SUMMARY
The discussion focuses on finding the values of x in the interval [-π/2, π/2] where the graph has horizontal tangents, specifically using the derivatives f'(x) = 9cos(x) - 2sin(x) and f'(x) = -5csc(x)(5cot(x) - csc(x)). The critical points are determined by solving the equations 9cos(x) = 2sin(x) and 5cot(x) = csc(x). The solution involves using the arctan function to find x such that tan(x) = 9/2, which is not a simple solution but can be approached using inverse trigonometric functions.
PREREQUISITES
- Understanding of trigonometric functions and their derivatives
- Knowledge of inverse trigonometric functions, specifically arctan
- Familiarity with solving equations involving trigonometric identities
- Basic calculus concepts, particularly differentiation
NEXT STEPS
- Learn how to solve trigonometric equations using inverse functions
- Study the properties and applications of the arctan function
- Explore the concept of horizontal tangents in calculus
- Review techniques for finding critical points in functions
USEFUL FOR
Students studying calculus, particularly those focusing on trigonometric functions and their applications in finding horizontal tangents, as well as educators looking for examples to illustrate these concepts.