math_ Use differentiation to calculate approximate value

Application of differentiation

$$\begin{gathered} \Delta x\to 0when, \\ IPeoplecanWithWithcutLinegenerationOn\; behalf\; ofsongLine \\ WithtinybranchdygenerationOn\; behalf\; of\Delta y(LetterCounttinybranchgenerationOn\; behalf\; ofLetterCountincreaseThe\; amount) \\ \Delta y\approx dy \\ dy=y^{\prime }dx=f^{\prime }(x)dx \end{gathered}$$

Approximate value

• Even the same function , The calculation difficulty of function value at different points is different

• For example, functions sin(x), stay

  • $$sin(30°)=sin(\frac{\pi }{6})=\frac{1}{2}$$

    and
    $$sin(33°)$$

    The evaluation of will be troublesome

  • $$\begin{gathered} f=(x)=f(x_0+\Delta x)=sin(x_0+\Delta x); \\ x=x_0+\Delta x \\ {x}_0=30°;\Delta x=3° \end{gathered}$$

• Practical application , The scene includes

  • Estimate function value increment $\Delta y=f(x_0+\Delta x)-f(x_0)$,($\Delta y$ More specific , Write it down as :$\Delta y(x_0,\Delta x))$)

    • Set the increment of the estimation function $\Delta y$ The function of is $dy(x_0,\Delta x)=f^{\prime }(x_0)\Delta x$,

    • From the perspective of programming , This is a two parameter function ,$\Delta yLetterCountPick\; upsuffertwoindividualoftenCountginsengCount:x_0,\Delta x;returnreturnvalueyesf(x_0+\Delta x)-f(x_0)Ofnearlikevalue$

      • requirement ,$\Delta x$ Small enough , Then the returned result is valid and reliable

    • $$\begin{gathered} OnelikeThe\; earth,\Delta y(x,\Delta x)=f(x+\Delta x)-f(x)=f^{\prime }(x)\Delta x+o(\Delta x)\approx f^{\prime }(x)\Delta x \\ Whenx=x_0,be \\ \Delta y(x_0,\Delta x)=f^{\prime }(x_0)\Delta x \end{gathered}$$

  • Estimate function value

    • In fact, the estimated value is divided into two parts :

      • $f(x)=f(x_0+\Delta x)=f(x_0)+\Delta y(x_0,\Delta x)$

        • among $\Delta y(x_0,\Delta x)\approx f^{\prime }(x_0)\Delta x$
        • Sum up ,$f(x)=f(x_0)+f^{\prime }(x_0)\Delta x$

      • What needs to be done is , Will give a x The value is expressed in two parts

      • $$\begin{gathered} x=x_0+\Delta x \\ \Delta xwantfootenoughSmall \\ andAnd,f(x)stayx=x_{0}It\text{'}s\; aboutThanagoodmetercount,Same\; aswhen{f}^{\prime }(x_0)alsogoodmetercount; \end{gathered}$$

      • thus it can be seen , In the estimation process ,$x_0$ It's the protagonist ,$\Delta x$ It's a supporting role.

    • according to $\Delta y(x,\Delta x)=f(x+\Delta x)-f(x)\approx f^{\prime }(x)\Delta x$, By moving towards deformation , You can get :$f(x)$ The approximate value of the solution function

    $$\begin{gathered} YesOnf(x)Ofnearlikevalue,stayfullfootOneNextstripPieces\; ofOfwhenHoumostYesuse: \\ (Such\; asfruitcanWithsurfaceinby(Demolition):x=x_0+\Delta x, \\ Itsin(x,x_0,\Delta x)allyesoftenCount \\ andAnd,f(x_0)and{f}^{\prime }(x_0)allThanaRongeasyseekExplain) \\ be,f(x)=f(x_0+\Delta x)\approx f(x_0)+f^{\prime }(x_0)\Delta xthisindividualsurfacereachtypealsoyesgoodseekExplainOf \end{gathered}$$

Estimate $f(x)$ example

• for example ,
  $$\begin{gathered} sin(33^{°})Ofnearlikevalue=? \\ x=x_0+\Delta x \\ {x}_0=30°=\frac{\pi }{6};\Delta x=3°=\frac{\pi }{60} \\ sin(30°+3°)\approx sin(\frac{\pi }{6})+si{n}^{\prime }(\frac{\pi }{6})\cdot \frac{\pi }{60}=\frac{1}{2}+\frac{\sqrt{3}}{2}\times \frac{\pi }{60}\approx 0.545 \end{gathered}$$

Differential supplement

From the perspective of differential approximation , We know :
$$\begin{gathered} \Delta y=f(x+\Delta x)-f(x) \\ \Delta y=f^{\prime }(x)\Delta x+\beta \\ \text{\hspace{1em}( from A few What horn degree (a1))} \end{gathered}$$

$$\begin{gathered} \beta =o(\Delta x),namely,o(\Delta x)yes\Delta xOfhighranknothingpoorSmall \\ (stayand\Delta xcommonSame\; asTrendnearOn0Oftoochengin,\beta =o(\Delta x)TrendnearOn0Ofspeeddegreemorefast \\ It\; meanstastethe,When\Delta xfootenoughSmall,thatWell: \end{gathered}$$

$$\Delta y=f^{\prime }(x)\Delta x+\beta \approx f^{\prime }(x)\Delta x\text{\hspace{1em}(a2)}$$

$$\begin{gathered} LetterCounty=f(x)Oftinybranchsurfaceinby: \\ dy=y^{\prime }dx=f^{\prime }(x)dx; \\ \Delta x=dx \\ \therefore f(x+\Delta x)=f(x_0)+\Delta y\approx f(x_0)+df(x); \end{gathered}$$