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A Standard Interview Question: Finding Integral and Derivative of a Graph without Knowing the Functional Form
Whenever you want to get into a research institute, almost in every case you need to appear for an interview. The interview board mainly tries to know how good your basics are and there are always questions to judge your mathematical skill and intuition. Here we will discuss one of such common but standard questions.
A y vs. x plot is given as shown in the figure 1 and it is said that it is the plot of velocity of a car as a function of time.
The graph is piecewise continuous and consists of three parts
A: linearly incresing
C: linearly decreasing
You are asked to find its plot of distance travelled and acceleration as a function of time. Since numerical calculations are not required, numbers along y axis are removed.
This is exactly equivalent to find the integral and derivative of the given graph because distance is the integral of velocity and acceleration is nothing but the derivative of velocity. The same situation can occur in case of a plot between force applied to a system as a function of distance and you are required to find the work done. So here we will discuss the key ideas.
In the part A, value of velocity is increasing monotonically and so the distance must increase at a faster rate. Now as we see that it is a linear function of time the distance must increase quadratically or in other words the plot of integral will be parabolic (proportional to x2), further the curvature will be downward (i.e. of concave type). In the region B velocity is constant, so the distance will be a linear function (proportional to x) of time. Finally in the third region C velocity is constantly decreasing so that distance traveled will still increase but at a smaller rate, further since the plot is decreasing linearly the distance will go quadratically (proportional to –x2) but now with a curvature pointing upward (i.e. of convex type). We also need to keep in mind that at the corner points the distance vs. time plot must have continuous slope. Hence our graph looks like figure 2.
Now to calculate the acceleration, first we notice that in region A velocity is linearly increasing so the acceleration should be constant and positive. In region B velocity is constant so the acceleration is zero and finally in region C it is linearly decreasing, so the acceleration is constant and negative. We should notice that at the junction points of regions A and B, and C and D, the given graph is not smooth, consequently the plot of acceleration will be discontinuous at those junctions. Actually it is a positive Dirac delta function at the junction of A and B, and a negative Dirac delta function at the junction of B and C. The resulting graph is shown in figure 3.
This is how you can work out integral and derivative of a graph without knowing its exact functional form. The given link will give you an idea how to plot derivative of an arbitrary graph.