Medical and Biological Physics (4th ed.) / Chalyi A. V.

MEDICAL AND BIOLOGICAL PHYSICS

MEDICAL AND BIOLOGICAL PHYSICS Edited by Prof. A. Chalyi

Practical Section 1

PRACTICAL WORK 1.1. Elements of differential calculation

1.

EDUCATIONAL AIMS – to master the notion of derivative and the notion of derivative of higher orders; – to learn to apply these notions for testing intervals of an increase and decrease of function, points of extrema, and to graph a function; – to master the notion of the differential of the function, notion of the partial derivative and the total differential, the means of gradient.

2.

ABILITIES AND SKILLS – to learn to find the derivatives of elementary functions; – to learn to find derivatives of higher orders; – to apply the derivative for testing intervals of increase and decrease of function, points of extrema, and to graph a function; – to be able to find the differential of elementary functions; – to be able to find the partial derivatives and the total differential of functions of many variables; – calculation of the gradient of functions.

3.

EQUIPMENT – tables; – textbooks; – task for the test control.

4.

CONTROL QUESTIONS FOR THE VERIFICATION OF THE INITIAL LEVEL OF KNOWLEDGE 1. Formulate the notion of the limit of function.

2. Define the derivative to function.

PRACTICAL WORK 1.1.

53

3. Give the physical interpretation of the derivative.

4. Give the geometrical interpretation of the derivative.

5. Formulate the basic rules of differentiation.

6. Which functions are named the composite functions? Give the examples.

7. How are the composite functions differentiated?

8. Give the determination of the derivative of the second and higher orders.

9. What is called the differential of function?

10. Formulate the basic rules of differentiation.

11. Give examples (desirably from biology and medicine) of functions of two and greater number of independent variables.

12. Give determination of function of two variables.

13. Write down the increase of function z = x + y2, which corresponds to the increase of argument: а) x; b) y.

14. Give the definition of partial derivative.

15. What is called the total differential?

5.

LITERATURE 1. Чалий О. В. Вища математика : навчальний посібник для студентів вищих медичних та фармацевтичних навчальних закладів / Чалий О. В., Стучинська Н. В., Меленєвська А. В. – К. : Техніка, 2001. – 204 с.

2. Медична і біологічна фізика : підручник для студентів вищих навчальних закладів освіти ІІІ–ІV рівнів акредитації / За ред. О. В. Чалого. – К. : Книга плюс, 2005. – 760 с.

3. Свердан П. Л. Вища математика: Аналіз інформації у фармації та медицині / Свердан П. Л. – Львів : Світ, 1998. – 332 c.

4. Hoffmann L. D. Calculus for business, economics, and the social and life sciences / Hoffmann L. D., Bradley G. L. – 5th. ed. – New York ; St. Louis ; San Francisco : McGraw-Hill Inc., 1992. – 778 p.

5. Calculus. A Committee of Lebanese authors.– 3th. ed. – Beirut, Librairie du Liban, 1997. – 291 p.

MATERIALS OF THE METHODICAL PROVIDING OF BASIC STAGE OF THE LESSON

Differentiation rules General notes. In the following, assume f and g are differentiable. Each rule should be viewed as saying that the function to be differentiated is differentiable on its domain and that the derivative is as given. For each, there is also a functional

PRACTICAL SECTION 1

6.

54

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

form, e.g. (cf)’= cf’, and a Leibniz form, e.g.

MODULUS 1

d du  cu   c . dx dx

d  f  x   g  x    f   x   g   x  dx 

Sum

Scalar multiple

d cf  x    cf   x  dx 

Product

d  f  x  g  x  f   x  g  x   f  x  g   x  dx 

Quotient

d  f  x  f ' x g  x  f  x g ' x   2 dx  g  x   g  x

The Chain Rule (for compositions) d f  g  x   f   g  x  g   x  dx

PRACTICAL SECTION 1

This says that a small change in input to the composition is scaled by g’(x), then by f’(g(x)). In Leibniz notation, if z = f(y) and y = g(x), and we thereby view z dz dz dy dz , a function of x, then  being evaluated at y = g(x). dx dy dx dy Inverse functions. If f is the inverse of a function g (and g' is continuous and nonzero), then 1 f  x  g   f  x  .

