Onderstaand overzicht volgt de nummering en de opgaven van de derde editie.
[antwoorden eerste editie | antwoorden tweede editie]
| a. $u_n = 2n+1$ |
| c. $u_n = \displaystyle \frac{2n-1}{n}$ |
| e. $u_n = n^2-1$ |
| g. $u_n = (-3)^{1-n}$ |
| b. $u_n = 2^{n-1}$ |
| d. $u_n = \displaystyle 7\frac{3^n}{4^n}$ |
| f. $u_n = \displaystyle \frac{n(n+1)}{2}$ |
| h. $u_n = \displaystyle \frac{(n-1)!}{10^{n-1}}$ |
Afkortingen: rk (rekenkundig), mk (meetkundig), div (divergent) en conv (convergent):
| a. rk ($v=2$), div | e. div |
| b. mk ($q=2$), div | f. div |
| c. conv ($u_n \to 2$) | g. mk ($q=-1/3$), conv ($u_n \to 0$) |
| d. mk ($q=3/4$), conv ($u_n \to 0$) | h. div |
| a. $u_{n+1} = 3u_n+1 \,,\; u_1=1$ | c. $u_{n+1} = -u_n/2 \,,\; u_1=8$ |
| b. $u_{n+1} = 2u_n+4 \,,\; u_1=3$ | d. $u_{n+1} = u_n+u_{n-1} \,,\; u_1=u_2=1$ |
| a. $r_7 = -5$ | c. $r_1=-5 \,,\; v = 2$ | e. $m_8 = -128$ |
| b. $r_9 = 16$ | d. $m_8 = 192$ | f. $q = -1/4$ |
| a. $-2/3$ | b. $0$ | c. $+\infty$ | d. $-2$ | e. $e^2$ | f. $e^3$ |
| a. $-1/6$ | e. $0$ | i. $0$ | m. $1/4$ | q. $0$ |
| b. $-1/4$ | f. $1$ | j. $0$ | n. $-1$ | r. $4/3$ |
| c. $+\infty$ | g. $+\infty$ | k. $-1$ | o. $\sqrt{2}/2$ | s. $-2$ |
| d. $4/9$ | h. $1/8$ | l. $0$ | p. $-\infty$ | t. $1/12$ |
| a. $\pi$ | b. $1$ | c. $0$ | d. $1$ | e. $3/7$ | f. $-4/3$ |
| a. overal continu | b. discontinu in $x=0$ | c. overal continu |
| a. $\displaystyle k = \frac{e^2-1}{3}$ | b. $\displaystyle k = \pm \sqrt{2}-1$ |
| a. overal afleidbaar, behalve in $x=0$ | b. overal afleidbaar |
| a. $\displaystyle a=-\frac{7}{108} \, , \; b = \frac{1}{12}$ |
| a. $\displaystyle 3x^2-8x$ |
| b. $\displaystyle 9x^8+\sec^2 x$ |
| c. $\displaystyle -1-2x^{-3}$ |
| d. $\displaystyle \tfrac{1}{3\sqrt[3]{x^2}}+\tfrac{3}{x^2}+4$ |
| e. $\displaystyle \cos x -x\sin x$ |
| f. $\displaystyle e^x+4x^{-5}$ |
| g. $\displaystyle 2.4^{2x} \ln 4+3.7^{-3x} \ln 7$ |
| h. $\displaystyle \frac{1}{x\ln 4}$ |
| i. $\displaystyle \frac{\sin x}{\cos^2 x}$ |
| j. $\displaystyle -\frac{2x}{\sqrt{1-x^4}}$ |
| k. $\displaystyle 2x\mbox{Bgsin} x^3 + \frac{3x^4}{\sqrt{1-x^6}}$ |
| l. $\displaystyle \frac{3x^2+16x-6}{(x^2+2)^2}$ |
| m. $\displaystyle \frac{2\cos x+\sin x}{\cos^2 x}e^{2x}$ |
| n. $\displaystyle 2x\cot x^2$ |
| o. $\displaystyle \frac{8x-3}{2\sqrt{4x^2-3x+1}}$ |
| p. $\displaystyle \frac{1-2\cos (2x)}{2\sqrt{x-\sin(2x)}}$ |
| q. $\displaystyle \frac{4x^3 \cos x^4-3\sin x^4}{e^{3x}}$ |
| r. $\displaystyle \sec x$ |
| s. $\displaystyle -3.5^{2x}\left( 2x\ln 5 \sin x+x\cos x+\sin x \right)$ |
| t. $\displaystyle -3^{\cos x}\ln 3 \sin x$ |
| u. $\displaystyle x^x (\ln x +1)$ |
| v. $\displaystyle \csc x$ |
| w. $\displaystyle \frac{x^2(2\cos(2x+7)-7x^6)}{3\left(\sin(2x+7)-x^7\right)^{2/3}}+2x\sqrt[3]{\sin(2x+7)-x^7}$ |
