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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?

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### 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.

This is a sample exam for the undergraduate course in Mathematical Analysis 1. It includes problems on algebraic equations, limits, derivatives, rational functions, tangent lines, circle equations, logarithmic inequalities, and function optimization. Each section requires detailed solutions without calculators. Ideal for undergraduate students.

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