Quiz

Solve this quiz and write your answers in comments.

Exercise 1.

Calculate:

$$\begin{gathered} a. \frac{5-1i}{3+i}; \\ b. (7+2i)-(3-2i)+(11+i)-(17-3i); \\ c. (2+i)*(5-2i)+(6+13i)*(7-i); \\ d. \frac{-13+6i}{2–i}+\; (\frac{8–i}{5+2i}{)}^2 \end{gathered}$$

Exercise 2. 

Calculate:

$\mathrm{Re}\left(\frac{12–i}{6+3i}\right), \mathrm{Im}\left(\frac{-2i}{5+i}\right)$

Exercise 3.

Find all a values for the equality: $-5+i+(\mathrm{2+}a{)}^2$ where a is an imaginary number.

Exercise 4.

Resolve the system of equations:

$$\begin{gathered} a. \left\{\begin{array}{l}{z}_1+i*z_2=5-i\\ 3+i*z_1=z_2-3\end{array}\right.; \\ b. \left\{\begin{array}{l}(2–i)z_1+7z_2=3-15i\\ (8-2i)z_1+(4+i)z_2=1+2i\end{array}\right. \end{gathered}$$

Exercise 5.

Find all the complex numbers z for the following statements:

$\frac{z-11i}{4z+i}=\frac{5}{2}\mathrm{and} \frac{z-2}{z+3i}=\frac{8}{3}$

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:

$$\begin{gathered} a.\mathrm{y=} \sqrt{x+\sqrt{x+\sqrt{x}}}; \\ \mathrm{b.} \mathrm{y=} {\sin }^2(\mathrm{cosx})+{\cos }^2(\mathrm{sinx}); \\ c. \mathrm{y=}{e}^{x^2}*\cos 2x; \\ \mathrm{d.} \mathrm{y=}{x}^{\mathrm{sinx}}; \end{gathered}$$

Exercise 11.

Find derivatives and differentials for the functions below:

$$\begin{gathered} a. y=e^{e^x}+x^{e^x}; \\ b. \mathrm{y=ln}(e^x+\sqrt{1+e^{2}x}); \\ c. \mathrm{y=} \sin x^{\mathrm{cosx}}; \\ d. y={\ln }^3({\ln }^2(\mathrm{lnx})); \\ e. \mathrm{y=}{\tan }^{-1}(x+\sqrt{1+x^2}) \end{gathered}$$

Exercise 12.

Find differentials for the following functions:

$$\begin{gathered} a. y=\ln (x^2+x); \\ b. {\mathrm{y=x}}^2*\sin 2x; \\ c. \mathrm{y=x}*e^{5}x; \\ d. y=\frac{(x-1)}{(x+1)} \end{gathered}$$

Exercise 13.

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

$$\begin{gathered} a. f(x)={\sin }^{-1})x; \\ b. \mathrm{f(x})={\tan }^{-1})x; \\ c. \mathrm{f(x})=\mathrm{lnx}. \end{gathered}$$

Exercise 14.

Calculate the integrals:

$$\begin{gathered} a. \int (x^3+1)*x^2\mathrm{dx}; \\ b.\int \frac{\mathrm{dx}}{\sqrt{2-5x}}); \\ c.\int \frac{x^2\mathrm{dx}}{1+x^2 }; \\ d. \int \frac{\mathrm{dx}}{\sqrt{3+8x^2}} \end{gathered}$$

Exercise 15.

Calculate the integrals:

$$\begin{gathered} a.\int {\sin }^3\mathrm{xdx}; \\ b.\int (x+1)*\cos 2x\mathrm{dx}; \\ c. \int {\mathrm{xe}}^{–x} \mathrm{dx}; \\ d.\int x^4{e}^{x^2} \mathrm{dx} \end{gathered}$$

Exercise 16.

Calculate the integrals:

$$\begin{gathered} a.\int \frac{e^x}{\sqrt{1+e^x}}\mathrm{dx}; \\ \mathrm{b.} \int \frac{\mathrm{xdx}}{(x+1)(x+2)(x+3)}; \\ c. \int e^x\cos x\mathrm{dx}; \\ \mathrm{d.} \int \sqrt{x}\mathrm{lnxdx}; \end{gathered}$$

Exercise 17.

Calculate the integrals:

$$\begin{gathered} a. \int \sqrt{\frac{x+a}{x–a}}\mathrm{dx}; \\ b.\int \ln \sqrt{x^2-1}\pm x) \mathrm{dx}; \\ c.\int \frac{x^2\mathrm{dx}}{x^2+x-2}; \\ d.\int \frac{\mathrm{dx}}{(2+\mathrm{cosx})\mathrm{sinx}} \end{gathered}$$

Exercise 18.

Calculate the integrals:

$$\begin{gathered} a.{{\int }_0}^1 \frac{\mathrm{dx}}{3+x^2}; \\ b.{{\int }_0}^{\frac{1}{2}} \frac{\mathrm{dx}}{(x^2+x+1)*(x-1)}; \\ c.{\int }_{0^1} x(1-x{)}^{10}\mathrm{dx}; \\ d.{{\int }^0}_{-3} \frac{\mathrm{dx}}{x^3-6} \end{gathered}$$

Exercise 19.

Calculate the integrals:

$$\begin{gathered} a. {{\int }_1}^e \ln x \mathrm{dx}; \\ b.{{\int }_0}^{\pi /2} {\sin }^5 x \mathrm{dx}; \\ c.{{\int }_0}^{\pi } {\sin }^4{x}^2\mathrm{dx}; \\ d.{{\int }_0}^e \frac{\mathrm{sinln}x}{x^2+1}\mathrm{dx} \end{gathered}$$

Exercise 20.

Calculate the derivatives for functions below:

$$\begin{gathered} a. \frac{d}{\mathrm{dx}}{{\int }_0}^x \sin t^2\mathrm{dt}; \\ b. \frac{d}{\mathrm{dx}}{{\int }_{x_2}}^{x^3} \frac{\mathrm{dt}}{\sqrt{1+t^2 }}; \\ c. \frac{d}{\mathrm{dx}}\int {\sin }^{-1}\sqrt{t}\mathrm{dt}; \\ d. \frac{d}{\mathrm{dx}}{{\int }_0}^{x^2}\sqrt{1+t^2} \mathrm{dt} \end{gathered}$$