The concept of the derivative

A content creator analyses the growth of their followers on a social network over the past week. The number of followers $S$ as a function of day $t$ is given by the expression $S(t) = t^2 + 10t + 500$. Calculate the Average Rate of Change (ARC) of their followers between day $t=1$ and day $t=4$. Explain in your own words what the result obtained means in the context of the problem.

The speed of an electric scooter during the first few seconds of a journey follows the function $v(t) = 0,5t^2 + 2$, where $v$ is the speed in m/s and $t$ is the time in seconds.

1. a) Calculate the ARC in the intervals $[2, 3]$ and $[2, 2,1]$.
2. b) What value do you think the rate of change approaches as the interval becomes smaller and smaller around $t=2$?

Given the linear function $f(x) = 4x - 3$, use the definition of a derivative at a point using the limit:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

to calculate the value of the derivative at the point $x=2$. Check whether the result coincides with the gradient of the line.

Calculate, using strictly the definition of a derivative at a point (using the limit as $h \to 0$), the value of $f'(3)$ for the function:

$$f(x) = 2x^2 - 5x + 1$$

Find the equation of the tangent line to the graph of the function $f(x) = x^2 + 1$ at the point with x-coordinate $x=1$, knowing that the value of the derivative at that point is $f'(1) = 2$.

(Remember that the equation of the tangent line is $y - f(a) = f'(a)(x - a)$).

The concept of a derivative allows us to approximate the value of a function near a known point. Observe the following diagram showing a function and its tangent line at the point $P(1, 2)$.

If we know that the equation of the orange tangent line is $y = 3x - 1$:

1. a) What is the value of the derivative of the function at $x = 1$? Justify your answer.
2. b) If we wanted to know the value of the function at $x = 1,1$, what value would we obtain if we used the tangent line to approximate it?
3. c) Do you think that for $x = 5$ the tangent line would still be a good approximation of the function? Explain why.

A function is said to be "differentiable" at a point if its graph does not have any jumps or sharp points at that location. Analyse the following piecewise function:

$$f(x) = \begin{cases} x^2 + x & \text{if } x \le 1 \\ 3x - 1 & \text{if } x > 1 \end{cases}$$

1. a) Check whether the function is continuous at $x = 1$.
2. b) Calculate the one-sided derivatives at $x = 1$ (that is, the derivative of each section at that point).
3. c) From the previous results, is the function differentiable at $x = 1$? Justify your answer by relating continuity and the existence of one-sided derivatives.

Match each of the points marked on the graph ($A$, $B$ and $C$) with the sign of the derivative of the function at that point (positive, negative or zero). Justify your answer by mentally drawing the tangent line at each point.

The following graph shows the profile of a roller coaster on a specific section. It is known that the slope of the track at a point determines the passengers' sensation of acceleration.

Based on the geometric concept of the derivative:

1. a) Is the derivative zero at any of the three marked points ($A$, $B$, or $C$)? What does it mean for the derivative to be zero with respect to the slope of the track?
2. b) State whether the derivative at point $B$ is positive or negative. Justify whether the roller coaster car is ascending or descending at that moment.
3. c) Order the values of the derivatives at points $A, B$, and $C$ from lowest to highest.