To get a specific formula directly, start with y = f (x) rewrite it g(y) = x; dif1 and put g'(y) in ferentiate with respect to x to get g’(y)y’ = 1; write this y   g y terms of x, using the relations y = f(x) and g(y) = x.

E.g., y = ln x; e y = x; e y y' = 1; y' = 1/e y = 1/x.

PRACTICAL WORK 1.1.

55

Derivative formulas d C0 dx

Constants. For any constant C, Reciprocal function

d 1 1   2 dx x x

d  1  1 f  x    2 dx  f  x   f  x d 1 x dx 2 x

The chain rule gives Square root

Powers. For any real value of n, The chain rule gives

d n x  nx n 1 , valid where x n 1 is defined. dx

n n 1 d  f  x    n  f  x   f   x  . dx 

Exponentials. An exponential function has the derivative proportional to itself, the proportionality factor being the natural logarithm of the base:

The chain rule gives Logarithms

d f  x  e f  x f   x  e dx d 1 ln x  , dx x

d 1 log a x  dx  ln a  x

Trig. functions sin  x  cos x

cos x   sin x

tan  x  sec x

cot  x   csc 2 x

2

PRACTICAL SECTION 1

Some rules hold without absolute value, but the domain is restricted to (0, ∞).

The chain rule gives f  x d ln f  x   . dx f  x

56

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

sec x  sec x tan x arcsin x 

1 1 x

2

MODULUS 1

csc x   csc x cot x   arccos x

arctan  x 

1  arc cot  x 1  x2

HOW TO USE DERIVATIVES TO GRAPH A FUNCTION Step 1. Compute the first derivative f'(x), factor it if possible, find the firstorder critical points, and plot them on the graph.

Step 2. Compute the second derivative f''(x), factor it if possible, find the second-order critical points, and plot them on the graph.

Step 3. Use the x coordinates of the first- and second-order critical points (and of any discontinuities) to divide the x axis into a collection of intervals. Check the signs of the first and second derivatives on each of these intervals.

Step 4. Draw the graph on each interval according to the following table: Sign of f'

Sign of f''

+ + – –

+ – + –

Increase or decrease of f Increasing Increasing Decreasing Decreasing

Concavity of f

Shape of f

C-up C-down C-up C-down

PRACTICAL SECTION 1

THE SECOND DERIVATIVE TEST The second derivative can be used to classify the first-order critical points of a function as relative maxima or relative minima. Here is a statement of the procedure, which is known as the second derivative test.

Suppose f' (a) = 0.

If f'' (a) > 0, then f has a relative minimum at x = a.

If f'' (a) < 0 then f has a relative maximum at x = a.

However, if f'' (a) = 0, the test is inconclusive and f may have a relative maximum, a relative minimum, or no relative extremum at all at x = a.

THE STEPS TO BE FOLLOWED FOR STUDYING THE VARIATION OF A FUNCTION 1. We determine the set of definition of the function, and we reduce if possible this set by considerations of symmetry.

2. We calculate the derivative and we study its sign which permits us to study the sense of variation of the function. We can also, by the study of the sign

PRACTICAL WORK 1.1.

57

of the second derivative, determine the sense of concavity of the graph of the function.

3. We study the values of the function at the bounds of the intervals of definition.

4. The obtained results are tabulated in what is called a table of variation.

5. We construct the graph of the function.

1. Differentiate the polynomial y = 5x3 – 4x2 + 12x – 8.

Solution Differentiate this sum term by term to get dy d   5x3   dxd  4 x 2   dxd 12 x   dxd  8  15x 2  8x1  12 x0  0  15x 2  8x  12 dx dx 2. Differentiate the function y = x2(3x + 1).

