Numerical integration and differentiationOften in an experiment, some quantity is measured as a function of time. For instance, by videotaping a moving object it is possible to determine the position of the object in every frame of the videotape. Using a Doppler radar, it is possible to record the velocity of an object as a function of time. Using the accelerometer in a mobile phone, it is possible to record the acceleration as a function of time. If one of these quantities is measured, the others can be calculated. If the measured data is in tabular form, the integration or differentiation must be done numerically. This page contains some programs for integrating and differentiating numerical data. A function $f(t)$ is can be specified either by inputting a mathematical formula at the top or by pasting two columns of data in the textbox at the top-left. When the "Calculate from formula" button is pressed, the formula is used to fill the table with 1000 equally spaced values of $f(t)$ equally spaced between $t_1$ and $t_2$. When the "calculate from table" button is pressed, the data is plotted on the right. Below the data and plot of $f(t)$, the derivative $\frac{df}{dt}$ and the second derivative $\frac{d^2f}{dt^2}$ are tabulated and plotted. Below the derivatives, the integral of $f(t)$ is shown as well as the integral of the integral. The integration routines assume that the measurements are equally spaced with an interval $\Delta t$. For problems involving position, velocity, acceleration, and force
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