Antiderivative of a function
A company knows its marginal cost: what it costs to produce one more unit. What it really cares about is the total cost. Going from one to the other is exactly the problem you are going to learn to solve: you are given a derivative and you have to recover the function it came from. It is the same table of derivatives as always, read in reverse.
Differentiating in reverse: what an antiderivative is
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…
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.
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…
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.
The family of antiderivatives: the constant $C$
To pick out one particular antiderivative from the whole family, one extra piece of data is enough: a point it passes through.
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…
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.
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…
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.
From rate of change to total: cost, revenue, position
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…
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 ($I(0) = 0$).
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 $s = 10$…
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.
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…
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 $s(t)$.
Conditions that fix an antiderivative: points, gradients and turning points
The more data you are given (a point, a gradient at another point, a turning point), the more conditions you can set up to solve also for parameters that appear inside $f$, not just the constant $C$.
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…
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 $y = 7$.
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…
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 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…
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:
$$C'(t) = \frac{4t^2}{2t+1}$$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.
Back to the start: If a company has a marginal cost $C'(x) = 4x + 10$ and fixed costs of $100$ € ($C(0) = 100$), its total cost is the antiderivative that satisfies that condition: $C(x) = 2x^2 + 10x + 100$.