Solution According to the product rule, d 2 d d  x  3 x  1   x 2  3 x  1   3 x  1  x 2   x 2  3   3 x  1 2 x   9 x 2  2 x dx dx dx Which is precisely the result that was obtained when the product was multiplied out and differentiated as a sum. x 2  2 x  21 3. Differentiate the rational function y  x3 Solution According to the quotient rule, d d x  3  x 2  2 x  21   x 2  2 x  21  x  3 dy  dx dx  2 dx  x  3

 x  3 2 x  2    x 2  2 x  21 1  2  x  3

2 x 2  4 x  6  x 2  2 x  21 x 2  6 x  15  2 2  x  3  x  3

y   5u 4  u   5sin 4 1  x 3   u  The function u is also a composite function u  sin v , where v  1  x 3

PRACTICAL SECTION 1

4. Differentiate the function y sin 5 1  x 3  Solution It’s a composite function y  u 5 , where u sin 1  x 3 

58

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

MODULUS 1

u cos v  v cos 1  x 3   v So,

 ' sin 1  x  5

3





5sin 4 1  x 3   sin 1  x 3   ' 5sin 4 1  x3   cos 1  x3   1  x3 '

 15 x  sin 1  x   cos 1  x  .

2

4

3

3

5. Determine where the function f  x  x 4  8 x 3  18 x 2  8 is increasing, decreasing, concave upward, and concave downward. Find the relative extrema and inflection points and draw the graph.

Solution The first derivative is f   x   4 x 3  24 x 2  36 x  4 x  x 2  6 x  9   4 x  x  3

2

which is zero when x = 0 and x = –3. Since f  0   8 and f  3  19 , the firstorder critical points are (0, –8) and (–3, 19).

The second derivative (computed from the unfactored form of the first derivative) is f   x  12 x 2  48 x  36 12  x  3 x  1

PRACTICAL SECTION 1

which is 0 when x = –3 and x = –1. Since f  3  19 and f  1 , 3 the corresponding second-order critical points are (–3, 19) and (–1, 3). Plot these critical points and check the signs of f   x  and f   x  on each of the intervals defined by their x coordinates.

Interval

f  x

f   x 

Increase or decrease of f

Concavity of f

x  3

–

+

Decreasing

C-up

3  x  1

–

–

Decreasing

C-down

1  x  0

–

+

Decreasing

C-up

x0

+

+

Increasing

C-up

Shape of f

Draw the graph as shown on Fig. 1.1 by connecting the critical points with a curve of appropriate shape on each interval.

Note, that the first-order critical point (0, –8) is a relative minimum, while the first-order critical point (–3, 19) is neither a relative minimum nor a relative

PRACTICAL WORK 1.1.

59

maximum, and that both of the second-order critical points (–3, 19) and (– 1, 3) are inflection points.

Fig. 1.1. The graph of f  x  x 4  8 x 3  18 x 2  8 EXAMPLES OF FINDING DIFFERENTIALS AND PARTIAL DERIVATIVES 1. Find the differential of function y  ln x 2 .

2x 2 . Taking y  Solution. We will find the derivative of the function  2 x x 2dx advantage of formula dy  y dx , we get dy  . x x 2. Find the partial derivatives of function z  lntg   .  y Solution. Considering y as a constant, we will find the partial derivative of the function z  f  x, y  with respect to the variable x

PRACTICAL SECTION 1

The same way we take the partial derivative of the function z  f  x, y  with respect to the variable y, considering x as the constant

60

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

MODULUS 1

3. Find the total differential of function of three independent variables  u x3 y 2 z  2 x  3 y  z  5 .

Solution. Considering as the constants y and x, we find the partial derivative of the function u with respect to the variable x and a similar way – the partial derivatives of u-function with respect to the variables y and z  u x 3 x 2 y 2 z  2  ; u y 2 x 3 yz  3 ;  u z x 3 y 2  1 .

