Facit
Sommarmatte 2
Övning 1.1:1
| a) | $f'(-4)>0, \,\,\,\, f'(1)<0$ |
| b) | $x=-3$ och $x=2$ |
| c) | $-3\le x \le 2$ |
Övning 1.1:2
| a) $f'(x)=2x-3$ |
| b) $f'(x)=-\sin x -\cos x$ |
| c) $f'(x)=e^x-\displaystyle\frac{1}{x}$ |
| d) $f'(x)=\displaystyle\frac{1}{2}x^{-1/2}=\frac{1}{2\sqrt x}$ |
| e) $f'(x)=4x(x^2-1)$ |
| f) $f'(x)=-\sin \left(x+\frac{\pi}{3}\right)$ |
Övning 1.1:3
| $14{,}0\,$ m/s |
Övning 1.1:4
|
Tangentens ekvation: $\ y=2x-1$ |
Övning 1.1:5
| $\bigl(1-\sqrt2, -3+2\sqrt2\bigr)\,$ och $\,\bigl(1+\sqrt2, -3-2\sqrt2\bigr)$ |
Övning 1.2:1
| a) | $\cos^2x-\sin^2x=\cos2x$ | b) | $2x\ln x+ x$ | c) | $\displaystyle\frac{x^2+2x-1}{(x+1)^2}=1-\frac{2}{(x+1)^2}$ |
| d) | $\displaystyle\frac{\cos x}{x}-\frac{\sin x}{x^2}$ | e) | $\displaystyle\frac{1}{\ln x}-\frac{1}{(\ln x)^2}$ | f) | $\displaystyle \frac{\ln x + 1}{\sin x}-\frac{x\ln x \cos x}{\sin^2x}$ |
Övning 1.2:2
| a) | $\cos x^2 \cdot 2x$ | b) | $e^{x^2+x}(2x+1)$ | c) | $\displaystyle - \frac{\sin x}{2\sqrt{\cos x}}$ |
| d) | $\displaystyle\frac{1}{x\ln x}$ | e) | $(2x+1)^3(10x+1)$ | f) | $\displaystyle\frac{\sin\sqrt{1-x}}{2\sqrt{1-x}}$ |
Övning 1.2:3
| a) | $\displaystyle\frac{1}{2\sqrt{x}\sqrt{x+1}}$ | b) | $\displaystyle - \frac{1}{(x-1)^{3/2}\sqrt{x+1}}$ | c) | $\displaystyle - \frac{1-2x^2}{x^2(1-x^2)^{3/2}}$ |
| d) | $-\cos\cos\sin x \cdot \sin\sin x \cdot \cos x$ | e) | $e^{\sin x^2}\cdot \cos x^2 \cdot 2x$ | f) | $\displaystyle x^{\tan x}\Bigl(\frac{\ln x}{\cos^2x}+\frac{\tan x}{x}\Bigr)$ |
Övning 1.2:4
| a) | $\displaystyle\frac{3x}{(1-x^2)^{5/2}}$ | b) | $\displaystyle - \frac{2\sin \ln x}{x}$ |
