This is a sample exam for a course in Mathematical Analysis 1 at the undergraduate level. It includes problems across various topics such as algebraic equations, limits, derivatives, rational functions, tangent lines, circle equations, logarithmic inequalities, and function optimization. Below is a summary of the key tasks:

Part 1: Fundamental Problems

1. Solve equations and simplify expressions.
2. Solve a set of algebraic and radical equations.
3. Find domains, compositions, and inverses of functions.
4. Evaluate a limit and find a derivative.
5. Analyze a rational function for stationary points, asymptotes, and sketch the graph.
6. Find the tangent to a trigonometric curve at a specific point.

Part 2: Advanced Problems 7. Analyze and plot a circle and line intersection, then find the exact points of intersection. 8. Identify where a logarithmic function is undefined and solve a related inequality. 9. Determine the maximum and minimum values of a trigonometric function over a given interval.

Each problem requires detailed solutions, and calculators are not permitted.

Would you like assistance with a specific problem or topic?

---

Related Questions:

1. How do you find the stationary points of a rational function?
2. What steps are needed to solve logarithmic inequalities?
3. How is the tangent to a curve calculated at a specific point?
4. What are typical methods to solve radical equations?
5. How can we determine the maximum and minimum values of functions on a closed interval?

Tip:

Always check for domain restrictions when working with logarithmic and rational functions, as undefined points may impact solutions.