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Asymptoty, bryły obrotowe i długość łuku

1. Asymptoty funkcji

y=f(x)y=f(x)

Pozioma:

lim⁡x→∞f(x)=L⇒y=L\lim_{x \to \infty} f(x) = L \quad \Rightarrow \quad y = L

Pionowa:

lim⁡x→a−f(x)=±∞lublim⁡x→a+f(x)=±∞⇒x=a\lim_{x \to a^-} f(x)=\pm\infty \quad \text{lub} \quad \lim_{x \to a^+} f(x)=\pm\infty \quad \Rightarrow \quad x = a

Ukośna:

a=lim⁡x→∞f(x)x,b=lim⁡x→∞(f(x)−ax)⇒y=ax+ba = \lim_{x \to \infty} \frac{f(x)}{x}, \quad b = \lim_{x \to \infty} \big(f(x) - ax\big) \quad \Rightarrow \quad y = ax + b

2. Objętość bryły obrotowej

Wokół OxOx:

V=π∫ab(f(x))2 dxV = \pi \int_a^b \big(f(x)\big)^2 \, dx

Wokół OyOy:

V=2π∫abx f(x) dxV = 2\pi \int_a^b x \, f(x) \, dx

3. Pole powierzchni bryły obrotowej

Wokół OxOx:

S=2π∫abf(x) 1+(f′(x))2 dxS = 2\pi \int_a^b f(x)\,\sqrt{1+\big(f'(x)\big)^2}\, dx

Wokół OyOy:

S=2π∫abx 1+(f′(x))2 dxS = 2\pi \int_a^b x\,\sqrt{1+\big(f'(x)\big)^2}\, dx

4. Długość łuku

Funkcja y=f(x)y=f(x) na [a,b][a,b]:

L=∫ab1+(f′(x))2 dxL = \int_a^b \sqrt{1+\big(f'(x)\big)^2}\, dx

Wzory pochodnych

  1. (C)′=0(C)' = 0

  2. (xn)′=nxn−1(x^n)' = nx^{n-1}

  3. (x)′=1(x)' = 1

  4. (1x)′=−1x2\left( \frac{1}{x} \right)' = -\frac{1}{x^2}

  5. (x)′=12x(\sqrt{x})' = \frac{1}{2\sqrt{x}}

  6. (ax)′=axln⁡a(a^x)' = a^x \ln a

  7. (ex)′=ex(e^x)' = e^x

  8. (log⁡ax)′=1xln⁡a(\log_a x)' = \frac{1}{x \ln a}

  9. (ln⁡x)′=1x(\ln x)' = \frac{1}{x}

  10. (sin⁡x)′=cos⁡x(\sin x)' = \cos x

  11. (cos⁡x)′=−sin⁡x(\cos x)' = -\sin x

  12. (tgx)′=1cos⁡2x(tgx)' = \frac{1}{\cos^2 x}

  13. (ctgx)′=−1sin⁡2x(ctgx)' = -\frac{1}{\sin^2 x}

  14. (arcsin⁡x)′=11−x2(\arcsin x)' = \frac{1}{\sqrt{1 - x^2}}

  15. (arccos⁡x)′=−11−x2(\arccos x)' = -\frac{1}{\sqrt{1 - x^2}}

  16. (arctgx)′=1x2+1(arctg x)' = \frac{1}{x^2 + 1}

  17. (arcctgx)′=−1x2+1(arcctgx)' = -\frac{1}{x^2 + 1}

Wzory na całki

  1. ∫dx=x+C\int dx = x + C

  2. ∫xndx=1n+1xn+1+C,n≠−1\int x^n dx = \frac{1}{n+1} x^{n+1} + C, \quad n \ne -1

  3. ∫x dx=12x2+C\int x\, dx = \frac{1}{2} x^2 + C

  4. ∫1xdx=ln⁡∣x∣+C\int \frac{1}{x} dx = \ln|x| + C

  5. ∫axdx=axln⁡a+C\int a^x dx = \frac{a^x}{\ln a} + C

  6. ∫exdx=ex+C\int e^x dx = e^x + C

  7. ∫sin⁡x dx=−cos⁡x+C\int \sin x\, dx = -\cos x + C

  8. ∫cos⁡x dx=sin⁡x+C\int \cos x\, dx = \sin x + C

  9. ∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int \tan x\, dx = -\ln|\cos x| + C

  10. ∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x\, dx = \ln|\sin x| + C

  11. ∫1cos⁡2xdx=tan⁡x+C\int \frac{1}{\cos^2 x} dx = \tan x + C

  12. ∫1sin⁡2xdx=−cot⁡x+C\int \frac{1}{\sin^2 x} dx = -\cot x + C

  13. ∫1x2+a2dx=1aarctan⁡(xa)+C\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C

  14. ∫1x2−a2dx=12aln⁡∣x−ax+a∣+C\int \frac{1}{x^2 - a^2} dx = \frac{1}{2a} \ln\left|\frac{x - a}{x + a}\right| + C

  15. ∫1a2−x2dx=arcsin⁡(xa)+C\int \frac{1}{\sqrt{a^2 - x^2}} dx = \arcsin\left(\frac{x}{a}\right) + C

  16. ∫1x2+qdx=ln⁡∣x+x2+q∣+C\int \frac{1}{\sqrt{x^2 + q}} dx = \ln\left|x + \sqrt{x^2 + q}\right| + C