SUMMARY
The function f is defined as f(x) = cx + 7 for x ≤ 2 and f(x) = (cx)² + 1 for x > 2. For f to be continuous at x = 2, the left-hand limit, \(\lim_{x \to 2^-} f(x) = 2c + 7\), must equal the right-hand limit, \(\lim_{x \to 2^+} f(x) = 4c^2 + 1\). Setting these two expressions equal results in the equation \(2c + 7 = 4c^2 + 1\). Solving this quadratic equation yields the values of c that ensure the function is continuous at x = 2.
PREREQUISITES
- Understanding of limits in calculus
- Knowledge of continuity in functions
- Ability to solve quadratic equations
- Familiarity with piecewise functions
NEXT STEPS
- Study the concept of limits and their properties in calculus
- Learn how to determine continuity for piecewise functions
- Practice solving quadratic equations using the quadratic formula
- Explore the implications of continuity in real-world applications
USEFUL FOR
Students studying calculus, particularly those focusing on limits and continuity, as well as educators seeking to explain these concepts effectively.