Kopertina e librit Matematika 12

Zgjidhja e ushtrimit 1

Zgjidhja e ushtrimit 1 të mësimit 13A në librin Matematika 12 nga shtëpia botuese Mediaprint me autorë Greg Attwood, Jack Barraclough, Ian Bettison, Alistair Macpherson, Bronwen Moran, Su Nicholson, Diane Oliver, Joe Petran, Keith Bledger, Harry Smith, Geoff Staley, Robert Ward-Penny dhe Dave Wilkins.

Pyetja

Gjej një shprehje për yy kur dydx\dfrac{dy}{dx} është si vijon:

  1. x5x^5
  2. 10x410x^4
  3. −x−2-x^{-2}
  4. −4x−3-4x^{-3}
  5. x23x^{\frac{2}{3}}
  6. 4x124x^{\frac{1}{2}}
  7. −2x6-2x^6
  8. x−12x^{-\frac{1}{2}}
  9. 5x−325x^{-\frac{3}{2}}
  10. 6x136x^{\frac{1}{3}}
  11. 36x1136x^{11}
  12. −14x−8-14x^{-8}
  13. −3x−23-3x^{-\frac{2}{3}}
  14. −5-5
  15. 6x6x
  16. 2x−0.42x^{-0.4}

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Zgjidhja

  1. y=∫x5dx=x66+cy=\int x^5 dx = \dfrac{x^6}{6}+c
  2. y=∫10x4dx=10x55+c=2x5+cy=\int 10x^4 dx = \dfrac{10x^5}{5}+c=2x^5+c
  3. y=−x−2dx=−x−1−1+c⇒y=x−1+cy=-x^{-2}dx=-\dfrac{x^{-1}}{-1}+c \rArr y=x^{-1}+c ose y=1x+cy=\dfrac{1}{x}+c
  4. y=∫−4x−3dx=−4x−2−2+c=2x−2+cy=\int-4x^{-3} dx = \dfrac{-4x^{-2}}{-2}+c=2x^{-2}+c ose y=2x2+cy=\dfrac{2}{x^2}+c
  5. y=∫x23dx=x5353+c=35x53+cy=\int x^{\frac{2}{3}}dx = \dfrac{x^{\frac{5}{3}}}{\frac{5}{3}}+c = \dfrac{3}{5}x^{\frac{5}{3}}+c
  6. y=∫4x12dx=4x3232+c=4×23x32+c=83x32+cy=\int 4x^{\frac{1}{2}}dx=4\dfrac{x^{\frac{3}{2}}}{\frac{3}{2}}+c= 4 \times \dfrac{2}{3}x^{\frac{3}{2}}+c= \dfrac{8}{3}x^{\frac{3}{2}}+c
  7. y=∫−2x6dx=−2×x77+c=−27x7+cy=\int-2x^6dx=-2 \times \dfrac{x^7}{7}+c = -\dfrac{2}{7}x^7+c
  8. y=∫x−12dx=x1212+c=2x12+cy=\int x^{-\frac{1}{2}}dx=\dfrac{x^{\frac{1}{2}}}{\frac{1}{2}}+c = 2x^{\frac{1}{2}}+c ose y=2x+cy=2\sqrt{x}+c
  9. y=∫5x−32dx=5x−12−12+c=−10x−12+cy=\int 5x^{-\frac{3}{2}}dx=\dfrac{5x^{-\frac{1}{2}}}{-\frac{1}{2}}+c=-10x^{-\frac{1}{2}}+c ose y=−10x+cy=-\dfrac{10}{\sqrt{x}}+c
  10. y=∫6x13dx=6x4343+c=34×6x43+c=184x43+c=92x43+cy=\int 6x^{\frac{1}{3}}dx = \dfrac{6x^{\frac{4}{3}}}{\frac{4}{3}}+c = \dfrac{3}{4} \times 6x^{\frac{4}{3}} + c= \dfrac{18}{4}x^{\frac{4}{3}}+c = \dfrac{9}{2}x^{\frac{4}{3}}+c
  11. y=∫36x11dx=36x1212+c=3x12+cy=\int 36x^{11}dx = \dfrac{36x^{12}}{12}+c=3x^{12}+c
  12. y=∫−14x−8dx=−14x−7−7+c=2x−7+cy=\int -14x^{-8}dx = \dfrac{-14x^{-7}}{-7}+c= 2x^{-7}+c ose y=2x7+cy=\dfrac{2}{x^7}+c
  13. y=∫−3x−23dx=−3x1313+c=3×(−3)×x13+c=−9x13+cy=\int -3x^{-\frac{2}{3}}dx=\dfrac{-3x^{\frac{1}{3}}}{\frac{1}{3}}+c= 3 \times (-3) \times x^{\frac{1}{3}}+c = -9x^{\frac{1}{3}}+c
  14. y=∫−5x0dx=−5x11+c=−5x+cy=\int -5x^0 dx = \dfrac{-5x^1}{1}+c = -5x+c
  15. y=∫6xdx=6x22+c=3x2+cy=\int 6x dx = \dfrac{6x^2}{2}+c=3x^2+c
  16. y=∫2x−0.4dx=2x0.60.6+c=2×106×x0.6+c=206x0.6+c=103x0.6+cy=\int 2x^{-0.4}dx = \dfrac{2x^{0.6}}{0.6}+c = 2 \times \dfrac{10}{6} \times x^{0.6}+c = \dfrac{20}{6}x^{0.6}+c=\dfrac{10}{3}x^{0.6}+c