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3 group task in calculus:

Edited at 1pm 26.11.2016.

1. Calculate the derivatives.

a. [pic]

b. [pic]

c. [pic]

2. Prove that (cf)’ = cf’

3. Prove that (fg)’ = f’g + fg’

4. Prove that (f/g)’=(f’g - fg’)/g2

5. Prove the Chain Rule.

6. Calculate the average value of each of these functions on the interval [0, 1]

a. x

b. x2

c. x3

d. x4

7. Compute the center of mass of a rod for each of these density functions from 0 to 1.

a. x

b. x2

c. x3

d. x4

8. Calculate the moment of inertia around the origin of rod with each of these density functions at [0, 1].

a. x

b. x2

c. x3

d. x4



















9. Calculate the curve length of f(x) = -0.006x2 + 0.3x @ [0, 11].

10. Calculate the curve length of f(x) = 1 + cos(x) @ [0, 1].









11. Calculate the revolutionary surface area of f(x) = 1 + cos(x) @ [0, 1].

12. Calculate the revolutionary volume of f(x) = 1 + cos(x) @ [0, 1].







13. Solve these inequalities.

a. x < 1

b. –x > 8

c. |x - 9| < -2

d. |4 - x| < 4



14. [pic] = . . .

15. [pic] = . . .

16. [pic]

17. Graph these functions.

a. [pic]

b. [pic]

c. x-1

d. x-2

e. |x|

18. Define an Indeterminate form.

19. Draw these graphs in polar coordinates (angle A and radius R).

a. R = A. b. R = sin 4A. c. R = 1 + sin A.

20. Find the dot-product and the cross-product of the vectors.

a. (1, 3) . (5, 9) = . . .

b. (2, 3) × (4, 11) = . . .

21. Prove that the lines y = sx + i and y = gx + I are perpendicular if sg = -1.

22. Find curvature (K) of each of these curves.

a. x2 + y2 = 4 b. x2 = y c. xy = 1

[pic]

d. Link the curvature radius (R) with the curvature (K).

23. R is the radius-vector on a circumference. Calculate the dot-products and the cross-product.

a. R.R' = . . . b. R'.R'' = . . . c. R×R'' = . . .

24. Perform the linear least squares fitting of these points (0, 0), (1, 0) and (0, 1). Use vertical offsets and the fitting line in the form y(x) = gx + i.

Check if for any 3 points (x1,y1), (x2,y2), (x3,y3), which are not on the same straight line,

[pic]

[pic].

Write the expressions for any number of points (n).

25. Work out.

a. div curl = . . . b. curl grad = . . . c. div grad = . . .

[pic], curl V = [pic], div V =[pic]. [pic], grad S = [pic]

26. Name and sketch the shapes described by these equations.

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

j. [pic] k. [pic] L. [pic]

m. [pic] n. [pic] o. –x2 – y2 – z2 = -37

p. –x2 – 7y2 + 2z2 = -65 q. x2 – y2 – z2 = 29 r.– y2 – z2 = - 2

27. Write the general equations of a linear surface and a quadratic surface.

28. Write the equation of a line in 3D space.

29. If your homework score is reduced 10% for each day of the delay, in how many days will your score be halved?

30. Fill in the blanks.

a. [pic]= . . .; b. [pic]= . . .; c. [pic] - [pic] = . . .; d. [pic] - [pic] = . . .

31. Write and explain the Euler’s formula.

32. Solve the Linear Programming problems.

33. Explain quantum computing, cryptography, entanglement, superposition and analogue computing.

34. How are Dirac delta-function and Heaviside function used in calculus?

35. Calculate Riemann sum for integral

[pic]

for 9 intervals.

36. Prove cone volume formula using revolutionary volumes.

We did it in our class on 22.11.2016.

37. Prepare to Dota2 Gaming Competition. Try to win millions US$.



(Dota_2)

Your digital signature:

kgroup =floor(8090+8089)/2),1)), take your kgroup as floor of the average of your k.

Calculate your secret random number between 1 and kgroup as your digital signature.

d =RANDBETWEEN(1, kgroup)



Do NOT share your d with anybody, including me.

My mode is 421, give your 3d mod 997 = . . . . . . . . . here d is your secret digital signature.

421d mod 997 = secret key.



Create new email account with password DiGsIg777, here instead of 777 must be the secret key.

Write the secret key only in the password to the new email, do not share the secret key we anybody, including me.

Deadline: 30.11.2016.

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