1. M. Gewert, Z Skoczylas: "Analiza Matematyczna cz. II"

2. R. Wrede, M. R. Spiegel: "Advanced Calculus" (Schaums Outlines)

3. J. Stewart, “Calculus, Concepts and Contexts”

4. W. Krysicki i L. Włodarski: "Analiza matematyczna w zadaniach. cz.II"

PROBLEMS

1. Sketch the following sets:

2. Give the domain of definition of the following function and then sketch this domain:

3. Sketch the following surfaces (use traces and level curves where necessary) and find level curves passing through (x,y) = (0.0), (1,0), (2,2) where possible.

4. Find the following limits (split the limits, use the substitution method or squeeze principle where necessary):

5. Show that the following limits do not exist

6. Show that the following limits do not exist:

7. Determine whether the following functions are continuous:

8. Can you define the following functions f(x,y) at (0,0) so that they are continuous at (0,0)?

9. Find the values of a, b for which the following functions are continuous?:

10. Use the definition to find the first partial derivatives of the following functions :

11. Calculate all the partial derivatives of the third order.

12. Calculate all the second order partial derivatives

13. Show that the following functions satisfy the corresponding equations

14. Find the equations of the tangent plane and normal line to the following surfaces

15. Find all the extrema in the natural domain of :

16. Determine the largest and smallest value of the following functions for points in the defined regions.

17. The combined length and girth (distance around) of a package sent through the mail cannot exceed 108 cm. If the package is a rectangular box, how large can its volume be?

18. Find the shortest distance from the point (1,2,3) to the plane x + 3y + 5z - 6 = 0.

19. (lecture) Show that:
for

20. Find the differential `df' of the following functions:

21. Calculate the approximation `df ` for the following data

22. Use the differential df to find the approximate value of
at point P(0.05, 0.98).

23. Find the approximate values of

24. Find the second degree Taylor Polynomials for f(x) at the given value of (a,b)

26. Calculate the following integrals (iterated integrals):

27. Sketch the region D. Find the volume below the surface given by the integrand and above D i.e. calculate the double integrals:

28. The region D is given by its boundaries, sketch the region D. Calculate the double integrals, rewriting them as iterated over a normal region.

29. Reverse the order of integration:

30. Change the variables to calculate:

31. Find the area of the regions bounded by the following curves :

a) x = 2, y = x, y = 1/x

b) x + y = 1, x + 3y = 1, x = y, x = 2y

c) y = lnx, x = 2, y = 0

d) y = 2x - x², y = x²

e) ρ = cosθ, ρ = 2cosθ

32. Calculate the following triple integrals

33. Find the volume of the regions bounded by the following surfaces:

a) z = 4 - x - y, z = 0, y = x², y = 1 b) x² + y² = 1, z = 0, z = y, x ≥ 0

c) x = 0, y = x, y = 2, z = y˛ - x˛ d) x˛ + y˛ = 4, x + y + z = 4, z = 0

e) y = x˛, z = x˛ + y˛, y = 1, z = 0 f) z = 1 + x + y, x = 0, y = 0 z = 0, x + y = 1

34. Rewrite the integral
as an equivalent iterated integral in the order dzdydx, if the region E is

a) the solid tetrahedron bounded by the planes x = 0, y = 0, z = 0, and

x +y + z = 1

b) the region bounded by the paraboloid z = x² + y², and the plane z = 0.

c) the region bounded by the ellipsoid 9x² + 4y² + z² = 1

d) the region bounded by the planes y = 0, y = 6, and the cylinder x² + z² = 4,

e) the region bounded by the planes x = 0, z = 0 y = 0, y = 1-x, and the parabolic cylinder

z = 1 - x²

35. Write the above integrals as an interated integral in the of the five other orders: dzdxdy, dydxdz, dydzdx, dxdydz, dxdzdy.

36. Sketch the solid whose volume is given by the iterated integral

37. Calculate the following improper integrals

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