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.

Look at the following graph, which represents the change in the price of a cryptocurrency over a morning. Read from it the values of $f(1)$ and $f(5)$ and calculate the Average Rate of Change over the interval $[1, 5]$, interpreting it as the slope of the secant line that passes through those two points.

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 $t=2$?

A streaming platform has analysed the growth of its subscribers during the first months of the year. The function estimating the number of subscribers (in thousands) is given by $S(t) = t^2 + 2t + 10$, where $t$ is the time measured in months from the beginning of the year ($t=0$ is 1 January).

1. a) Calculate the Average Rate of Change (ARC) of the number of subscribers between the second month ($t=2$) and the fifth month ($t=5$).
2. b) Explain the meaning of the result obtained in the context of the problem.
3. c) Would it be correct to state that the platform gained exactly that number of subscribers each month during that period? Reason your answer.

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

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

A small cell phone case manufacturing company has estimated that its daily costs (in pesos) depend on the number of units manufactured $x$ according to the function $C(x) = x^2 - 4x + 100$.

Calculate the Instantaneous Rate of Change of the cost when exactly 5 units are produced, applying the definition of derivative:

Calculate, using the limit definition, the derivative function of the constant function $f(x) = 7$. Based on the geometric interpretation of the derivative as the "rate of change" or "gradient", explain why it makes sense for the result to be $0$.

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:

Given the function $f(x) = \frac{4}{x}$:

1. a) Find the Instantaneous Rate of Change (IRC) at the point with x-coordinate $x = 2$ using the limit of the Average Rate of Change.
2. b) Determine the equation of the tangent line to the graph of the function at that point $x = 2$.

The number of followers of a video game streaming channel follows the function $S(t) = 2t^2 - 4t + 500$, where $t$ is the time in months.

1. a) Calculate the Average Rate of Change ($TVM$) in the generic interval $[2, 2 + h]$.
2. b) Using the previous result, deduce the value of the limit as $h \to 0$. What does this value physically represent for the streamer in month $t = 2$?
3. c) Is there any value of $h$ for which the $TVM$ in the interval $[2, 2 + h]$ is equal to $10$ followers/month?

Calculate, using strictly the definition of a derivative as a limit, the derivative function of:

From the result obtained, determine the coordinates of the point or points on the graph where the gradient of the tangent line is exactly $-3$. Explain why there cannot be points where the gradient of the tangent line is positive.

At a hot air balloon festival, the height of a hot air balloon (in meters) with respect to time (in seconds) since it begins its ascent is given by the function $H(t) = \sqrt{t + 4}$.

1. a) Calculate the derivative $H'(t)$ using the definition of a limit.
2. b) Find the equation of the tangent line to the trajectory at the instant $t = 5$.
3. c) For safety reasons, the festival regulations require that the balloon does not exceed an ascent rate of $0.25$ m/s. Determine at what instant the instantaneous speed is exactly that value.

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.

Consider the function $f(x) = ax^2 + bx + c$. Determine the values of $a$, $b$ and $c$ knowing that:

1. The graph of $f(x)$ passes through the point $(0, 5)$.

2. The tangent line to the curve at the point with x-coordinate $x = 1$ is the line $y = 2x + 1$.

3. Use the derivative to check whether the vertex of the resulting parabola lies to the left or to the right of the $Y$-axis.

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:

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.

Study the differentiability of the following function at the point $x = 2$. In order for the function to represent the production cost of a new electronic component smoothly (without jumps or sharp points), determine the values of the parameters $a$ and $b$:

Remember that for it to be differentiable at $x = 2$, it must first be continuous at that point.

Given the function $f(x) = |x^2 - 4|$, analyse its differentiability at the points $x = 2$ and $x = -2$. The figure shows its graph.

1. a) Express the function as a piecewise-defined function.

1. b) Calculate the one-sided derivatives at $x = 2$ using the limit definition. (At $x = -2$ the reasoning is analogous, by the symmetry of the graph.)

1. c) Explain graphically what happens at those points (the concept of a "corner point") and why the derivative does not exist even though the function is continuous.

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.

A small clothing store has estimated that its daily profits (in pesos) depend on the number of discounted items ($x$) according to the function $B(x) = -x^2 + 40x - 300$.

1. a) Calculate the derivative function $B'(x)$ using differentiation rules.
2. b) Calculate $B'(15)$ and $B'(25)$.
3. c) Interpret the results from the previous section: Is it advantageous for the store owner to increase the number of discounted items if they currently have 15? What if they have 25? Justify your answer using the concept of instantaneous change.

Find the points on the graph of the function $f(x) = x^3 - 3x^2 + 5$ where the tangent line is horizontal.

Hint: Remember what value the gradient (and therefore the derivative) must have for a line to be horizontal.

Find the points of the curve $y = x^3 - 3x^2 - 9x + 2$ at which the tangent line is parallel to the x-axis. Once you have found them, use the interpretation of the derivative to reason whether at those points the function has a relative maximum, a relative minimum, or an inflection point, by analyzing the sign of the derivative near those points.

It is known that the revenue function of an agricultural cooperative is $I(x) = -x^2 + 12x$, where $x$ are the metric tons of product sold.

1. a) Calculate the derivative function $I'(x)$ and interpret it as the "Marginal Revenue" (the approximate revenue from selling one additional unit).
2. b) Show that the revenue is maximum when the Marginal Revenue is equal to zero.
3. c) What happens to the revenue function if the derivative is negative? Justify your answer in terms of increasing and decreasing.