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?

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 package (in pesos) depends on its weight $x$ (in kg) according to the function $C(x) = \sqrt{4x + 9}$. Calculate, using the definition of the derivative as a limit, the instantaneous rate of change of the cost for a package 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.

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 hectometers). 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 $f(x) = x \times |x - 2|$ 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 $I(x) = \frac{8\,000x}{x+40}$, where $x$ is the marketing investment in thousands of pesos and $I(x)$ is the revenue in thousands of pesos.

1. Calculate the Average Rate of Change (AROC) in the interval $[40; 50]$.

2. Calculate the Instantaneous Rate of Change (derivative) at $x = 40$.

3. What error is made if we use the derivative at $x=40$ to estimate the revenue increase when going from 40 to 50 thousand pesos 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 $x = 2$.