Exercises: Antiderivative of a function

Given the function $f(x) = 4x^3 - 6x + 1$, check with justification whether the function $F(x) = x^4 - 3x^2 + x + 5$ is one of its antiderivatives. To do this, use the definition of an antiderivative based on differentiation.

Find the set of all antiderivatives of the polynomial function $f(x) = 3x^2 + 2x - 5$. Remember to include the constant of integration $C$ and explain what this value represents geometrically in the graph of the resulting functions.

Calculate the antiderivative of the function $f(x) = e^x + \cos(x)$ that passes through the origin, that is, through the point $(0, 0)$.

In the marketing department of a well-known trainer brand, they have determined that the marginal revenue (the derivative of total revenue) is given by the function $I'(x) = 100 - 4x$, where $x$ represents the number of units sold. Calculate the total revenue function $I(x)$, knowing that if no units are sold, the revenue is zero ($I(0) = 0$).

Find the indefinite integral of the rational function $f(x) = \frac{3}{x} + \frac{1}{x^2}$ for values of $x > 0$. Use the linearity properties of integrals to solve each term separately.

Determine the function $F(x)$ whose derivative is $f(x) = \sqrt{x}$ and which satisfies the condition $F(1) = 2$. Express the final result with the exponent as a fraction.

A drone moves such that its velocity in metres per second follows the function $v(t) = 2t + 3$. If at the initial time ($t = 0$) the drone is at position $s = 10$ metres, obtain the position function $s(t)$ by integrating the velocity function.

Find a function $G(x)$ such that $G'(x) = 5 \cdot e^x - 2x$ and that passes through the point $(0, 7)$. Once found, verify your result by differentiating the obtained function.

Carlos claims that the functions $F(x) = \ln(x^2+1)$ and $G(x) = \ln(3x^2+3)$ cannot be antiderivatives of the same function because there is no explicit numerical constant added between their algebraic expressions, that is, he does not observe the form $G(x) = F(x) + C$.

1. a) Differentiate both functions and check whether Carlos is right or if, on the contrary, both differentiate into the same function $f(x)$.
2. b) Identify Carlos's conceptual error by applying the properties of logarithms to $G(x)$ and determine the exact value of the real constant $C$ that satisfies $G(x) = F(x) + C$.

You want to check whether the function $F(x) = \dfrac{2x-1}{x^2+1}$ is an antiderivative of $f(x) = \dfrac{-2x^2 + 2x + 2}{(x^2+1)^2}$. To solve this, two classmates debate which strategy to follow:

• Mateo proposes directly calculating the indefinite integral $\int \dfrac{-2x^2 + 2x + 2}{(x^2+1)^2} \, dx$ using integration methods for rational functions.

• Sara proposes differentiating the function $F(x)$ by applying the quotient rule and comparing the result obtained with the expression for $f(x)$.

1. a) Reason which of the two methods is more direct and convenient according to the mathematical definition of an antiderivative.
2. b) Apply the most appropriate method to give a justified ruling on whether or not $F(x)$ is an antiderivative of $f(x)$.

Given the function $f(x) = \dfrac{6x}{x^2+3} + 4e^{2x}$, check whether the function $F(x) = 3 \ln(x^2+3) + 2e^{2x} + \pi$ is one of its antiderivatives. Justify your answer using the definition of an antiderivative and explain what role the constant $\pi$ plays in this context.

A surveillance drone moves in a straight line, and its velocity in metres per second, after $t$ seconds, is given by the function $v(t) = 3t^2 - 6t + 10$. Given that at the initial time ($t=0$) the drone is at a distance of 15 metres from the control point, find the expression for the position function $s(t)$.

A mobile app development startup estimates that the marginal revenue (in thousands of euros) for every additional thousand downloads of its new app is given by the function:

$I'(x) = \dfrac{1}{x+1} + 2x$

where $x$ represents the number of thousands of downloads ($x > 0$).

Calculate the total revenue function $I(x)$, knowing that if there are no downloads ($x=0$), the revenue is zero.