| x. $\displaystyle \frac{2e^{\mbox{Bgtan}\, x}}{(x^2+1)^{3/2}}$ |
| $\displaystyle f^{(10)}(x)=2^{10}e^{2x}-5\sin x$ |
| $\displaystyle e^{-2\sqrt{x}}x^{5/4}$ |
| $\displaystyle a = 0 \,,\; b = \frac{1}{3}$ |
| a. $\displaystyle y=x-1$ | b. $\displaystyle y=-\frac{7}{18}x+\frac{4}{9}$ | c. $\displaystyle y=\frac{\sqrt{2}}{4}x+\frac{\sqrt{2}}{2}$ |
| a. $\displaystyle \frac{3}{8}x^8-\frac{2}{3}x^3+c$ | e. $\displaystyle \tan x + c$ | i. $\displaystyle 4\,\mbox{Bgtan} x + c$ |
| b. $\displaystyle \frac{2}{3}x^{3/2}+c$ | f. $\displaystyle \frac{3}{2}x^2-2\ln |x|+c$ | j. $\displaystyle x-5\,\mbox{Bgtan} x + c$ |
| c. $\displaystyle \frac{2^x}{\ln 2}+\cos x +c$ | g. $\displaystyle 4x^{3/4}-\frac{12}{5}x^{5/6}+c$ | k. $\displaystyle \frac{1}{\ln \tfrac{3}{4}}\frac{3^{x-1}}{4^{x+2}}+c$ |
| d. $\displaystyle 2 \ln |x| + \frac{3}{x}+c$ | h. $\displaystyle \tan x - x + c$ | l. $\displaystyle -4\cot x +c$ |
| a. $\displaystyle \frac{(3x+7)^{6}}{18}+c$ | j. $\displaystyle \ln|e^x-7|+c$ |
| b. $\displaystyle -\frac{4}{3}(3-x)^{\frac{3}{2}}+c$ | k. $\displaystyle \frac{e^{x^2}}{2}+c$ |
| c. $\displaystyle -\frac{\cos(3x)}{3}+c $ | l. $\displaystyle \frac{2\sqrt{x-1}}{105}(15x^3-3x^2-4x-8)+c$ |
| d. $\displaystyle -\frac{\ln|4x-7|}{4}+c$ | m. $\displaystyle -3\sqrt{5-x^2}+c$ |
| e. $\displaystyle \frac{3\sqrt[3]{2}}{4}(x-2)^{\frac{4}{3}}+c$ | n. $\displaystyle \frac{(x+10)\sqrt{2x-1}}{3}+c$ |
| f. $\displaystyle \frac{\mbox{Bgtan}(3x)}{3}+c$ | o. $\displaystyle \frac{6\sqrt[6]{x-1}}{1729}(91x^3-7x^2-12x-72)+c$ |
| g. $\displaystyle -\frac{e^{2-3x}}{3}+c$ | p. $\displaystyle \mbox{Bgtan}(e^x)+c$ |
| h. $\displaystyle \frac{3}{\sqrt{35}}\mbox{Bgtan}\frac{\sqrt{35}x}{5}+c$ | q. $\displaystyle 4\left(x^{\frac{1}{12}}-1\right)^3+18\left(x^{\frac{1}{12}}-1\right)^2+36\left(x^{\frac{1}{12}}-1\right)+12\cdot\ln\left(\left|x^{\frac{1}{12}}-1\right|\right)+c$ |
| i. $\displaystyle 2\ln|2x^2-3x+1|+c$ | r. $\displaystyle \frac{1}{\ln\frac{5}{3}} \ln \left(\left(\frac{5}{3}\right)^x +1\right)+c$ |
| a. $\displaystyle -e^{-x}(x^2-x-1)+c$ | d. $\displaystyle x\mbox{Bgsin}x+\sqrt{1-x^2}+c$ |
| b. $\displaystyle -\frac{\ln|x|}{x}-\frac{1}{x}+c$ | e. $\displaystyle \frac{e^{ax}}{a^2+b^2}(a\cos(bx)+b\sin(bx))+c$ |
| c. $\displaystyle \frac{e^x}{2}(\sin x - \cos x)+c $ | f. $\displaystyle \ln|\sin x|-x\cot x +c$ |
| a. $\displaystyle \frac{3}{2}\ln(x^2+2x+5)-\frac{7}{2}\mbox{Bgtan}\left(\frac{x+1}{2}\right)+c$ |
| b. $\displaystyle \frac{5\sqrt{2}}{8}\mbox{Bgtan}\left(\frac{x}{\sqrt{2}}\right)-\frac{7x}{4(x^2+2)}+c$ |
| c. $\displaystyle \ln|(x-3)(x+2)^2|+c $ |
| d. $\displaystyle 2\mbox{Bgtan}x+\frac{\ln(x^2+1)}{2} - \ln|x-1| +c$ |
| e. $\displaystyle \mbox{Bgtan}(x+1) + \frac{\ln(x^2+2x+2)}{2} +\frac{x^3}{3} - \frac{x^2}{2} +x +c$ |
