To find the intersection points of the function $y = \sin(e^x)$ and a circle with radius 5 centered at $(1,1)$, we solve the system of equations:

1. Circle equation: $(x - 1)^2 + (y - 1)^2 = 25$
2. Function equation: $y = \sin(e^x)$

Substitution

Substitute $y = \sin(e^x)$ into the circle's equation: $(x - 1)^2 + (\sin(e^x) - 1)^2 = 25$

Solve Numerically

This equation does not have a closed-form solution and must be solved numerically for $x$. Let me calculate the $x$-values of the intersection points and the corresponding $y$-values.import numpy as np from scipy.optimize import fsolve

Define the system of equations

def circle_intersection(x): # Circle equation with substitution of y = sin(e^x) return (x - 1)**2 + (np.sin(np.exp(x)) - 1)**2 - 25

Find the root closest to the origin by testing nearby initial guesses

initial_guesses = np.linspace(-10, 10, 100) # Wide range of guesses roots = [fsolve(circle_intersection, guess)[0] for guess in initial_guesses]

Filter unique solutions and calculate corresponding y-values

unique_roots = np.unique(np.round(roots, decimals=5)) intersection_points = [(x, np.sin(np.exp(x))) for x in unique_roots]

Sort by distance from the origin

nearest_point = sorted(intersection_points, key=lambda p: np.sqrt(p[0]**2 + p[1]**2))[0] nearest_point/home/sandbox/.local/lib/python3.11/site-packages/scipy/optimize/_minpack_py.py:175: RuntimeWarning: The iteration is not making good progress, as measured by the improvement from the last ten iterations. warnings.warn(msg, RuntimeWarning) /home/sandbox/.local/lib/python3.11/site-packages/scipy/optimize/_minpack_py.py:175: RuntimeWarning: The iteration is not making good progress, as measured by the improvement from the last five Jacobian evaluations. warnings.warn(msg, RuntimeWarning) (1.56248, -0.9983040189645134)(1.56248, -0.9983040189645134)successThe intersection point of $y = \sin(e^x)$ and the circle with radius 5 centered at $(1,1)$ that is nearest to the origin is approximately:

$(x, y) = (1.56248, -0.99830)$

Would you like details on the calculations or further analysis?

Related Questions:

1. How can we verify intersection points geometrically?
2. Can the solution be generalized for different circle radii or centers?
3. How does the behavior of $\sin(e^x)$ influence the number of intersections?
4. How do numerical methods (e.g., $fsolve$) work to find solutions?
5. What is the impact of choosing different initial guesses in this numerical method?

Tip:

For complex systems like these, always visualize the functions (e.g., using graphs) to better understand the behavior and intersections.