We find the total differential of function du = (3x2y2z + 2)dx + (2xyz – 3)dy + (x3y2 + 1)dz.

PRACTICAL SECTION 1

7.

TASKS FOR THE VERIFICATION OF THE EVENTUAL LEVEL OF KNOWLEDGE Find the first derivatives of the next functions: 3 x sin x  cos 2 x  2 x  3 9. y  1. y  5 x 3

2.

y  e3  x

3.

y

4.

y  sin x 

5.

y  ln x

6.

y

7.

y 3  tg x  lg x  3 4 x 2  1

8.

y  5x

10. y  3  tg x  lg x  3 4 x 2  1

x2  2 ctgx

11. y  5 x 2x 3

x

 e2 x 

1 4

tgx 3x

2

12. y 

2

1

sin x x

13. y  sin 2 x .

14. y  x 2 3x 1

1

Confront functions with their differentials а) x 1) 8dx b) ex 2) 3х2dx с) x + 10 3) exdx

PRACTICAL WORK 1.1.

d) x3 e) 8х

61

4) dx 5) 8x7dx

Confront differentials with their functions a) x3dx 1) x2 2x b) 2e dx 2) sin x c) sin x dx 3) ln |cos x| d) 2x dx 4) x4/4 e) –tgx dx 5) e2x Give the definition of gradient.

What physical maintenance does a gradient have?

Give the definition of the total differential.

Equation of the plane wave is X (t,s) = A sin(ωt –

8.

2π s 2 x 2 x ). Find 2 and 2 . λ t s

TASK FOR INDEPENDENT WORK Find the derivatives of the next functions.

1.

y  x5  4 x3  2 x  3 .

2.

y

1 x   x 2  0,5 x 4 .

4 3

3.

y  ax 2  bx  c .

4.

y

x2 . x

5.

y

5x 3 . a

6.

y

7.

y

π  ln x . x

8.

y

9.

y

2x  3 . x  5x  5

10. y  3 3 x 2  2 x 5 

2

13.  y

2 1  .

2x 1 x

15.  y 5sin x  3cos x .

12.  y 14. y 

a 2  b2

.

a  bx . c  bx

x ( x  1) .

1 z 1 z

.

16.  y tg x  ctg x .

1 . x3

PRACTICAL SECTION 1

11. y  x 2 3 x 2 .

ax 6  b

62

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

17. y 

sin x  cos x . sin x  cos x

20. y  x ctg x .

21. y  x arcsin x .

22. y (1  x 2 ) arctg x .

23. y  x 7 e x .

24.  y ( x  1) e x .

ex . x2

26. y 

x5 . ex 2

27. f ( x)  e x cos x .

28.  y ( x  1) e x .

29. y  e x arcsin x .

30. y 

 y x 3 ln x  31.

x3 .

3

x2 . ln x

1 ln x 32. y  2 ln x  . x x

33. y ln x lg x  ln a log a x . 

34. y (3  2 x 2 ) 4 .

35.  y

36.  y

1  x2 .

2

PRACTICAL SECTION 1

18.  y 2t sin t  (t 2  2) cos t .

19.  y arctg x  arcctg x .

25. y 

2

2

3

a  bx 3 .

37.  y (a 3  x 3 ) 3 .

38. y (3  2sin x)5 .

1 1 tg x  tg 3 x  tg 5 x .

39. y  3 5

40.  y

41.  y 2 x  5cos3 x .

42. y  esin x .

43. f ( x) (1  3cos 2 x) 2 .

 44. y

1 1 cos3 x  .

3 cos x

3sin x  2 cos x .

46.  y

1  arcsin x .

arctg x  arcsin 3 x .

48. y 

45. y  47. y  y 49.

2

ex  x .

MODULUS 1

ctg x  ctg α .

