Let's address each problem step-by-step:

Problem №2: Interpretation of the graph

The graph shows the dependence of a variable $y$ on $x$. To answer the given questions:

• a) What is the value of $y$ if $x = -3; -1; 2; 5$?

  To find these values, we need to check where the graph crosses or touches these specific $x$-coordinates.

  • For $x = -3$, trace vertically from $x = -3$ to the curve. The corresponding value of $y$ is $y = 1$.
  • For $x = -1$, trace vertically from $x = -1$. The corresponding $y$-value is $y = -3$.
  • For $x = 2$, trace vertically from $x = 2$. The value of $y$ is $y = 0$.
  • For $x = 5$, trace vertically from $x = 5$. The corresponding $y$-value is $y = 2$.

• b) What are the values of $x$ if $y = 3; 0; 1$?

  • For $y = 3$, trace horizontally from $y = 3$. The graph crosses at $x = 4$.
  • For $y = 0$, trace horizontally from $y = 0$. The values of $x$ are $x = 1$ and $x = 2$ (since the graph touches the $x$-axis at these points).
  • For $y = 1$, trace horizontally from $y = 1$. The values of $x$ are $x = -3$ and $x = 5$.

• c) What is the minimum and maximum value that $y$ takes?

  By observing the graph:

  • The minimum value of $y$ is $y = -3$ (occurs at $x = -1$).
  • The maximum value of $y$ is $y = 3$ (occurs at $x = 4$).

Problem №3: Bicycle motion based on the graph

The graph in this problem shows the cyclist’s motion over time (distance vs time).

• a) How much time did the cyclist take to cover 1 km on the descent?

  From the graph, the descent (distance decreasing from 2 km to 0 km) happens between 25 and 35 minutes. The total descent distance is 2 km over 10 minutes. Therefore:

  • The cyclist descends 1 km in $\frac{10}{2} = 5$ minutes.

• b) What was the cyclist's speed (in km/h) on the ascent?

  The ascent happens from 5 minutes to 15 minutes, where the cyclist travels from 0 km to 2 km. Thus, the time for ascent is $15 - 5 = 10$ minutes, which is $\frac{10}{60} = \frac{1}{6}$ hours.

  • The distance covered is 2 km, so the speed is: $\text{Speed} = \frac{2 \text{ km}}{1/6 \text{ hours}} = 2 \times 6 = 12 \text{ km/h}.$

Would you like any more details or have any questions about these solutions?

Here are 5 additional related questions to deepen your understanding:

1. How would the graph change if the cyclist stopped for a while?
2. Can we determine the average speed of the cyclist for the entire journey?
3. How can the shape of the graph tell us about the cyclist's acceleration?
4. How would the graph look if the cyclist maintained a constant speed?
5. How can we use derivatives to analyze changes in motion based on this graph?

Tip: In problems involving motion graphs, always check the slopes of the graph. The steeper the line, the higher the speed.