https://file.aiquickdraw.com/mathbot/file/1727364471547uddnnov6.pdf

The document you uploaded seems to be a seminar worksheet for a **Cálculo I** (Calculus I) course. Below is a summary of the main questions covered:

1. **Relating Functions to Graphs**: 
   Match the given graphs with the functions:
   \[
   f_1(x) = x^3, \quad f_2(x) = -x^2, \quad f_3(x) = e^x, \quad f_4(x) = x^4, \quad f_5(x) = e^{-x}
   \]

2. **Curves of Logarithmic Functions**:
   Identify the curves of the logarithmic functions \(L(-x)\), \(L(x)\), and \(L(-x) - x\).

3. **Domain of the Logarithmic Functions**:
   Determine the domains for the curves identified above.

4. **Identifying Trigonometric Functions**:
   Based on graphs, distinguish between:
   - \(\sin(-x)\), \(\sin(x)\)
   - \(\cos(-x)\), \(\cos(x)\)
   Discuss any relationships between them.

5. **True/False Statements**:
   Analyze and reason about the following:
   - \(L(-x)\) is undefined because the variable is negative.
   - \(e^{-x}\) has negative values when \(x < 0\).
   - \(1 \leq \sin(x) \leq 1\) for all \(x \in \mathbb{R}\).
   - \(1 \leq \arcsin(x) \leq 1\) for all \(x \in \mathbb{R}\).

6. **Identifying Sine Functions**:
   Identify functions from graphs:
   - \(2\sin(x)\), \(\sin(2x)\), \(\sin(x/2)\), \(\frac{\sin(x)}{2}\).

7. **Step Functions**:
   Identify and define the step functions, including \(h(x) = \frac{1}{x}\).

8. **Comparing Growth of Functions**:
   Identify functions like \(f(x) = x^3\) and \(g(x) = \frac{1}{x^3}\) and discuss their growth behavior.

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Do you want further explanations for any specific question or help with solving any of these problems?

Here are 5 follow-up questions to expand on the content:

1. How do we find the domain of the logarithmic functions in question 3?
2. What is the geometric interpretation of even and odd functions from the graphs in question 4?
3. Can you prove why \(L(-x)\) might be undefined for negative values of \(x\)?
4. What transformations occur between \(\sin(x)\), \(\sin(2x)\), and \(\frac{\sin(x)}{2}\)?
5. How do \(f(x) = x^3\) and \(g(x) = \frac{1}{x^3}\) compare in terms of increasing and decreasing behavior?

**Tip:** When analyzing trigonometric functions, recall that even functions are symmetric around the y-axis, while odd functions are symmetric around the origin.

The document uploaded by the user appears to be a seminar worksheet for a Calculus I course covering various mathematical concepts including functions, graphs, logarithmic functions, trigonometric functions, and growth behavior of functions.

Explore a detailed seminar worksheet from Calculus I covering topics such as identifying functions from graphs, analyzing logarithmic and trigonometric functions, determining domains, and comparing growth behavior of functions like x^3 and 1/x^3. Ideal for university-level students.

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