Concepts: 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).

A Year 13 student trains for the 100-metre sprint. The distance covered (in metres) 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 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?

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:

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 ($OY$).

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.

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.

In a Leaving Certificate exam, Seán must analyze the differentiability at $x = 1$ of the following piecewise-defined function:

Seán writes in his solution:

"I differentiate each branch directly: $f'(x) = 2x + 2$ for $x \le 1$, and $f'(x) = 4$ for $x > 1$.

I calculate the one-sided derivatives by substituting $x = 1$:

$f'(1^-) = 2(1) + 2 = 4$

$f'(1^+) = 4$

Since the one-sided derivatives exist and coincide ($4 = 4$), I conclude that the function is differentiable at $x = 1$ and that the tangent line at that point has slope $m = 4$".

1. a) Identify and explain in detail the serious conceptual error Seán makes in his reasoning.
2. b) Using the formal definition of the derivative via the limit of the difference quotient from the right, $\lim_{h \to 0^+} \frac{f(1+h) - f(1)}{h}$, calculate the real value of this limit and show what actually happens to $f$ at $x = 1$.
3. c) Correct Seán's conclusion by stating whether the function is continuous, whether it is differentiable, and whether it makes sense to speak of a tangent line at $x = 1$.

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.

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.

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 behaviour of the graph near the origin.