REVIEW 2

1. Find all the asymptotes of the following functions

ANS:

1. Find the derivatives of the following functions:

a)
, b)
c)

d)
e)
f)

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

a)
b)
c)
d) y = ln(x)^ln(x)

e)
f)

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

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

5. Find
for

a)
b)
c)

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

a)
b)
c)

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

(0/0)
:
,
,
,
,
,

,

(
:
,
,
,

:
,
,
,

:
,
,
,
,

,
,

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

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

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

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

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

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

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

1. 
   b)
   

13. Find the local extrema of the functions:

a)
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)
b)

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)
b)

a)
b)
Dół formularza

---