p. 942 #5
This is a three-part question:
a. the graph y = f(x) in the xy-plane has parametrization x=x, y=f(x), and vector formula r(x) = xi + f(x)j. Use this to show that if f is twice-differentiable, then
k(x) = ((absvalue f''(x))/[1+((f'(x))^2]^3/2
b. use the formula for k in part a to find the curvature of y = ln(cosx) when -pi/2 < x < pi/2.
c. show that the curvature is zero at the point of inflection.
To answer these questions, we need to understand the concepts of parametrization, vector formula, second derivative, and curvature.
a. The graph of a function y = f(x) in the xy-plane can be parametrized using the variables x and y. In this case, we have x = x and y = f(x). The vector formula for this graph is r(x) = xi + f(x)j, where xi represents the x-component and f(x)j represents the y-component.
To find the curvature, we need to consider the second derivative of the function f(x). The second derivative represents the rate of change of the slope of the graph at each point. Let's calculate it.
First, we find the first derivative of f(x):
f'(x) = d/dx [f(x)]
Then, we find the second derivative:
f''(x) = d/dx [f'(x)]
Now, let's calculate the expression for k(x):
k(x) = (|f''(x)|) / [1 + (f'(x))^2]^(3/2)
This expression represents the curvature of the graph.
b. To find the curvature of y = ln(cosx) when -π/2 < x < π/2, we need to calculate f(x), f'(x), and f''(x).
Given y = ln(cosx), we can find the derivatives:
f(x) = ln(cosx)
f'(x) = d/dx [ln(cosx)]
f''(x) = d/dx [f'(x)]
After calculating f'(x) and f''(x), we can substitute the values into the expression for k(x) to find the curvature at each point within the given range.
c. To show that the curvature is zero at the point of inflection, we need to find the point of inflection first. The point of inflection occurs when f''(x) = 0 and there is a change of concavity in the graph.
Once we find the x-value(s) of the point of inflection, we can calculate the curvature k(x) at those points and demonstrate that the curvature is zero.
Remember to substitute the values obtained from f'(x) and f''(x) into the expression for k(x).
These steps should help you answer parts a, b, and c by understanding the concepts of parametrization, vector formula, second derivative, and curvature.