QExample - 6: HC Verma: Concepts of Physics - Volume 1 Physics and Mathematics Example - 6 Solution: Given the graph of a function y = f(x), determine the value of…
Students frequently confuse the geometric slope of the curve $y=f(x)$, which is $dy/dx$, with the requested value $dx/dy$. You must recognize that $dx/dy$ is the reciprocal of the tangent slope, so if the tangent is horizontal ($dy/dx=0$), $dx/dy$ is undefined, not zero.
Given the graph of a function y = f(x), determine the value of the derivative, dx/dy, at x = 2, 6 and 10
Step-by-Step Verified Solution
Step 1: Understand the Relationship Between dy/dx and dx/dy
The derivative dy/dx is the reciprocal of dx/dy. Mathematically:
dy/dx = 1 / (dx/dy)
Thus, if we can find dx/dy (the slope of the tangent to the curve at a given point), we can compute dy/dx by taking its reciprocal.
Step 2: Find dx/dy from the Graph
From the graph of y = f(x), the slope of the tangent at any point x is given by:
dx/dy = tan(θ)
where θ is the angle the tangent makes with the positive x-axis. This slope can be estimated visually or calculated using the rise-over-run method if specific data points are provided.
Step 3: Compute dy/dx for Each Point
Using the relationship dy/dx = 1 / (dx/dy), we compute dy/dx for each specified x-value.
Example Solution Based on Hypothetical Data
Let’s assume the following observations from the graph (as no graph is explicitly provided in your question):
At x = 2:
- The tangent to the curve at x = 2 has a slope of dx/dy = 4/5 (rise = 5, run = 4).
- Therefore: dy/dx = 1 / (dx/dy) = 1 / (4/5) = 5/4
At x = 6:
- The tangent to the curve at x = 6 is parallel to the x-axis, meaning the slope dx/dy = 0.
- Since dy/dx = 1 / (dx/dy), division by zero is undefined.
- Thus, dy/dx is undefined at x = 6.
At x = 10:
- The tangent to the curve at x = 10 has a slope of dx/dy = -2 (negative slope).
- Therefore: dy/dx = 1 / (dx/dy) = 1 / (-2) = -1/2