3/2003 Rev 1 I.2.10 – slide 1 of 36 Part I Review of Fundamentals Module 2Basic Physics and Mathematics Used in Radiation Protection Basic Mathematics Session 10 Basic Mathematics Session I.2.10 IAEA Post Graduate Educational Course Radiation Protection and Safe Use of Radiation Sources

3/2003 Rev 1 I.2.10 – slide 2 of 36 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

3/2003 Rev 1 I.2.10 – slide 3 of 36 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

3/2003 Rev 1 I.2.10 – slide 4 of 36 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

3/2003 Rev 1 I.2.10 – slide 5 of 36 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xxx f(x +  x) – f(x)

3/2003 Rev 1 I.2.10 – slide 6 of 36 Constant Rule for Differentiation ddx (c) = 0 where c is a constant ddx (cu) = c du dx

3/2003 Rev 1 I.2.10 – slide 7 of 36 Sum and Difference Rule and Power Rule for Differentiation ddx (u  v) =  du dxdvdx ddx (x n ) = nx n-1

3/2003 Rev 1 I.2.10 – slide 8 of 36 Product and Quotient Rules for Differentiation ddx (uv) = u + v dv dxdudx d dx = u vdvdxdudx u v - v2v2v2v2

3/2003 Rev 1 I.2.10 – slide 9 of 36 Chain and Power Rules for Differentiation dydx =dydu dudx dydx u n = nu n-1 du dx

3/2003 Rev 1 I.2.10 – slide 10 of 36 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)

3/2003 Rev 1 I.2.10 – slide 11 of 36 Definition of Integral Notation for 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)

3/2003 Rev 1 I.2.10 – slide 12 of 36 Inverse Relationship Between Differentiation and Integration  f (x) dx = f(x) + C ddx  f(x) dx = f(x)

3/2003 Rev 1 I.2.10 – slide 13 of 36 Basic Integration Rules  k dx = kx + C, where k is a constant  k f(x) dx = k  f(x) dx

3/2003 Rev 1 I.2.10 – slide 14 of 36  [ f(x)  g(x) ] dx =  f(x) dx   g(x) dx Basic Integration Rules

3/2003 Rev 1 I.2.10 – slide 15 of 36  x n dx = Basic Integration Rules x n+1 n+1

3/2003 Rev 1 I.2.10 – slide 16 of 36 a b f(x) dx = F(b) - F(a) Basic Integration Rules

3/2003 Rev 1 I.2.10 – slide 17 of 36 Definition of An Exponential Function If a > 0 and a  1, then the exponential function with base “a” is given by Y = a x

3/2003 Rev 1 I.2.10 – slide 18 of 36 Properties of Exponents a 0 = 1 a 0 = 1 (ab) x = a x b x a x a y = a x+y a x a y = a x+y (a x ) y = a xy (a x ) y = a xy axaxaxax ayayayay = a x-y 1 axaxaxax a -x = ab x = axaxaxax bxbxbxbx

3/2003 Rev 1 I.2.10 – slide 19 of 36 Natural Exponential Function e = lim (1 + x) x  0 1x Let y = e x where “e” is the base of the natural logarithms (e = )

3/2003 Rev 1 I.2.10 – slide 20 of 36 Definition of the Natural Logarithmic Function notation: ln(x) = log e (x) ln(x) = b if and only if e b = x ln(e x ) = x and e ln(x) = x e x and ln(x) are inverse functions of each other

3/2003 Rev 1 I.2.10 – slide 21 of 36 Properties of Exponentials and Natural Logarithms ln(1) = 0 ln(1) = 0 ln(e) = 1 ln(e) = 1 ln(e -1 ) = -1 ln(2)  ln(2)  e 0 = 1 e 0 = 1 e 1 = e e 1 = e e ln(2) = e = 2 1e e -1 =

3/2003 Rev 1 I.2.10 – slide 22 of 36 Additional Properties of Natural Logarithms ln(xy) = ln(x) + ln(y) xy ln( ) = ln(x) – ln(y) ln(x y ) = y ln(x)

3/2003 Rev 1 I.2.10 – slide 23 of 36 Derivative of the Natural Logarithmic Function ddx ln(x) = 1x ddx ln(u) = 1ududx

3/2003 Rev 1 I.2.10 – slide 24 of 36 Log Rule for Integration  ( )( )dx = ln(u) + C 1x dudx1u  ( )dx = ln(x) + C where C is a constant

3/2003 Rev 1 I.2.10 – slide 25 of 36 Sample Problem No. 1 Solve (by integration) the basic differential equation for radioactive decay = - N = - N where N is the number of radioactive atoms of a given radionuclide present at time t dNdt

3/2003 Rev 1 I.2.10 – slide 26 of 36 Is the radioactive decay constant, in units of sec -1 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 N 0 Sample Problem No. 1

3/2003 Rev 1 I.2.10 – slide 27 of 36  ( ) =  - dt Solution to Sample Problem No. 1 dNdt = - N dN = - Ndt dN N = - dt dN N

3/2003 Rev 1 I.2.10 – slide 28 of 36 ln(N) = - t + C let C = ln(N 0 ) ln(N) = - t + ln(N 0 ) ln(N) - ln(N 0 ) = - t Solution to Sample Problem No. 1

3/2003 Rev 1 I.2.10 – slide 29 of 36 ln ( ) = - t Solution to Sample Problem No. 1 N NoNoNoNo N NoNoNoNo N NoNoNoNo N(t) = N 0 e - t N(t) = N 0 e - t = e - t = e - t e ln( ) = e (- t )

3/2003 Rev 1 I.2.10 – slide 30 of 36 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 A 0 is the initial activity at time t = 0 A AoAoAoAo =12n

3/2003 Rev 1 I.2.10 – slide 31 of 36 Recall from the previous problem that N(t) = N 0 e - t Multiply both sides of the equation by Multiply both sides of the equation by N(t) = N 0 e - t N(t) = N 0 e - t Solution to Sample Problem No. 2

3/2003 Rev 1 I.2.10 – slide 32 of 36 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 = A 0 e - t Solution to Sample Problem No. 2

3/2003 Rev 1 I.2.10 – slide 33 of 36 Solve the equation for an elapsed decay time “t” equal to “n” half-lives where T ½ is the half-life A = A 0 e - t and recall = Solution to Sample Problem No. 2 ln(2) T½T½T½T½ A AoAoAoAo = e - ln(2) T½T½T½T½ nT ½

3/2003 Rev 1 I.2.10 – slide 34 of 36 = e ln(2 ) = e ln(2 ) = 2 -n = = 2 -n = Solution to Sample Problem No. 2 A AoAoAoAo = e - nln(2) -n12n

3/2003 Rev 1 I.2.10 – slide 35 of 36 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

3/2003 Rev 1 I.2.10 – slide 36 of 36 Where to Get More Information  Camber, H., Introduction to Health Physics, 3 rd Edition, McGraw-Hill, New York (2000)  Firestone, R.B., Baglin, C.M., Frank-Chu, S.Y., Eds., Table of Isotopes (8 th Edition, 1999 update), Wiley, New York (1999)  International Atomic Energy Agency, The Safe Use of Radiation Sources, Training Course Series No. 6, IAEA, Vienna (1995)