Exercise 1.


a. 5-1i3+i;b. (7+2i)-(3-2i)+(11+i)-(17-3i);c. (2+i)*(5-2i)+(6+13i)*(7-i);d. -13+6i2i+ (8i5+2i)2


Exercise 2. 


Re(12i6+3i), Im(-2i5+i)


Exercise 3.

Find all a values for the equality: -5+i+(2+a)2 where a is an imaginary number.


Exercise 4.

Resolve the system of equations:

a. z1+i*z2=5-i3+i*z1=z2-3;b. (2i)z1+7z2=3-15i(8-2i)z1+(4+i)z2=1+2i


Exercise 5.

Find all the complex numbers z for the following statements:

z-11i4z+i=52and z-2z+3i=83


Exercise 6.

Show the unity of points z on the complex plane what correspond to an inequity |z-2i+1|≤5.


Exercise 7.

Represent the complex number z=√3 i-1 with its trigonometric form.


Exercise 8.

Find the following derivatives using sum, residual, composition formula for the functions:

  • tan ⁡x;
  • cot ⁡x;
  • sinh ⁡x;
  • cosh ⁡x.


Exercise 9.

Find derivative of function in the point x.

a. y=√x, x=4
b. y=x*|x|, x=0


Exercise 10.

Find derivatives and differentials for the functions below:

a.y= x+x+x;b. y= sin2(cosx)+cos2(sinx);c. y=ex2*cos2x; d. y=xsinx;


Exercise 11.

Find derivatives and differentials for the functions below:

a. y=eex+xex;b. y=ln(ex+1+e2x);c. y= sin xcosx;d. y=ln3(ln2(lnx));e. y=tan-1(x+1+x2)


Exercise 12.

Find differentials for the following functions:

a. y=ln (x2+x);b. y=x2*sin2x;c. y=x*e5x;d. y=(x-1)(x+1)


Exercise 13.

Find differentials of the functions using theorem of inverse function derivative:

a. f(x)=sin-1)x;b. f(x)=tan-1)x;c. f(x)=lnx.


Exercise 14.

Calculate the integrals:

a. (x3+1)*x2dx; b.dx2-5x);c.x2dx1+x2 ;d. dx3+8x2


Exercise 15.

Calculate the integrals:

a.sin3xdx;b.(x+1)*cos 2xdx;c. xex dx;d.x4ex2 dx


Exercise 16.

Calculate the integrals:

a.ex1+exdx;b. xdx(x+1)(x+2)(x+3);c. excos xdx;d.  xlnxdx;


Exercise 17.

Calculate the integrals:

a. x+axadx;b.lnx2-1±x) dx;c.x2dxx2+x-2;d.dx(2+cosx)sinx


Exercise 18.

Calculate the integrals:

a.01 dx3+x2;b.012 dx(x2+x+1)*(x-1); c.01 x(1-x)10dx;d.0-3 dxx3-6


Exercise 19.

Calculate the integrals:

a. 1e ln x dx;b.0π/2 sin5 x dx;c.0π sin4x2dx;d.0e sinlnxx2+1dx


Exercise 20.

Calculate the derivatives for functions below:

a. ddx0x sin t2dt;b. ddxx2x3 dt1+t2 ;c. ddxsin-1tdt;d. ddx0x21+t2 dt


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