Exercises: Derivative of a function at a point

A content creator has observed that the number of followers of their channel during the first week follows the function $f(t) = t^2 + 2t + 100$, where $t$ is time in days.

Calculate the Average Rate of Change (AROC) of the number of followers between the second day ($t = 2$) and the fifth day ($t = 5$). What does this value mean in the context of the problem?

Observe the following graph of the function $f(x)$ and the tangent line to it at point $P$.

From the visual information, state with reasoning what the value of the derivative of the function is at the point with x-coordinate $x = 2$, that is, find $f'(2)$.

Given the polynomial function $f(x) = x^2 - 4x + 5$, calculate the value of its derivative at the point $x = 3$ using the definition of derivative using limits:

Find the equation of the tangent line to the graph of the function $f(x) = 3x^2 - 5$ at the point with x-coordinate $x = 1$.

Note: Remember that the equation of the tangent line is given by $y - f(a) = f'(a)(x - a)$.

The number of downloads of a new trending app, in thousands, is given by the function $D(t) = 2t^2 - t$, where $t$ is the time in hours since its launch.

Calculate the instantaneous download speed (the derivative) at $t = 4$ hours using the limit:

Given the rational function $f(x) = \frac{2}{x}$, calculate $f'(2)$ by applying the definition of derivative at a point. Explain whether the function is increasing or decreasing at that point based on the sign of the result.

Determine the points on the graph of the function $f(x) = x^2 - 6x$ at which the tangent line is horizontal.

Hint: A line is horizontal when its slope is equal to zero.

Consider the piecewise function:

Study the differentiability of the function at the point $x = 2$ by checking whether the one-sided derivatives exist and are equal. Is the function continuous at that point?

In a Mathematics Applied to the Social Sciences II exam, Sergio must calculate the equation of the normal line to the graph of the function $f(x) = 2x^2 - 6x + 5$ at the point with abscissa $x = 2$. His work on paper is as follows:

1. He calculates the ordinate of the point: $f(2) = 2\cdot 2^2 - 6\cdot 2 + 5 = 1$. The point of tangency is $(2, 1)$.

2. He differentiates the function: $f'(x) = 4x - 6$, from which he gets the slope of the curve: $f'(2) = 4\cdot 2 - 6 = 2$.

3. Sergio writes: "Since the normal line is perpendicular to the tangent line, its slope is the opposite, that is, $m_n = -2$".

4. He writes the point-slope equation: $y - 1 = -2(x - 2)$, which simplified gives $y = -2x + 5$.

1. a) Locate and explain with reasoning the conceptual error made by Sergio in step 3, recalling the geometric relationship that the slopes of two perpendicular lines must satisfy.
2. b) Correct the exercise by determining the correct equation of the normal line at that point.
3. c) Find out whether the normal line you obtained passes through the point $(4, 0)$.

An agricultural cooperative in Castilla-La Mancha has estimated that the total daily cost (in euros) of bottling $x$ jugs of extra virgin olive oil is given by the function:

Currently, the cooperative bottles $x = 50$ jugs per day.

1. a) Estimate: Without using derivatives, calculate the actual cost of bottling the 51st jug using the difference $C(51) - C(50)$. From this increment, make an estimate of what the marginal cost or instantaneous rate of change $C'(50)$ should approximately be.
2. b) Check: Calculate $C'(50)$ exactly by rigorously applying the definition of derivative using the limit of the difference quotient:

1. c) Evaluate: Compare the actual cost obtained in part (a) with the instantaneous value calculated in (b). Explain why in economics the derivative is an excellent approximation for estimating the cost of an additional unit and what the exact difference between the two values is.

Use the definition of derivative at a point by calculating the limit:

to find the value of the derivative of the function $f(x) = \frac{3}{x-2}$ at the point with x-coordinate $x = 4$. Explain the geometric meaning of the result obtained in relation to the gradient of the curve at that point.

A courier company has found that the cost of shipping a parcel (in euros) depends on its weight $x$ (in kg) according to the function $C(x) = \sqrt{4x + 9}$. Calculate, using the definition of derivative as a limit, the instantaneous rate of change of the cost for a parcel weighing exactly $4$ kg and interpret the result in economic terms (marginal cost).

