Question

Consider the curve \( C_1 \) given by \( y = e^{-x} \quad \text{for } x \in [0, 10\pi], \) and the curve \( C_2 \) given by \(y = e^{-x}(\sin x + \cos x) \quad \text{for x \in [0, 10\pi]. \) Let \( n \) be the total number of points of intersection of the curves \( C_1 \) and \( C_2 \).
Suppose that \( \alpha_1, \alpha_2, \dots, \alpha_n \in [0, 10\pi] \) are the \( x \)-coordinates of the points of intersection of the curves \( C_1 \) and \( C_2 \) such that \(\alpha_1 < \alpha_2 < \dots < \alpha_n. \)} 15. Then the value of $n$ is {

Hint:

When solving equations like $\sin x + \cos x = 1$ over large intervals, use the periodic nature of the trigonometric functions. There are exactly 2 solutions in each $2\pi$ interval, but check the boundaries of the closed interval carefully.

Answer 1

Correct Answer: 11