2

1 . arctg x

50. y  ( x 2  2 x  2) e x .

PRACTICAL WORK 1.1.

63

x 51. y  sin 3 x  cos  tg x .

5

52.  y sin( x 2  5 x  1)  tg

53. f ( x) cos(α x  β ) .

54. f (t ) sin ω t  sin(ω t  ϕ ) .

55. y 

1  cos 2 x .

1  cos 2 x

56. f ( x)  arctg

57. y cos 5 x 2  4 cos x 2 .  59. y  arcsin

1 . x2

a . x

x . a

58. y  arcsin 2 x .

60. y  ln 2 x 2 .

61. f (t )  t sin 2t .

62. f ( x)  arccos x 3 .

63. y  arccose x .

64. y  arctg

65.  y ln(2 x  7) .

66. y  arcctg

1 . x 1 x .

1 x

Find derivatives of the the second order of functions.

2

67. y  x8  7 x 6  5 x  4 .

68. y  e x .

69. y  sin x 2 .

70. y ln 3 1  x 2 .





71. y  ln x  a 2  x 2 .

72. f ( x) (1  x 2 ) arctg x .

73. y  arcsin 2 x .

74. y  a ln

x . a

1.

y  2  x  x2 .

3.

y

ex . x

2. y  2 x 3  3 x 2  12 x  5 .

4. y 

1 σ 2π

 ( x  a )2

e

2σ 2

.

PRACTICAL SECTION 1

Investigate the extremum of functions.

64

MATHEMATICAL PROCESSING OF MEDICAL AND BIOLOGICAL DATA

5.

8 x  y ( x  2) 2 . x

6. y 

16  4  x 2  x

MODULUS 1

.

Investigate the functions and draw the graphs: 4 x

1.

y x 

2.

y  x2  4x  5

3.

y  x3  3x  4

Find the differentials of functions.

1.

y ln(e x  5sin x  4 arcsin x) .

2.  y

3.

y  ln(arcsin 5 x) .

4.

 y ln sin x  x 3 .

5.

7.

y

x3 . sin 2 3 x

PRACTICAL SECTION 1

 9. y arctg(ln x)  ln(arctg x) .

ln( x  1)  ln x  1 .

y

x3 . x 1 2

6.  y sin 2 3 x  ln x 2 .

8.

y

cos x .

1  sin 2 x

10. y  sin 3 5 x cos 2

11. y  arcsin(ln x) .

12. y  ln 2 cos x .

x x 13. y  e 2 cos .

2

14. y  esin x sin x .

15. y  cose x .

16. y  ln 3 x 2 .

Find partial derivatives of the next functions.

1.

z  x 3  y 3  3 xy .

3.  z

x2  y 2 .

2. z 

y . x

4. z  x y .

x .

3

PRACTICAL WORK 1.1.

65

5. z =y ln x. sin

y x

.

8. z  arctg

9. u = (xy)z.

10. u = zxy.

7.

ze

6. u= x tg y.

11. z 

x x  y2 2

13. u  sin

.

y . x

12. u = ln(x2y).

x . y2 z

Find the total differentials of functions.

1. u  ln( xy ) .

2.

z = x3 + y3 – 3xy.

3. z = x2y3.

4.

u  cos

5. u sin( x  y ) .

6.

z

7. z = sin2 y + cos2 2x.

8.

z = yxx.

9. z= arcsin 11. z 

x . y2

x . y

10. u 

x2  y 2 . x2  y 2

x2  y 2  z 2 .

12. u = z · x · y.

13. z = ln (x2+y2).

14. u= arctg

15. z = ln cos3(x2y3).

16. u 

x2  y 2 .

2. grad z in the point (1; 1), if z  ln x 2 y .

xy . z2

5x  3 y .

9x  2 y PRACTICAL SECTION 1

Find gradients of the functions.

1. grad z in the point (5; 3), if z

x . y