The concept of the derivative

The following table shows the number of registered users on a new mobile app during its first months after launch:

• Month 1: $100$ users

• Month 2: $250$ users

• Month 3: $500$ users

• Month 4: $850$ users

• Month 5: $1300$ users

Calculate the average rate of change (ARC) of the number of users between months $2$ and $4$.

Given the function $f(x) = x^2$, calculate the average rate of change in the intervals $[3; 3,1]$, $[3; 3,01]$ and $[3; 3,001]$. In light of the results obtained, deduce what numerical value the ARC tends to as the width of the interval approaches zero, and state what this value is called mathematically.

Using the definition of derivative as the limit of the average rate of change, $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, calculate the derivative of the function $f(x) = x^2 + 3$ at the point $x=2$.

Calculate the derivative of the linear function $f(x) = 5x - 7$ at the point $x=10$ by applying the definition of derivative using limits. Does the result depend on the chosen point?

Making strict use of the definition of derivative by calculating the limit, find the value of $f'(1)$ for the function $f(x) = 2x^2 - 3x + 1$.

Observe the graph of the function $f(x)$ and the secant line drawn between the points with x-coordinates $x=1$ and $x=3$. Calculate visually the average rate of change (ARC) in the interval $[1, 3]$ using the values projected on the axes.

From the graph of the function $f(x)$ and its tangent line at the point with x-coordinate $x=2$, determine the value of the derivative $f'(2)$.

Find the equation of the secant line to the curve $f(x) = -x^2 + 6x$ passing through the points of intersection of this curve with the x-axis. (Note: calculate these x-intercepts first).

Find the equation of the tangent line to the graph of the parabola $f(x) = x^2 - 4x + 5$ at the point with x-coordinate $x=3$. First calculate the gradient using the definition of derivative with limits.

Knowing that the derivative of a function $f(x)$ at the point $x=4$ is $f'(4) = -2$, state with reasons whether the function is increasing or decreasing at that point. Briefly justify the geometric meaning of this value with respect to the tangent line to the curve at that point.

Given the function $f(x) = -2x^2 + 12x - 5$, determine whether the function is increasing or decreasing at $x=4$, having first calculated the instantaneous rate of change at that point using the definition of limit.

Observe the graph of the function $f(x)$ and its tangent line drawn at the point with x-coordinate $x=4$. Write the equation of the tangent line and deduce the value of $f'(4)$. What happens to the growth of the function at that exact point?

The position of a cyclist on a straight track is given by the function $s(t) = 3t^2 + 2t$ (where $s$ is measured in metres and $t$ in seconds). Calculate the average speed of the cyclist in the time interval between $t=1$ and $t=4$ seconds.

The total distance covered by a cyclist descending a mountain pass (in metres) as a function of time is given by the function $e(t) = 150 + 2t + 0,5t^2$ ($t$ in seconds). Calculate the average rate of change of the distance from $t = 20$ to $t = 30$ seconds and interpret the physical meaning of the result obtained (the average speed over that section).

An object is dropped from the top of a building. Its height in metres after $t$ seconds is given by the expression $h(t) = 45 - 5t^2$. Calculate the instantaneous speed of the object at exactly $t=2$ seconds using the definition of derivative.

The number of subscribers to a video channel is given by the expression $S(x) = -0,1x^2 + 50x - 1000$, where $x$ is the number of days elapsed since its launch. Calculate $S'(200)$ using strictly the definition of derivative via limits, and interpret what this value indicates about the rate of growth of the channel at that moment.

Find the equation of the tangent line to the curve $f(x) = \frac{6}{x}$ at the point with x-coordinate $x=3$. To do so, first calculate the gradient by applying the definition of derivative as the limit of the average rate of change.

Find the exact point on the parabola $f(x) = x^2 - 4x + 7$ where the tangent line to the curve is parallel to the line with equation $y = 2x - 5$. Write down the full coordinates $(x, y)$ of this point.

Determine the values of the parameters $a$ and $b$ of the polynomial function $f(x) = ax^2 + bx$ knowing that its graph passes through the point $(1, 2)$ and that the slope of its tangent line at $x=1$ is equal to $3$.

Calculate the derivative of the radical function $f(x) = \sqrt{x+5}$ at the point $x=4$ using the definition of derivative via limits. (Hint: to evaluate the resulting limit, multiply and divide by the conjugate expression).

The vertex of a parabola corresponds to the point where its tangent line is horizontal (gradient equal to zero). Using the definition of derivative via limits, find the $x$-coordinate of the vertex of the parabola $f(x) = -2x^2 + 12x - 9$ by setting its derivative to zero.

The volume of a cube as a function of the length of its edge $x$ is given by $V(x) = x^3$. Calculate the instantaneous rate of change of the volume with respect to the edge when $x=5$ cm, using the definition of limit. If the error in measuring the edge is very small, how does it proportionally affect the calculation of the volume?

From the graph of the function $f(x)$ and the three tangent lines drawn at the points $A$, $B$ and $C$, order the values of the derivatives $f'(x_A)$, $f'(x_B)$ and $f'(x_C)$ from smallest to largest. Justify your answer based on the geometric gradient of each tangent line.