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The document you uploaded contains a midterm assignment for a Calculus-II (MAT 1103) course at United International University. Here are the questions in the assignment:

1. **Graph of Second Derivative:**
   - (a) Determine the intervals where the function \( f \) is concave upward or concave downward based on the given graph of \( f''(x) \).
   - (b) Find the x-coordinates of the inflection points of \( f \).

2. **Absolute Maximum and Minimum:**
   - Find the absolute maximum and minimum values of the function \( f(x) = 12 - 4x + x^2 \) on the interval \([0, 5]\) using the closed interval method. Also, express the function as a perfect square and use graph transformations to sketch it.

3. **Curve Sketching:**
   - Sketch the curve of \( y = \frac{x}{x - 1} \) following these guidelines:
     1. Domain
     2. Intercepts
     3. Symmetry
     4. Asymptotes (vertical and horizontal)
     5. Intervals of increase or decrease
     6. Local maximum or minimum values
     7. Concavity and points of inflection
     8. Sketch the curve

4. **Limit Calculation:**
   - Find \( \lim_{x \to 0^+} \frac{\tan x - \sin x}{x^2} \).

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Would you like a detailed solution or explanation for one of the questions? Here are five related questions you might find interesting:

1. How do you determine concavity and inflection points from \( f''(x) \)?
2. What is the closed interval method, and how is it used to find absolute extrema?
3. How do you find asymptotes and sketch the graph of a rational function?
4. What is the difference between vertical and horizontal asymptotes?
5. How can the limit of a difference between trigonometric functions be calculated near zero?

**Tip:** When sketching curves, always identify key features like domain, intercepts, and asymptotes to guide your sketch effectively.

The document contains a midterm assignment for a Calculus-II (MAT 1103) course at United International University. The questions involve concepts like concavity, inflection points, absolute maximum and minimum values, curve sketching, and limit calculation.

This assignment covers key calculus concepts such as finding concavity and inflection points using second derivatives, calculating absolute maximum and minimum using the closed interval method, and sketching curves with domain, intercepts, asymptotes, and more. Suitable for university-level calculus students.

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