SUMMARY
The function g(x) = x is differentiable on the closed interval [-π, π]. The derivative g'(a) is calculated using the limit definition of the derivative, resulting in g'(a) = 1 for all a within the interval. This confirms that the function is linear and has a constant slope of 1 across the specified range. Therefore, the conclusion is that g(x) = x is differentiable on [-π, π] with a derivative of 1.
PREREQUISITES
- Understanding of the limit definition of a derivative
- Familiarity with basic calculus concepts
- Knowledge of continuous functions
- Ability to work with closed intervals in real analysis
NEXT STEPS
- Study the properties of differentiable functions on closed intervals
- Learn about the Mean Value Theorem and its applications
- Explore the implications of differentiability on continuity
- Investigate higher-order derivatives and their significance
USEFUL FOR
Students studying calculus, educators teaching differentiation, and anyone interested in the properties of linear functions and their derivatives.