Rate of Change and Derivative of a Function at a Point

Calculate the average rate of change (ARC) of the function $f(x) = x^2 - 4x + 5$ on the interval $[1, 4]$. Explain graphically what the value obtained represents in relation to the secant line that passes through the points $A$ and $B$ of the curve corresponding to the endpoints of that interval (they are marked in the figure).

The growth in followers of a new account on a popular social network follows the function $S(t) = t^2 + 4t + 100$, where $t$ is the time in days since its creation and $S(t)$ is the number of followers. The figure shows the graph of $S$ and the lines joining the points of the curve corresponding to $t=0$, $t=5$ and $t=10$.

1. a) Calculate the average rate of change (ARC) of the number of followers over the intervals $[0, 5]$ and $[5, 10]$.
2. b) Explain the meaning of the results you obtained in the context of the account's growth. In which interval has the growth been faster?

Observe the following graph of a function $f(x)$. Without performing calculations, order the following values from smallest to largest, justifying your answer based on the steepness of the lines:

• The average rate of change in the interval $[1, 3]$.

• The derivative of the function at the point $x = 1$.

• The derivative of the function at the point $x = 3$.

Given the function $f(x) = 3x^2 - 1$, use the definition of derivative at a point via the limit:

to find the value of the derivative at the point with x-coordinate $x = 2$.

Consider the rational function $f(x) = \frac{2}{x}$.

Show, applying the definition of derivative as a limit, that the value of the instantaneous rate of change at $x = 1$ is $f'(1) = -2$. Interpret the sign of the result obtained.

Calculate the derivative of the function $f(x) = \dfrac{3}{x + 1}$ at the point with x-coordinate $x = 2$ using strictly the definition of derivative via the limit:

A Year 13 student trains for the 100-meter sprint. The distance covered (in meters) as a function of time (in seconds) is given by the expression $s(t) = 0.5t^2 + 2t$.

Calculate the average speed of the athlete between $t = 2$ s and $t = 6$ s. Which mathematical concept does this value correspond to?

The number of subscribers of a new YouTube channel evolves according to the function $S(t) = t^2 + 10t + 100$, where $t$ is the time in days since its creation.

1. a) Calculate the average rate of change of subscribers between day $t = 0$ and day $t = 10$.
2. b) What is the instantaneous growth of subscribers on exactly day $t = 5$? (Calculate $S'(5)$ using the definition).

The temperature of a metal plate (in degrees Celsius) varies with time $t$ (in minutes) following the model $T(t) = 20 + 4t - 0.1t^2$.

Calculate the instantaneous rate of change of the temperature at time $t = 10$ minutes. State the units of measurement of the result and explain whether the plate is heating up or cooling down at that moment.

A skater moves along a track whose position (in meters) with respect to the start is given by the expression $e(t) = \sqrt{4t + 9}$, where $t$ is the time in seconds.

1. a) Calculate the average velocity of the skater between $t = 0$ and $t = 4$ seconds.
2. b) Find the instantaneous velocity at $t = 4$ seconds using the definition of derivative.

Given the function $f(x) = \dfrac{x^2}{x - 1}$:

1. a) Calculate the average rate of change in the interval $[2, 4]$.
2. b) Determine whether there is any point $c$ within the interval $(2, 4)$ such that the instantaneous derivative $f'(c)$ coincides with the average rate of change calculated in the previous part.

The temperature of an electronic component during the first hours of operation follows the function $T(t) = 20 + t \times e^{-t}$, where $T$ is the temperature in degrees Celsius and $t$ is the time in hours.

Calculate the instantaneous rate of change of temperature when exactly 1 hour has elapsed ($t = 1$). Is the temperature increasing, decreasing, or neither at that instant? Justify your answer based on the sign of the derivative.

A forest surveillance drone follows an approach trajectory whose altitude $h$ (in meters) with respect to time $t$ (in seconds) is given by the function:

Find:

1. Calculate the average rate of change of the height in the interval $[0, 5]$. Explain its physical meaning.

2. Using the definition of a derivative at a point, obtain the instantaneous rate of ascent at time $t = 5$ seconds.

3. Check whether there is any time $t > 0$ at which the instantaneous rate of ascent matches exactly the average rate of change calculated in part 1.

In a biotechnology laboratory, the population of a bacterial colony (in millions) is modelled by the function:

where $t$ is the time in hours.

1. Find the instantaneous rate of change of the population at $t = 1$ hour and at $t = 4$ hours.

2. Determine at which of these two times the population is growing faster.

3. Is there any time at which the rate of growth is zero? If so, calculate that value of $t$ and interpret what happens to the population at that moment.

Determine the gradient of the tangent line to the graph of the function $f(x) = \sqrt{x+1}$ at the point where $x = 3$, applying the limit of the instantaneous rate of change:

Consider the function $f(x) = x^2 - 6x + 5$. Determine the point on the graph where the tangent line has a gradient of $m = 2$. Once the point has been found, write the complete equation of this tangent line in point-slope form.

Find the equation of the tangent line and of the normal line to the curve $f(x) = x^3 - 3x^2 + 2$ at the point where the graph crosses the y-axis ($OY$).

Find the equations of the tangent lines to the curve $y = x^2 - 4x + 5$ that pass through the external point $A(0, 0)$ (the figure shows only one of them).

Note: Note that point A does not lie on the curve. You will need to set the gradient of the tangent at a general point $(a, f(a))$ equal to the gradient of the line joining that point to the origin.

Let $f(x) = \frac{1}{x+1}$ be the function. Prove, applying the definition of derivative, that its derivative function is $f'(x) = \frac{-1}{(x+1)^2}$ for any value of $x$ belonging to its domain.

From this result, determine the coordinates of the points on the graph where the tangent line is parallel to the line $y = -x$ (the bisector of the second quadrant).

Given the piecewise function:

Calculate the values of $a$ and $b$ so that the function is differentiable at the point $x = 1$. Remember that for a function to be differentiable, it must first be continuous.

Given the function $f(x) = x \times |x - 3|$, study its differentiability at the point $x = 3$.

To do this:

1. Express the function as a piecewise function.

2. Calculate the one-sided derivatives $f'(3^-)$ and $f'(3^+)$ using the limit of the difference quotient.

3. Interpret the result geometrically: does the graph have a sharp point (corner) or a smooth tangent at $x=3$? Justify your answer.

Consider the function $f(x) = \sqrt[3]{x^2}$.

1. Study the continuity of the function at the origin ($x=0$).

2. Analyse differentiability at $x=0$ by calculating the limit of the difference quotient.

3. If the limit tends to infinity, what type of tangent line does the function have at that point? Describe the visual behavior of the graph near the origin.

Consider the piecewise function:

Determine the values of the real parameters $a$ and $b$ so that the function is differentiable at the point $x = 2$. To do this, you must first ensure continuity and then the existence of equal, finite one-sided derivatives at that point using the definition of derivative.

Calculate, using exclusively the definition of a derivative as the limit of the difference quotient, the value of the following limit:

Hint: Identify this limit with the derivative of a known function at a given point and solve using standard differentiation rules to verify your result.