To find the point at which the curve given by the equation y = 7 tan(x) has maximum curvature, we need to calculate the curvature of the function and then analyze its critical points.
The curvature K of a function y = f(x) can be expressed using the formula:
K = rac{f”(x)}{(1 + (f'(x))^2)^{3/2}}
1. **First, we find the first derivative y’:**
y’ = 7 sec²(x)
2. **Now, we find the second derivative y”:**
y” = 14 sec²(x) tan(x)
3. **Substituting in the curvature formula:**
K = rac{14 sec²(x) tan(x)}{(1 + (7 sec²(x))^2)^{3/2}}
4. **To maximize curvature, we need to look for critical points of K:**
This involves setting the derivative of the curvature function to zero and solving for x. This is typically done using calculus techniques. The maximum curvature will occur at values where the rate of change of curvature switches direction.
5. **Finally, solve for x to find the corresponding point on the curve:**
The specific points can be calculated to find exact locations for maximum curvature within the interval (-π/2, π/2) due to the nature of the tan(x) function. Proper computational tools or numerical methods can pinpoint the exact value.
In conclusion, to determine where the curve has maximum curvature, we can utilize the curvature formula and calculus techniques to find critical points, leading us to the required coordinates on the curve.