Graphics Representation of Differentiation and Integration Physics Notes
Graphics Representation of Differentiation:
A slope field is the graphical representation of a differential equation. It is a graph of short line segments whose slope is determined by evaluating the derivative at the midpoint of the segment.
To And the derivative graphically, we must take a tangent line (the point where we have to calculate the derivative) and then find out the slope of this tangent line. To find out the instantaneous rate of change, we must repeat the process of calculating the slope.
Graphical Representation of Derivatives
Let y be a function of x so that y = f(x). This dependency of y on x is shown in the figure. On this graph, the two points P and Q are represented by (x, y) and (x + δx, y + δy) respectively. If P is closer to Q, then δx → 0. Hence, PQ becomes a straight line with
slope, tan θ = \(\frac{Q R}{P R}\)
If the angle is acute (< 90°), then tan0 or the slope is positive and if the angle is obtuse (> 90°), then tan0 or the slope is negative.
Graphical Representation of Integration:
The value of y changes with different values of x, for the function, y = f(x)
Graphical Representation of integration
The graph in figure 2.30 shows the values of dependent variables x and (x + dx). The value of dx is very small (dx → 0). So, the value of y remains constant, as the value of change in x is very small. The area of this small segment (PQRS) will be y dx. If we imagine similar segments from x = a to x – b, then the area will be the sum of area of all the segments. The process of adding up considering that the value of dx is very small is called integration.
∴ ∫ba y dx = area between Y-X plane and X-axis (from x = a to x = b)