Find the values of the parameters $a$ and $b$ so that the function $F(x) = a \cdot e^{x^2} + b \cdot \ln(x)$ is an antiderivative of the function $f(x) = 4x \cdot e^{x^2} + \dfrac{5}{x}$. Once found, write down the expression for all possible antiderivatives of $f(x)$.

The derivative of a function $f(x)$ is $f'(x) = 6x^2 - 10x + 1$. Find the expression for $f(x)$ knowing that its graph crosses the y-axis (the $Y$ axis) at the point $y = 7$.

Determine the function $f(x)$ whose second derivative is known to be $f''(x) = 12x - 4$. Furthermore, it is known that the tangent line to the graph of $f$ at the point with x-coordinate $x = 1$ is horizontal, and that the function passes through the point $(0, 5)$.

The marginal cost of manufacturing a new model of wireless headphones is defined by the function $C'(q) = \sqrt{q+4}$, where $q$ is the number of units produced. If the fixed costs of the factory (costs when production is zero) amount to $5000$ euros, find the total cost function $C(q)$.

Of all the antiderivatives of the function $f(x) = \dfrac{2}{x-2} + 3x^2$, find the one whose graph passes through the point $(3, 10)$. Express the result in simplified form using natural logarithms.

A last-mile delivery cooperative in a Spanish city is renewing its fleet by purchasing electric vans. The economic team estimates that the rate of change of the total weekly cost is modelled by the marginal cost function $C'(x) = 12x + 50$ (in euros per week and additional vehicle), where $x \geq 0$ is the number of active vehicles. It is known that the weekly fixed costs (structural costs when there are no vehicles delivering, $x = 0$) are $1\,800$ euros.

1. a) Model the total weekly cost function $C(x)$ by calculating its corresponding antiderivative and justifying how the value of the integration constant is determined.
2. b) The average cost function per vehicle is defined as $\bar{C}(x) = \dfrac{C(x)}{x}$. Calculate the average cost if the cooperative operates with a fleet of $10$ vans and explain with reasoning why the term of the average cost due to the integration constant (average fixed cost) decreases as the fleet size increases.

Find an affine function $f(x) = ax + b$ knowing that one of its infinite antiderivatives, which we will call $F_1(x)$, simultaneously satisfies the following three conditions:

1. It has a relative extremum at the point with abscissa $x = 2$.

2. Its graph intersects the vertical $y$-axis at the point $(0, -5)$.

3. Its value at the aforementioned relative extremum is exactly $3$, that is, $F_1(2) = 3$.

1. a) Deduce the analytical expression of $f(x)$ and the formula of the antiderivative $F_1(x)$, explaining clearly what information the existence of a relative extremum of $F_1$ provides about the value of $f(2)$.
2. b) Determine whether the relative extremum at $x = 2$ is a local maximum or a local minimum, arguing from the sign of the function $f(x)$ on either side of that point.

Consider the marginal profit function of a software development startup, defined by the following piecewise function:

Where $x$ represents the number of years since the creation of the company and $f(x)$ the rate of growth of profits in thousands of euros per year.

Find the accumulated profit function $F(x)$ knowing that it is a continuous function across its entire domain and that at the time of its launch ($x=0$) the company had initial funds of $10$ thousand euros.

A well-known streaming platform has analysed that the rate of acquisition of new subscribers in Spain follows the function $v(t) = \frac{\ln(t+1)}{(t+1)^2}$, where $t$ is the time in months since a viral marketing campaign.

Find the function that determines the total number of additional subscribers $S(t)$ generated by the campaign, knowing that at the initial moment ($t=0$) there were no subscribers attributable to that campaign. Express the result in simplified form using the integration by parts technique.

Given the function $f(x) = \frac{ax^2 + b}{x^2}$, determine the values of the real parameters $a$ and $b$ knowing that one of its antiderivatives, $F(x)$, satisfies the following three conditions simultaneously:

1. It passes through the point $P(1, 4)$.

2. It has a relative extremum at the point with x-coordinate $x = 2$.

3. The tangent line to the graph of $F(x)$ at $x = -1$ has a gradient equal to $3$.

A team of sociologists is studying the rate of spread of fake news on a social network. They determine that the rate of shares per hour is given by the function:

Find the expression of the function $C(t)$ representing the total number of accumulated shares, knowing that when the study started ($t=0$) $100$ shares had already been made. Use partial fraction decomposition to evaluate the integral and give the result in terms of a single logarithm.

