Review of basics

1. State the fundamental theorem of algebra (d'Alambert's theorem). [answer]

   Every non-constant single-variable polynomial \(a_nx^n + a_{n-1}x^{n-1} + \dots+a_0 =0\) with complex coefficients \(a_n, a_{n-1}, \dots, a_0\) has at least one complex root.

2. State the rational root theorem. [answer]

   Each rational root of \(a_nx^n + a_{n-1}x^{n-1} + \dots+a_0 =0\) is given by \(x= p/q\), where \(p\) is an integer factor of \(a_0\), and where \(q\) is an integer factor of \(a_n\).

3. Solve \(x^3 - x^2 - 5x -3 = 0.\) [answer]

   \(x=-1,-1,3\)

4. Solve \(x^3 - x^2 - 8 x - 6 = 0\). [answer]

   \(x=-1,1+\sqrt{7},1-\sqrt{7}\)

5. Solve \(x^4 - 2x^3 - 3x^2 + 8x -4 = 0\). [answer]

   \(x=-2,1,1,2\)

6. Decompose \(1/(x^2 + 2x-3)\) into partial fractions. [answer]

   \[\frac{1}{4}\bigg(\frac{-1}{x+3} + \frac{1}{x-1}\bigg)\]

7. Decompose \((x^3+16)/(x^3 - 4x^2 +8x)\) into partial fractions. [answer]

   See example 2 here.

8. Decompose \(1/(x^3-1)\) into partial fractions. [answer]

   See example 5 here and note that the limit method need not be used.

9. Find the roots of \(\exp(2z)=2i\). [answer]

   Write \(i = e^{i\arctan \infty} = e^{i\pi/4}\). Then take the log of both sides, giving \(z =\frac{\ln 2}{2} + i\frac{\pi}{4}\).

10. Evaluate \((-i)^{1/3}\). [answer]

    \begin{align*} (-i)^{1/3} &= (e^{-i\pi/2})^{1/3} \\ &=e^{-i\pi/6} \\ &=(\cos \pi/6 - i\sin \pi/6)\\ &=(\sqrt{3}/2 - i/2) \end{align*}

11. Find the real part of \(e^{-ix}/(1+e^{a+ib})\). [answer]

    \begin{align*} \frac{e^{-ix}}{1+ e^{a+ib}}&=\frac{\cos x - i\sin x}{1 + e^ae^{ib}}\\ &=\frac{\cos x -i\sin x}{1 + e^a(\cos b + i\sin b)}\\ &=\frac{\cos x -i\sin x}{1 + e^a(\cos b + i\sin b)}\frac{1 + e^a(\cos b - i\sin b)}{1 + e^a(\cos b - i\sin b)}\\ &= \frac{\cos x +e^a\cos x(\cos b - i\sin b)- i\sin x - ie^a\sin x(\cos b - i\sin b)}{1 +e^{a}(\cos b +i\sin b) + e^{a}(\cos b -i\sin b) + e^{2a}(\cos b +i\sin b)(\cos b -i\sin b)}\\ &= \frac{\cos x +e^a\cos x\cos b - ie^a\cos x\sin b- i\sin x - ie^a\sin x\cos b -e^a\sin x \sin b}{1 +e^{a}(\cos b) + e^{a}(\cos b) + e^{2a}(\cos^2 b + \sin^2 b)}\\ &=\frac{\cos x +e^a\cos x\cos b -e^a\sin x \sin b}{1 + e^{2a}(\cos^2 b + \sin^2 b)} -i\frac{e^a\cos x\sin b+\sin x +e^a\sin x\cos b}{1 + 2e^{a}\cos b + e^{2a}(\cos^2 b + \sin^2 b)} \end{align*} The real part is \(\frac{\cos x +e^a\cos x\cos b -e^a\sin x \sin b}{1 + e^{2a}(\cos^2 b + \sin^2 b)}\).

12. State the fundamental theorem of calculus. [answer]

    The derivative and integral are inverses.

13. What is the relationship between differentiability and continuity? [answer]

    Differentiability implies continuity, but continuity does not imply differentiability. For an example of a continuous function that is not differentiable, consider \(\sqrt{|x|}\) at \(x= 0\).

14. \(\lim_{x\to 0}\frac{e^x-1}{x^2+x}=\) [answer]

    \(1\) by L'Hopital's rule.

15. \(\lim_{x\to 0}\frac{2\sin x - \sin 2x}{x - \sin x}=\) [answer]

    \(6\) by repetitive use of L'Hopital's rule.

16. \(\lim_{x\to \infty} x^n\, e^{-x} =\) [answer]

    \(0\) by L'Hopital's rule. Repeat l'Hopital's rule until \(x^{n-1} = x^0\).

17. \(d\sinh x/dx = \) [answer]

    \(\cosh x\)

18. \(d\cosh x/dx = \) [answer]

    \(\sinh x\)

19. \(d\tan x/dx = \) [answer]

    \(\sec^2 x\)

20. \(d\cot x/dx = \) [answer]

    \(-\csc^2 x\)

21. \(d\sec x/dx = \) [answer]

    \(\sec x\tan x\)

22. \(d\csc x/dx = \) [answer]

    \(-\cot x\csc x\)

23. \(\int dx(x^4 + x^3 + x^2 + 1)/(x^2 + x - 2)=\) [answer]

    See example 6 here.

24. Show that \(\int u dv = uv - \int v du \). [answer]

    Start with the product rule, \(\frac{d}{dx} (uv) = v\frac{du}{dx} + u\frac{dv}{dx}\), integrate both sides over \(x\), and rearrange.

25. \(\int \ln x dx = \) [answer]

    \(x\ln (x) -x\)

26. \( \int e^x \sin x dx = \) [answer]

    Integrate by parts twice to get \( \int e^x \sin x dx = e^x[\sin(x)- \cos(x)]/2\).

27. What are some general guidelines about trigonometric subtitution? [answer]

    If a quantity \(a^2-x^2\) is involved, set \(x = a\sin\theta\). If a quantity \(x^2+ a^2\) is involved, set \(x = a\tan\theta\). If a quantity \(x^2-a^2\) is involved, set \(x = a\sec\theta\).

28. \( \int \sqrt{a^2-x^2} dx = \) [answer]

    \[\frac{a^2}{2}\arcsin{\frac{x}{a}} + \frac{a^2}{4}\frac{x}{a}[1-(x/a)^2]^{1/2} +C.\] See this page for more on trigonometric substitution.

29. \( \int \frac{dx}{\sqrt{a^2-x^2}} = \) [answer]

    \[\arcsin(x/a) + C\]

30. \( \int \frac{dx}{a^2+x^2} = \) [answer]

    \[\frac{1}{a}\arctan (x/a)+C \]

31. ☸ \( \int \sqrt{a^2+x^2}dx = \) [answer]

    See this page for the solution. The integral of secant cubed is needed.