Find the equation of the tangent line to the graph of the function $f(x) = x^2 - 5x + 2$ at the point where the function crosses the y-axis (the $Y$ axis).

Given the piecewise function:

Calculate the values of $a$ and $b$ so that the function is differentiable at the point $x = 2$. Remember that for a function to be differentiable at a point, it must first be continuous at that point.

The number of users of a new social network (in thousands) evolves according to the function $U(t) = \frac{20t}{t+2}$, where $t$ is the time in months since its launch. Find:

1. The average rate of change of users between month $t=1$ and month $t=3$.

2. The instantaneous growth rate (derivative) in month $t=2$ using differentiation rules and compare both values.

Determine the points on the graph of the function $f(x) = \frac{1}{3}x^3 - x^2 - 2x + 5$ where the tangent line is parallel to the line $y = 1 - 2x$.

Study the differentiability of the function $f(x) = |x^2 - 4|$ at the point $x = 2$ by calculating its one-sided derivatives. Interpret graphically why the function is not differentiable at that x-value.

Find the equation of the normal line to the curve $f(x) = e^{x-1} + 2x$ at the point with x-coordinate $x = 1$. Remember that the normal line is perpendicular to the tangent line at the point of contact.

During a sunny spring day, the instantaneous electric power (in megawatts, MW) generated by a photovoltaic solar farm located in Albacete is modeled by the function:

where $t$ represents the solar hour of the day, valid in the effective radiation interval $6 \le t \le 20$.

1. a) Calculate the instantaneous rate of change of the power at 10:00 ($t = 10$) and at 16:00 ($t = 16$). Interpret the physical and operational meaning of the sign of the result obtained in each case.
2. b) Determine the equation of the tangent line to the power curve at 11:00 ($t = 11$). What information does the slope of this line provide to power grid operators about the rate of energy input at that moment?
3. c) Control technicians at Red Eléctrica de España need to schedule storage systems for the instant when the power stops increasing and begins to decrease, that is, when the instantaneous rate of change becomes zero ($P'(t) = 0$). Find the exact time this occurs and calculate the maximum power generated.

Consider the economic function $f(x) = \dfrac{a}{x} + bx$, defined for all $x \ne 0$, where $a$ and $b$ are constant real parameters. It is known that at the point with abscissa $x = 1$, the tangent line to the graph of $f$ is parallel to the line $r: 5x - y + 3 = 0$, and also that the graph of $f$ passes through the point $P(1, 7)$.

1. a) Set up the appropriate system of equations using the information from the statement and determine, with justification, the values of $a$ and $b$.
2. b) Write the equation of the normal line to the graph of $f$ at that point with abscissa $x = 1$.
3. c) Is there any other point on the graph of $f$ where the tangent line is also parallel to line $r$? If so, calculate its coordinates and the equation of that tangent line; if not, justify mathematically why it is not possible.

A logistics company has modelled the maintenance cost of its fleet of electric vans using a piecewise function $f(x)$. To ensure that there are no abrupt changes in operating costs, the function must be differentiable across its entire domain. Determine the values of $a$ and $b$ so that the function is differentiable at $x = 1$:

Calculate, using exclusively the definition of the derivative via the limit of the difference quotient ($h \to 0$), the value of $f'(4)$ for the function:

Justify each step of the process, especially the elimination of the indeterminate form.

The trajectory of a delivery drone follows the curve $f(x) = x^2 - 6x + 8$ (in hectometres). The drone operator wants to program a straight-line approach manoeuvre from a point $P(a, f(a))$ on the trajectory, such that the tangent line to the curve at that point passes exactly through the charging station, located at the origin $(0,0)$.

Find the coordinates of the possible points of tangency and the equations of these lines.