| f. $\displaystyle 19 \ln|x+2| + 2 \ln|x+1| + \frac{22}{x+2}+ \frac{x^2}{2} -6x +c$ |
| a. $\displaystyle \frac{x}{2}-\frac{\sin(2x)}{4}+c$ | d. $\displaystyle \frac{2\sqrt{3}}{9} \mbox{Bgtan} \left(\frac{1}{\sqrt{3}} \tan \left( \frac{3x}{2} \right)\right) +c $ |
| b. $\displaystyle -\frac{\cos^5 x}{5} + c$ | e. $\displaystyle \frac{1}{\sqrt{3}}\mbox{Bgtan} \left(\frac{2 \sin x - 1}{\sqrt{3}}\right) + \frac{1}{6} \ln|\frac{1 - \sin x + \sin ^2 x}{(1+\sin x)^2}|+c$ |
| c. $\displaystyle \frac{\sin ^3 x}{3} - \frac{\sin ^5 x}{5} + c $ | f. $\displaystyle \frac{\cos(2x)}{4} - \frac{\cos(8x)}{16} +c$ |
| a. $\displaystyle 6 \ln|\sqrt[6]{x}-1|+2\sqrt{x}+3\sqrt[3]{x}+6\sqrt[6]{x} + c $ |
| b. $\displaystyle \frac{x^2\cdot\mbox{Bgsin}\left(x\right)}{2}-\frac{1}{4}\left(\mbox{Bgsin}\left(x\right)-x\sqrt{1-x^2}\right)+c $ |
| c. $ \displaystyle \frac{x^3}{3}\mbox{Bgcos}\left(x\right)+\frac{\left(1-x^2\right)^{\frac{3}{2}}}{9}-\frac{1}{3}\sqrt{1-x^2}+c $ |
| d. $\displaystyle \frac{1}{3} \mbox{Bgsin}\left( \frac{3x+1}{\sqrt{5}} \right) +c$ |
| e. $\displaystyle -\ln\left(9-\sin^2\left(x\right)\right)+c$ |
| f. $\displaystyle \frac{1}{4}\left(\ln\left(\left|\sqrt{1-x^4}-1\right|\right)-\ln\left(\sqrt{1-x^4}+1\right)\right)+c$ |
| g. $\displaystyle \ln\left|\tan\left(\frac{x}{2}\right)\right| +c$ |
| h. $\displaystyle \frac{x^3}{3} \mbox{Bgcos} x - \frac{\sqrt{1-x^2}(x^2+2)}{9}+c$ |
| i. $\displaystyle \frac{3}{7}(1+\sqrt[4]{x})^{\frac{1}{3}}(4\sqrt{x}+\sqrt[4]{x}-3 +c$ |
| a. $\displaystyle \frac{1}{2}\tan\left(x\right)\:-\:\frac{1}{2\sqrt{2}} \mbox{Bgtan} \left(\frac{1}{\sqrt{2}}\cdot\:\cot\left(x\right)\right)+C$ |
| b. $\displaystyle \frac{2}{3}\mbox{Bgtan}\left(\sin\left(x\right)-\cos\left(x\right)\right)+\:\frac{1}{3\sqrt{2}}\ln\left|\frac{\sin\left(x\right)-\cos\left(x\right)+\sqrt{2}}{\sin\left(x\right)-\cos\left(x\right)-\sqrt{2}}\right|+C$ |
| c. $\frac{\sqrt{2}}{2}\mbox{Bgsin}\left(\sin\left(x\right)-\cos\left(x\right)\right)\:+\:\frac{\sqrt{2}}{2}\ln\left|\cos\left(x\right)+\sin\left(x\right)+\:\sqrt{\left(\cos\left(x\right)+\sin\left(x\right)\right)^2-1}\right|+C$ |
| d. $\displaystyle \frac{4}{3}\sqrt[4]{\frac{x}{x-3}}-2\sqrt[6]{\frac{x}{x-3}}+c$ |
| a. $-\frac{4}{3}$ | c. $\displaystyle \frac{3\pi}{8}-\frac{1}{4}$ | e. $\displaystyle -\ln\frac{\sqrt{2}}{2}$ |
| b. $\frac{14}{3}$ | d. $3$ | f. $\displaystyle \frac{\ln10}{2}+\mbox{Bgtan} \: 3$ |
| a. $\frac{4}{3}$ | c. $\displaystyle \frac{27}{e^3}$ | e. $24$ | g. $\displaystyle \frac{\ln2}{2}+\frac{\pi}{4}-1$ |
| b. $\displaystyle 12\sqrt{2}-\frac{13}{4}$ | d. $\frac{64}{3}$ | f. $1$ | h. $\frac{4}{3}$ |
| a. $\frac{2}{3\ln(2)}-\frac{4}{5\ln(5)}$ | b. $2\sqrt{2}$ | c. $\frac{445}{48}$ |
| a. $\frac{\mbox{Bgtan} \: x}{2}$ | b. $\pi$ |