https://file.aiquickdraw.com/mathbot/file/1726647696269gfhpxwnw.jpg

Let's go through the questions visible in the image one by one. 

### Task 10:
- **Situation**: A car accelerates from rest. In the first 10 seconds, its speed increases uniformly, then between 10 and 20 seconds, the speed increases less strongly, after which it remains constant.
- **Questions**:
  - **a)** Sketch a possible graph of velocity \(v(t)\).
  - **b)** Investigate the sign of \(v'(t)\) (the derivative of the velocity).
  - **c)** Interpret the given statements \(v(2) = 3\), \(\frac{v(15) - v(12)}{15 - 12} = 1.2\), and \(v'(15) = 1\) in the context of the problem.

### Task 11:
- **Situation**: The function \(f(x)\) and a point \(P\) on the graph \(G_f\) of \(f\) are given. Determine the equation of the tangent to \(G_f\) at \(P\), and calculate the slope angle.
- **Questions**:
  - **a)** \(f(x) = \sin(x) + x^2\); \(P(0 | f(0))\)
  - **b)** \(f(x) = x^2 - 4x + 5\); \(P(3 | f(3))\)
  - **c)** \(f(x) = 2 \cdot \cos(x) - x\); \(P\left(\frac{\pi}{2} \middle| f\left(\frac{\pi}{2}\right)\right)\)

### Task 12:
- **Situation**: Investigate whether the graphs of functions \(f(x)\) and \(g(x)\) touch at the given points.
- **Questions**:
  - **a)** \(f(x) = x^3 - 3x\), \(g(x) = x^2 - 2x - 1\) at \(x_1 = 1\) and \(x_2 = -1\).
  - **b)** \(f(x) = 0.5x^4\), \(g(x) = x^4 - 2x^2\) at \(x_1 = -2\), \(x_2 = 0\), and \(x_3 = 2\).

### Task 13:
- **Situation**: A 5-meter-wide canal is represented by the graph \(G_f\) of a function \(f(x) = ax^2 + bx\), where \(a\) and \(b\) are real numbers. The task is to determine values for \(a\) and \(b\) such that the edges of the canal in the model are represented by points \(P(0 | 0)\) and \(Q(5 | 0)\), and the canal forms a 30-degree angle with the ground at this point.

Would you like to work on any particular question or part of these problems in more detail? If so, I can help guide you through the steps or explain the solution. 

Here are five relative questions:
1. What does the graph of a uniformly increasing speed look like?
2. How do you compute the slope of a tangent at a given point for a function?
3. What conditions must be met for two graphs to touch at a point?
4. How is the equation of a tangent related to the derivative of a function?
5. How can trigonometry be used to solve the problem involving the canal and angles?

**Tip**: When interpreting a velocity-time graph, the slope of the graph at any point represents the acceleration at that moment.

10. A car accelerates from rest. In the first 10 seconds, its speed increases uniformly, then between 10 and 20 seconds, the speed increases less strongly, after which it remains constant. 
a) Sketch a possible graph of velocity v(t). 
b) Investigate the sign of v'(t) (the derivative of the velocity). 
c) Interpret the given statements v(2) = 3, (v(15) - v(12)) / (15 - 12) = 1.2, and v'(15) = 1 in the context of the problem.

Explore a kinematics problem involving a car's acceleration. Learn how to sketch a velocity-time graph, analyze the sign of the velocity derivative, and interpret key statements in the context of motion. Suitable for students in Grades 11-12.

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