REVIEW 2


REVIEW 2

1. Find all the asymptotes of the following functions

0x01 graphic

0x01 graphic

0x01 graphic

ANS:

0x01 graphic

1. Find the derivatives of the following functions:

a) 0x01 graphic
, b) 0x01 graphic
c) 0x01 graphic

d) 0x01 graphic
e) 0x01 graphic
f) 0x01 graphic

2. Find the derivatives of [hint: use the equality 0x01 graphic
]

a) 0x01 graphic
b) 0x01 graphic
c) 0x01 graphic
d) y = ln(x)ln(x)

e) 0x01 graphic
f)0x01 graphic

3. Determine the equation of the tangent line to the curve 0x01 graphic
at point (1 , 3).

4* The function0x01 graphic
is equal to zero for 0x01 graphic
and 0x01 graphic
, nevertheless 0x01 graphic
for all 0x01 graphic
. Explain this seeming inconsistency with the Rolle's Theorem.

5. Find 0x01 graphic
for

a) 0x01 graphic
b) 0x01 graphic
c) 0x01 graphic

6. Let f(x) be three times differentiable. Find 0x01 graphic
for

a) 0x01 graphic
b) 0x01 graphic
c) 0x01 graphic

7. Apply the L'Hospital's Rule to find the following limits

(0/0) 0x01 graphic
: 0x01 graphic
, 0x01 graphic
, 0x01 graphic
, 0x01 graphic
, 0x01 graphic
,

0x01 graphic
, 0x01 graphic

(0x01 graphic
: 0x01 graphic
, 0x01 graphic
, 0x01 graphic
, 0x01 graphic

0x01 graphic
: 0x01 graphic
, 0x01 graphic
, 0x01 graphic
,

0x01 graphic
: 0x01 graphic
, 0x01 graphic
, 0x01 graphic
, 0x01 graphic
,

0x01 graphic
, 0x01 graphic
, 0x01 graphic

8. a) Use the Taylor's Polynomial for the function 0x01 graphic
at a = 0 to calculate 0x01 graphic
to an accuracy of 0,01 [hint: estimate the remainder Rn to find the degree n of the polynomial].

b) Use the Taylor polynomial of order 5 for the function 0x01 graphic
at a = 0 to calculate the approximate value of the number 0x01 graphic
and estimate the approximation error.

9. Find the Taylor polynomial of the 3-rd degree at x = 0 for the function 0x01 graphic
, next calculate an approximation of the number 0x01 graphic
. Write the expression for the remainder.

10. a) Find the Taylor polynomial (n-th degree) at x = 1 for the function 0x01 graphic

b) Write the Taylor polynomial for the function 0x01 graphic
at 0x01 graphic
with remainder.0x01 graphic

11*. Use the Newton method to calculate the roots of the equation 0x01 graphic
in the interval 0x01 graphic
.

12. Determine the intervals where the following functions are monotone:

  1. 0x01 graphic
    b) 0x01 graphic

13. Find the local extrema of the functions:

a) 0x01 graphic
b) f(x) = sinx - ln(sinx)

14. a) A wire 24 cm long is cut in two, and then one part is bent into the shape of a circle and the other into the shape of a square. How should it be cut if the sum of the areas of the circle and the square is to be a minimum?

b) Which is the more efficient container, a cube, or a circular cylinder ? ( Find the most efficient cylinder and then compare with the cube).

15. Find the inflection points of :

a) 0x01 graphic
b) 0x01 graphic

16. Find the intervals where the following functions are convex or concave

a) f (x) =x ln(cosx) b) f (x) = arcsin (1/x)

Dół formularza

17. Sketch the graphs of the functions a) 0x01 graphic
b) 0x01 graphic

a) 0x01 graphic
b) 0x01 graphic
Dół formularza



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