Practical training 3.1
The Derivative. Technique of differentiation
. . . . . . . . . .
Home tasks 3.1
Task 1. Find 
in term of 
, if 
, 
.
The answer: 
.
Task 2. Calculate , if 
, 
, 
.
The answer: 
.
Task 3. Using the definition of the derivative, find 
, if at the point 
. The answer:1/12.
Find the derivatives of the functions given in tasks 4 7.
Task 4. ) 
b) 
c) 
Task 5. ) 
b) (
c) 
find 
;
d) 
; e) 
.
The answer: c) 
; 
; d) 
.
Task 6. ) 
; b) 
; c) 
; d) 
.
The answer: d) 
.
Task 7. ) 
b) 
c) 
d) 
; e) 
.
The answer: b) 
; d) 9 
;
e) 
Task 8. Taking the logarithms find the derivatives of the functions:
) 
; b) 
.
The answer: ) 
b) 
Practical training 3.2
The differential of a function
. . ? ? . . .
Home tasks 3.2
Task 1. Given the function 
. Calculate the increment and its linear part when 
changes from 
to 
.
The answer: 
.
Task 2. Calculate the increment and the differential of the function 
when 
and 
. Calculate the absolute and relative errors as the increment is replaced by the differential.
The answer: 
absolute error 0,04;
relative error 
.
Task 3. Find the differentials of the functions:
) 
; b) 
; b) 5 
; c) 
.
Task 4. Find the differentials of the functions: ) 
; b) 
.
The answer: ) 
; b) 
.
Task 5. Find the differentials of the functions: ) 
; b) 
.
Task 6. Calculate the approximate value of the function 
as changes from 
to 
. What is the value of 
?
The answer: 
.
Task 7. Calculate approximately: ) 
; b) 
.
The answer: ) 
; b) 
.
8. Represent the differential of the following functionsin terms ofindependent variable and its differential:
) 
, 
; ) 
, 
.
Answer: ) 
; b) 
.
Practical training 3.3
The Derivatives and differentials of the higher orders.
Ttlors formula
. . - . . 
. 
. 
.
Home tasks 3.3
Task 1. Find the derivatives of the second order for the functions:
) 
b) 
The answer: ) 
; b) 
.
Task 2. Find 
, if 
The answer: 
.
Task 3. Find the differential of the second order for the function
The answer: 
;
Task 4. Find 
if 
. The answer: 
.
Task 5. Make the expansion of the polynomial 
in terms of the powers of the bynomial 
.
|
|
|
The answer: 
Task 6. If 
is a polynomial of the fourth degree. Calculate 
, if it is known 
.
The answer: 
Task 7. Calculate to within 
: a) 
; b) 
; c) 
.
The answer: a) 
; b) 
; c) 
.
Task 8. Using Maclaurins expansion calculate the limit 
.
The answer: 
.
Practical training 3.4
investigation of functions and construction of its graphs
( ) . () () . (). (). () . () . . . . . .
Home tasks 3.4
Task 1. Show that the function 
is decreasing everywhere.
Task 2. Find the extrema and intervals of monotonisity:
) 
; ) 
; ) 
.
The answer: ) 
as 
as 
- decreases; 
- increases;
) 
- decreases; 
- increases;
) 
- increases, 
- decreases.
Task 3. Prove the inequality: 
.
Task 4. Find the greatest and the least values of the function: 
on
. The answer: 
and 
.
Task 5. Represent the number 
as two summands so that the sum of their cubs will be the least. The answer: 
and .
Task 6. What should be the height of the cone inscribed in the sphere or the radius 
if the cones lateral is the greatest? The answer: 
.
Task 7. Show that the graph of the function 
is convex everywhere.
Task 8. Find the points of inflection and the interval of convexity of the graph of the function 
.
The answer: 
the point of inflection; 
the interval of concavity; 
the interval of convexity.
Task 9. Find the asymptotes of the functions graphs: 1) 
;
2) 
. Answer: 1) 
; 2) 
.
Task 10. Investigate the functions and construct their draphs:
) 
; ) 
; ) 
.
3.5