The attached graph is a parabola representing the derivative function $f'(x)$ of a function $f(x)$ that is continuous throughout the interval $[-2, 2]$.

Knowing that $k = 4$ and that the original function $f(x)$ passes through the point $(0, 1)$, obtain the analytical expression of $f(x)$ and reason, based on the geometry of the derivative, how many inflection points the antiderivative will have in that interval.

You can do this on paper and submit a photo of your answer.

Find the antiderivative $F(x)$ of the function $f(x) = \frac{1}{\sqrt{x}(1+\sqrt{x})}$ satisfying the condition $F(1) = 2 \cdot \ln(2)$.

Hint: Use the substitution $t = \sqrt{x}$.

It is known that the gradient of the tangent line to the graph of a function $f(x)$ at any point $(x, f(x))$ is given by the expression $m(x) = (x^2 - 1) \cdot \ln(x)$ for $x > 0$.

Find the expression for the function $f(x)$ knowing that its graph crosses the x-axis at the point $x = 1$. You must apply the method of integration by parts and simplify the resulting polynomial.

A logistics company has calculated that the marginal cost of keeping a parcel in its central warehouse (in euros per day) depends on the time $t$ (in days) according to the function:

Calculate the total cost function $C(t)$ if it is known that the fixed entry costs to the warehouse (cost at $t=0$) amount to $50$ euros.

Note: Perform polynomial division first to obtain an immediate or near-immediate integral.

In a report on net monthly job creation in a technology sector during the first semester of the year ($0 \leq t \leq 6$, measured in months), the rate of change is modelled by the continuous function $f(t) = 3t^2 - 12t + 9$ (in hundreds of jobs per month). The function $E(t)$ represents the accumulated net employment since the start of the semester, satisfying $E(0) = 0$.

1. a) Without previously calculating the analytical expression of the antiderivative $E(t)$:

• Solve the equation $f(t) = 0$ and analyse the sign of $f(t)$ on the interval $[0, 6]$ to estimate in which months a relative maximum and a relative minimum of accumulated employment will be reached.

• Qualitatively analyse the periods where $f(t) > 0$ versus those where $f(t) < 0$ to make a reasoned estimate of whether the final balance at $t = 6$, $E(6)$, will be positive, negative, or zero.

1. b) Analytically find the antiderivative $E(t)$ that satisfies the initial condition $E(0) = 0$. Calculate the exact values $E(1)$, $E(3)$, and $E(6)$, compare them with the predictions you made in the previous section, and justify whether your qualitative estimates were correct.

In a photovoltaic plant in southern Spain, the hourly rate of energy generation and feed-in to the grid during the morning is described by the continuous function $f(t)$ (in megawatts), where $t$ represents the time in hours elapsed since sunrise ($0 \leq t \leq 8$):

To calculate the accumulated energy $E(t)$, a student proposes the following function:

Since at sunrise ($t = 0$) the accumulated energy is zero ($E(0) = 0$), the student deduces that $C = 0$ in the first section and keeps that same value for the second section, concluding that after 5 hours the accumulated energy is $E(5) = 6 \cdot 5 + 0 = 30\text{ MWh}$.

1. a) Explain with mathematical rigour the mistake made by the student in assigning a single constant $C$ to both intervals. Justify why every antiderivative must be differentiable and, consequently, continuous at $t = 3$, and show that the function proposed by the student has a jump discontinuity at that point.
2. b) Correct the approach: write the family of antiderivatives using two independent constants ($C_1$ and $C_2$), impose the continuity condition at $t = 3$, and determine the correct accumulated function $E(t)$ that satisfies $E(0) = 0$. Using it, calculate the actual energy generated after 5 hours.