Exercises: 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?

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 $x = 2$.

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:

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).

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.

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 $x = 3$.

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 $x = 1$.

A model toy rocket is launched vertically. Its height above the ground in meters is described by the function $h(t) = 24t - 4t^2$, where $t$ is the time elapsed in seconds since launch ($t \in [0, 6]$).

1. a) Without performing algebraic calculations, estimate with justification the sign that the average rate of change ($\text{ARC}$) will have on the interval $[2, 4]$. To do so, consider the parabolic shape of the trajectory and identify the instant at which the rocket reaches its maximum height. Do you expect the $\text{ARC}[2, 4]$ to be positive, negative, or zero?
2. b) Analytically calculate the $\text{ARC}[2, 4]$ using its formula and check whether your initial estimate was correct. In view of the result, is it correct to deduce that the rocket remained stationary during those two seconds? Explain the physical and geometric meaning of the value obtained in relation to the secant line.
3. c) Using the definition of derivative as the limit of the difference quotient, calculate the instantaneous velocity $h'(2)$. Check whether the sign and value obtained are consistent with the rocket's position at that instant.

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?

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:

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.

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.

A skater moves along a track whose position (in metres) 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.

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 ($OY$).

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.

During an extreme heat event in Cork, Met Éireann records that the temperature $T$ (in degrees Celsius) during daytime hours is modeled with high accuracy by the function:

where $t$ represents the solar hour of the day, with $t \in [10, 18]$ (from 10:00 to 18:00).

1. a) Calculate the average rate of change of the temperature on the interval $[10, 14]$. Express the result with its corresponding units and interpret what information it provides about the overall thermal evolution during those four hours.
2. b) Strictly applying the definition of derivative via the limit of the difference quotient, calculate the instantaneous rate of change of the temperature at exactly 14:00 ($t = 14$).
3. c) Numerically compare the $\text{ARC}[10, 14]$ with $T'(14)$. Explain with justification why the instantaneous rate at 14:00 is lower than the average of the four preceding hours, relating this change to the proximity to the afternoon's maximum temperature point.

Determine the real coefficients $a$, $b$, and $c$ of a second-degree polynomial function $f(x) = ax^2 + bx + c$ from the following experimental information about its graph:

1. Its graph intersects the y-axis at the point $P(0, 2)$.

2. The average rate of change on the interval $[0, 3]$ is $\text{ARC}[0, 3] = 5$.

3. The normal line to the graph of $f$ at the point with abscissa $x = 1$ is the line with equation $x + 3y - 10 = 0$.

1. a) From the equation of the normal line, deduce with justification the value of the ordinate $f(1)$ and the value of the slope of the tangent line $f'(1)$.
2. b) Obtain the expression for the derivative $f'(1)$ in terms of the parameters by applying the definition of derivative as the limit of the difference quotient $\lim_{h \to 0} \frac{f(1+h)-f(1)}{h}$, and set up the system of equations that allows $a$, $b$, and $c$ to be uniquely determined.
3. c) Write the explicit equation of the tangent line to the curve at the point with abscissa $x = 1$.

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.

A forest surveillance drone follows an approach trajectory whose altitude $h$ (in metres) 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.

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.

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).

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.

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.

Consider the continuous function $f(x) = \sqrt[3]{x - 2} + 1$. Two 6th Year students discuss the existence of the tangent line to the graph of $f$ at the point with abscissa $x = 2$:

• Aoife (Differentiation rules method): She applies the power and chain rules, obtaining $f'(x) = \frac{1}{3\sqrt[3]{(x-2)^2}}$. Substituting $x = 2$, she observes that the denominator becomes zero ($\frac{1}{0}$), so she concludes: "Since there is no real value for $f'(2)$, the graph has no tangent line at $x = 2$".

• David (Formal definition method): He states that mechanical rules are insufficient when dealing with singularities and sets up the limit of the difference quotient:

1. a) Carry out David's method by calculating this limit as $h \to 0^+$ and as $h \to 0^-$. What behavior do the slopes of the secant lines show as they approach the point $(2, 1)$?
2. b) Compare both methods and critically assess Aoife's conclusion. Is it true that there is no tangent line, or is there a vertical tangent? Write, with justification, the equation of the tangent line and that of the normal line at that point.
3. c) Justify for what type of functions or singular points it is essential to resort to the formal definition method instead of merely applying the mechanical differentiation rules.

In the design project for a high-speed railway line, engineers must design a smooth transition between a straight section and a parabolic curved section to avoid abrupt accelerations on the train. The profile of the route is modeled using the function:

To ensure safety and comfort, the technical specifications require three conditions:

1. That the connection at $x = 1$ be continuous and differentiable (transition without sharp bends).

2. That on the curved section, the average rate of change between $x = 1$ and $x = 3$ be zero: $\text{ARC}[1, 3] = 0$.

3. That the tangent line to the curve at $x = 3$ be perpendicular to the line $y = \frac{1}{2}x$.

1. a) Formally model the three requirements of the specifications: express the differentiability at $x = 1$ via the equality of one-sided derivatives using the definition of the difference quotient, translate the ARC condition, and formulate the perpendicular tangency condition at $x = 3$.
2. b) Solve the system obtained and determine the values of the parameters $a$, $b$, and $c$.
3. c) Reason geometrically why, in a second-degree polynomial function, the condition $\text{ARC}[1, 3] = 0$ together with $f'(3) = -2$ necessarily forces $f'(1^+) = 2$, making differentiability at the junction point automatically satisfied without the need to enforce it independently.