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- https://phys.libretexts.org/Courses/Berea_College/Introductory_Physics%3A_Berea_College/26%3A_Calculus/26.03%3A_Anti-derivatives_and_integralsUsing the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\] We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\), as illustrated by the black arrow in Figure A2.3.1 \[\begin{aligned} F_1&=F(x_1)\\ &=F(x+\Delta x) \\ &\approx F_0 + \Delta F_0\\ &\approx F_0+f(x_0)\Delta x\end{aligned}…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_II_(2212)/02%3A_Math_Review/2.10%3A_Anti_derivatives_and_integralsUsing the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\] We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\), as illustrated by the black arrow in Figure A2.3.1 \[\begin{aligned} F_1&=F(x_1)\\ &=F(x+\Delta x) \\ &\approx F_0 + \Delta F_0\\ &\approx F_0+f(x_0)\Delta x\end{aligned}…
- https://phys.libretexts.org/Courses/Georgia_State_University/GSU-TM-Physics_I_(2211)/02%3A_Vectors_and_Math_Review_Topics/2.07%3A_Math_Review_of_Other_Topics/2.7.21%3A_Anti_derivatives_and_integralsUsing the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0...Using the derivative \(f(x)\) evaluated at \(x_0\), we have: \[\begin{aligned} \frac{\Delta F_0}{\Delta x} &\approx f(x_0)\;\;\;\; (\text{in the limit} \Delta x\to 0 )\\ \therefore \Delta F_0 &= f(x_0) \Delta x\end{aligned}\] We can then estimate the value of the function \(F_1=F(x_1)\) at the next point, \(x_1=x_0+\Delta x\), as illustrated by the black arrow in Figure A2.3.1 \[\begin{aligned} F_1&=F(x_1)\\ &=F(x+\Delta x) \\ &\approx F_0 + \Delta F_0\\ &\approx F_0+f(x_0)\Delta x\end{aligned}…
