Derivative vs Integral
Differentiation thiab kev koom ua ke yog ob txoj haujlwm tseem ceeb hauv Calculus. Lawv muaj ntau daim ntawv thov hauv ntau qhov chaw, xws li lej, engineering thiab Physics. Ob qho tib si derivative thiab ib qho sib tham txog tus cwj pwm ntawm kev ua haujlwm lossis tus cwj pwm ntawm lub cev uas peb xav txog.
Derivative yog dab tsi?
Suppose y=ƒ(x) and x0 is in the domain of ƒ. Then limΔx→∞Δy/Δx=limΔx→∞[ƒ(x 0+Δx) − ƒ(x0)]/ Δx yog hu ua tus nqi hloov pauv tam sim ntawm ƒ ntawm x0, muab qhov txwv no tshwm sim finitely. Qhov kev txwv no tseem hu ua derivative ntawm ntawm thiab yog denoted los ntawm ƒ(x).
Tus nqi ntawm qhov kev ua haujlwm ntawm f ntawm qhov chaw tsis txaus ntseeg x hauv lub luag haujlwm yog muab los ntawm limΔx→∞ [ƒ(x+Δx) − ƒ(x)]/Δx. Qhov no yog txhais los ntawm ib qho ntawm cov kab lus hauv qab no: y, ƒ(x), ƒ, dƒ(x)/dx, dƒ/dx, Dxy.
Rau kev ua haujlwm nrog ntau qhov sib txawv, peb txhais ib feem derivative. Ib feem derivative ntawm ib tug muaj nuj nqi nrog ob peb variables yog nws derivative nrog hwm rau ib tug ntawm cov variables, piv txwv li tias lwm yam variables yog tas mus li. Lub cim ntawm ib feem derivative yog ∂.
Geometrically tus derivative ntawm ib tug muaj nuj nqi tuaj yeem txhais raws li txoj kab nqes ntawm txoj kab nkhaus ntawm txoj haujlwm ƒ(x).
Dab tsi yog qhov tseem ceeb?
Kev sib koom ua ke lossis kev tawm tsam kev sib txawv yog cov txheej txheem rov qab ntawm kev sib txawv. Hauv lwm lo lus, nws yog tus txheej txheem ntawm kev nrhiav ib qho tseem ceeb muaj nuj nqi thaum muab cov txiaj ntsig ntawm kev ua haujlwm. Yog li ntawd, ib qho tseem ceeb los yog ib qho kev tiv thaiv ntawm ib qho kev ua ƒ(x) yog, ƒ(x)=F (x) tuaj yeem txhais tau tias yog txoj haujlwm F (x), rau tag nrho x hauv thaj tsam ntawm ƒ(x).
Cov lus ∫ƒ(x) dx qhia txog qhov kev ua haujlwm ntawm ƒ(x). Yog tias ƒ(x)=F (x), ces ∫ƒ(x) dx=F (x) + C, qhov C yog qhov tsis tu ncua, ∫ƒ(x) dx yog hu ua indefinite integral ntawm ƒ(x).
Rau txhua txoj haujlwm ƒ, uas tsis tas yuav tsis yog qhov tsis zoo, thiab txhais ntawm lub sijhawm [a, b], a ∫ b ƒ(x) dx yog hu ua qhov tseeb ƒ ntawm [a, b].
Cov ntsiab lus tseeb a∫bƒ(x) dx ntawm ib txoj haujlwm ƒ(x) tuaj yeem txhais geometrically raws li thaj tsam ntawm cheeb tsam bounded los ntawm nkhaus ƒ(x), x-axis, thiab kab x=a thiab x=b.
Derivative thiab Integral yog dab tsi?
• Derivative yog qhov tshwm sim ntawm kev sib txawv ntawm cov txheej txheem, thaum ib qho yog qhov tshwm sim ntawm kev sib koom ua ke.
• Derivative of a function sawv cev rau txoj kab nqes ntawm txoj kab nkhaus ntawm ib qho twg, thaum ib qho sawv cev rau thaj tsam hauv qab nkhaus.
