Download Self-training 1

Ministry of Health Care of Ukraine
O.O.Bohomolets National Medical University
EXERCISE BOOK 1
for independent work of students
Study discipline «Medical and biological Physics»
Specialization «Medical occupation», «Stomatology»
Department of Medical and biological Physics
Authors: A.V.Chalyi, N.V.Stuchynska, I.F.Margolych
Approved by methodical counsel of the Department of Medical and Biological
Physics on 31.08.2015., proceeding № 1
Considered and approved by CMC of physical and chemical disciplines on
1.09.2015., proceeding № 1
Topic 1. Elements of differential calculation
Education aims:
- to master notion of derivative and notion of derivative of higher orders;
- to study to apply these notions for testing intervals of increase and decrease of
function, points of extrema, and to graph a function;
- to master notion of the differential of function, notion of the partial derivative
and the total differential, the means of gradient.
Abilities and skills:
- to study to find the derivatives of elementary functions;
- to study 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.
Literature:
1.Medical and Biological Physics: Textbook for the students of higher medical
institutions of the IV accreditation level/ Chalyi A.V., Tsekhmister Ya.V., Agapov
B.T.[et al.]. Vinnitsja, Nova Knyha, 2010. - 480 pp.
2. Chalyi O.V., Stuchynska N.V., Melenevska A.V. Higher mathematics. Train
aid for the students of higher medical and pharmaceutical institutes. – K., Теhnica.2001.
3. Hoffmann L.D., Bradley G.L. Calculus for business, economics, and the
social and life sciences New York, St. Louis, San Francisco, McGraw-Hill Inc. –
5th. ed. , 1992.-778 pp.
2
4. Calculus. A Committee of Lebanese authors. Beirut, Librairie du Liban.3th.ed., 1997.-291 pp.
Control questions for theoretical study:
1. Formulate the notation of the limit of function.
2. Define the derivative to function.
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. Give the definition of partial derivative.
14. What is called the total differential?
Tasks for the independent work:
Task 1. Find the derivatives of the next functions:
1. y  lg(e  5 sin 3x  4 arccos x)
x
2. y 
ln( x 3  4)  ln 2 x  10
3. y  ln(arcsin 5 x)
2
4. y 
x3
x2  1
5. y  ln sin x  x
3
3
3
5
6. y  x sin 3 x  tg x
7. y 
8. y 
3
x
sin 2 3x
cos x
1  sin 2 x
x
4
9. y  arctg (ln )  ln(arctg 4 x)
2
x3
10. y  sin 6 x cos
3
2
2
11. y  arcsin (log 4 x)
2
2
12. y  ln ctg 2 x
x
2
13. y  5 arccos
x
cos  arcctgx 2
4
sin
14. y  e
3x
15. y  4
cose x
2
16.
x
x3
7
y  ln 6 x 4  7 arcsin8 x
Task 2. Find the differentials of functions:
1.
4 x
y  arcsin 2
x
2.
f ( x)  (3 sin 2 x  x 2 )arctg
3.
y  2 x 3  3x 4  5 x 2  4
4
x2
4
y  e x ctg(6 x 3  2 x)
2
4.
5. y  (2  4 x ) ln 3 1  x 2

2
6. y  ln x  10 x  x

2
x3
7.
y  arcsin 2 x  log 2
8.
y  arctg
9.
y  ln 5 4 x 2
10.
f ( x)  arccos x 3 
11.
y  ln 2 (e 3 x  15 sin 2 x  5 arcsin 4 x 4 )
12.
y  ln sin 4 x  x 3  tg5 x
13.
y  sin 4 8 x  ln 3 x 4
14.
y  ln( x 5  4)  ctg x  1
15.
y  arccos e x 
5  3x
 10 sin 5 x
5  3x
x
3
sin 3 x
3x
Task 3. Find partial derivatives of the next functions:
1. z  x  y  2 x y
6
5
2
3
x2  y
2. z  2
x y
3.
4.
2 y3
z
x
z  x2  y2
+
x2 y2
5
5.
z  xy  yx
6.
u= 2x ln y+ e
7.
z = y 3 ln x
8.
u= x tg 4 y
ze
9.
sin 2
y
2x
y
x
 cos 2 y  sin 3x
x2  y
x2  y2
10.
z3
11.
y2
z  arctg
 arcctg3 x
2x2
12.
u = (xy)z
3
13.
u = zxy + y z
x
14.
z
15.
u = log 3 (x3y) - tgy
16.
u  arcsin
x  y2
2
z
x

6
y2 z
Task 4. Find the total differentials of functions:
1. u  ln ( x  y )  5
4
2
3
xy
2. z =2 x3 + 3y3 – 3xy
3. z = x2y3 - arccos 2 2 x
4.
u  cos
x y
e
y
6
5.
u  sin 3 (2 x  4 y)  tg( xy)
6.
x2  y 2 x2  y 2
z 2

x  y 2 x2  y 2
7.
z = 4sin2y + cos 3 2 x
8.
z=arctg xy - yxy
9.
z= arcsin
5
x2
y4
10.
u  x 2  y 2  z 2 - log 5 ( x 2  y 2  z 2 )
11.
z
12.
u = z x y - z xy
13.
z = ln 2 (x2+y2)  7 xy
14.
u= arctg
15.
z = ln cos3(x4y3) – sin lg
16.
8x 4  3 y 2
u
.
5x  2 y 7
x
x2
4
y2 + y
xy
z
 z2 
1
xy
xy
7