Study the differentiability of the function $f(x) = x \cdot |x - 2|$ at the point $x = 2$. To do this:

1. Express the function as a piecewise function.

2. Calculate the one-sided derivatives at $x = 2$ using the definition of derivative.

3. Interpret graphically whether the point $(2, f(2))$ is a "sharp point" or a point of "smooth transition".

The revenue function of a tech startup is given by $I(x) = \frac{400x}{x+2}$, where $x$ is the marketing investment in thousands of euros and $I(x)$ is the revenue in thousands of euros.

1. Calculate the Average Rate of Change (AROC) in the interval $[2; 2,5]$.

2. Calculate the Instantaneous Rate of Change (derivative) at $x = 2$.

3. What error is made if we use the derivative at $x=2$ to estimate the revenue increase when going from 2 to 2.5 thousand euros of investment?

Find the equation of the normal line to the graph of the function $f(x) = e^{2x-4}$ at the point where the graph intersects the horizontal line $y = 1$. Remember that the normal line is perpendicular to the tangent at the point of contact.

Consider the function $f(x) = \frac{k}{x^2+1}$. Determine the value of the parameter $k \neq 0$ so that the tangent line to the curve at the point with x-coordinate $x = 1$ makes an angle of $135^\circ$ with the positive direction of the $OX$-axis. Once $k$ is found, write the equation of that tangent line.

Given the function $f(x) = x^3 + ax^2 + bx + c$, determine the values of the parameters $a, b$ and $c$ knowing that:

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

2. The gradient of the tangent line at $x = 0$ is minimum.

3. There is a stationary point (zero derivative) at $x = 2$.

Two 2nd-year Bachillerato students are debating how to analyze whether the function $f(x) = |x - 2| + x^2 - 3x$ is differentiable at the point $x = 2$:

• Elena proposes Method A (Formal definition using limits of the difference quotient): calculate the one-sided derivatives directly using:

• Carlos proposes Method B (Piecewise decomposition and functional differentiation): express $f(x)$ as a piecewise function by eliminating the absolute value, check continuity at $x = 2$, differentiate each branch using differentiation rules, and calculate the one-sided limits $\lim_{x\to 2^-} f'(x)$ and $\lim_{x\to 2^+} f'(x)$.

1. a) Apply Method A in detail by calculating both one-sided limits to determine whether $f$ is differentiable at $x = 2$.
2. b) Apply Method B in detail, first checking continuity and then finding the one-sided derivatives using differentiation rules. Verify that the same conclusion is reached.
3. c) Choose and justify the method: Critically compare both approaches. Why is checking the continuity of $f(x)$ an indispensable logical requirement in Method B before equating the derivatives of the branches? Explain what error would be made by directly applying differentiation rules if the function had a jump discontinuity at $x = 2$.
4. d) In light of the result, explain whether the curve admits a single tangent line at $(2, f(2))$ and reason geometrically what shape the tangent half-lines form at that point.

In the strategic launch plan for a financial services application in Spain, the estimated cumulative net profit (in thousands of euros) follows the curve:

where $x$ is the expenditure allocated to digital acquisition (in tens of thousands of euros), with a budgetary feasibility range $x \in [1, 7]$.

An investment fund offers a structured financing agreement whose contribution follows a linear relationship. This financing starts from a guaranteed initial balance of $20\,000$ euros when the investment in acquisition is zero (that is, the line passes through the point $(0, 2)$ in the model's units). To ensure that the injection of capital does not create financial strain or profitability jumps, the financing line is required to be strictly tangent to the profit curve at some point with unknown abscissa $x = a \in [1, 7]$.

1. a) Write the equation of the family of lines passing through the external point $(0, 2)$ in terms of its slope $m$.
2. b) Formulate the system of equations that relates the abscissa of tangency $a$, the function value $B(a)$, and the instantaneous derivative $B'(a)$ to ensure that the line is tangent to the curve at $(a, B(a))$.
3. c) Solve the previous system. Reason which of the obtained mathematical solutions is the only one economically admissible within the feasibility interval $[1, 7]$, and determine the exact coordinates of the point of tangency and the explicit equation of the financing line.
4. d) Find the equation of the normal line to the profit curve at that point of tangency and explain the geometric meaning of its slope with respect to the set financing trajectory.