

ddx (uv) = u + v

dvdx

dudx

ddx =

uv

dvdx

dudx

uv -

v2

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Chain and Power Rules for Differentiation

dydx =

dydu

dudx

dydx un = nun-1 du

dx

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Definition of the Anti-Derivative

A function F(x) is called an anti-derivative of a function f(x) if for every x in the domain of f:

F(x) = f(x)

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Definition of Integral Notationfor the Anti-Derivative

The notation for the anti-derivative (called the integral) of f(x):

f(x) dx = F(x) + C

where C is an arbitrary constant

F(x) is the anti-derivative of f(x)

That is, F (x) = f(x) for all x in the domain of f(x)

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Inverse Relationship Between Differentiation and Integration

f (x) dx = f(x) + C

ddx

f(x) dx = f(x)

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Basic Integration Rules

k dx = kx + C, where k is a constant

k f(x) dx = k f(x) dx

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[ f(x) g(x) ] dx = f(x) dx g(x) dx

Basic Integration Rules

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xn dx =

Basic Integration Rules

xn+1

n+1

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a

b

f(x) dx = F(b) - F(a)

Basic Integration Rules

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Definition of An Exponential Function

If a > 0 and a 1, then the exponential function with base “a” is given by

Y = ax

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Properties of Exponents

a0 = 1

(ab)x = axbx

axay = ax+y

(ax)y = axy

ax

ay = ax-y

1axa-x =

ab

x= ax

bx

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Natural Exponential Function

e = lim (1 + x)x 0

1x

Let y = ex

where “e” is the base of the natural logarithms (e = 2.71828. . .)

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Definition of the Natural Logarithmic Function

notation: ln(x) = loge(x)

ln(x) = b if and only if eb = x

ln(ex) = x and eln(x) = x

ex and ln(x) are inverse functions of each other

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Properties of Exponentialsand Natural Logarithms

ln(1) = 0

ln(e) = 1

ln(e-1) = -1

ln(2) 0.693

e0 = 1

e1 = e

eln(2) = e0.693 = 2

1ee-1 =

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Additional Propertiesof Natural Logarithms

ln(xy) = ln(x) + ln(y)

xyln( ) = ln(x) – ln(y)

ln(xy) = y ln(x)

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Derivative of theNatural Logarithmic Function

ddx

ln(x) = 1x

ddx

ln(u) = 1u

dudx

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Log Rule for Integration

( )( )dx = ln(u) + C

1x

dudx

1u

( )dx = ln(x) + C

where C is a constant

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Sample Problem No. 1

Solve (by integration) the basic differential equation for radioactive decay

= -N

where N is the number of radioactive atoms of a given radionuclide present at time t

dNdt

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is the radioactive decay constant, in units of sec-1

t is the elapsed decay time in seconds

Assume that the initial number of radioactive atoms at t = 0 is N0

Sample Problem No. 1

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( ) = - dt

Solution toSample Problem No. 1

dNdt = -N

dN = - Ndt

dN N = - dt

dN N

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ln(N) = -t + C

let C = ln(N0)

ln(N) = - t + ln(N0)

ln(N) - ln(N0) = - t

Solution toSample Problem No. 1

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ln ( ) = - t

Solution toSample Problem No. 1

NNo

NNo

NNo

N(t) = N0 e- t

= e- t

eln( ) = e(-t )

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Derive the rule of thumb:

Sample Problem No. 2

where A is the remaining activity of any radionuclide after an elapsed time of “n” half-lives and A0 is the initial activity at time t = 0

AAo

=12

n

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Recall from the previous problem that

N(t) = N0 e- t

Multiply both sides of the equation by

N(t) = N0 e- t

Solution toSample Problem No. 2

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Now recall that activity is simply A = N, so that the previous equation (which was in terms of radioactive atoms) can be written in terms of activity, as:

A = A0 e-t

Solution toSample Problem No. 2

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Solve the equation for an elapsed decay time “t” equal to “n” half-lives where T½ is the half-life

A = A0 e-t and recall =

Solution toSample Problem No. 2

ln(2)T½

AAo

= e-ln(2)

T½ nT½

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= e ln(2 )

= 2-n =

Solution toSample Problem No. 2

AAo

= e-nln(2)

-n

12

n

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Summary

Basic mathematics needed to perform health physics calculations was reviewed

Students learned about differentiation and integration; exponential and natural logarithmic functions; properties of logs and exponents; properties of differentials and integrals; and worked example health-physics related problems

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Where to Get More Information

Cember, H., Johnson, T. E., Introduction to Health Physics, 4th Edition, McGraw-Hill, New York (2008)

Martin, A., Harbison, S. A., Beach, K., Cole, P., An Introduction to Radiation Protection, 6th Edition, Hodder Arnold, London (2012)

Jelley, N. A., Fundamentals of Nuclear Physics, Cambridge University Press, Cambridge (1990)

Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds., Table of Isotopes (8th Edition, 1999 update), Wiley, New York (1999)