Exercises: Concepts: Rate of change and derivative of a function at a point

Calculate the average rate of change (ARC) of the function $f(x) = x^2 - 4x + 5$ in the interval $[1, 4]$. Explain graphically what the obtained value represents in relation to the secant line passing through the points on the curve corresponding to the endpoints of this interval.

A high school senior athlete is training for the 100 foot sprint. The distance covered (in feet) as a function of time (in seconds) is given by the expression $s(t) = 0.5t^2 + 2t$.

Calculate the average velocity of the athlete between the instants $t = 2$ s and $t = 6$ s. Which mathematical concept does this value correspond to?

Given the function $f(x) = 3x^2 - 1$, use the definition of the derivative at a point via the limit:

to find the value of the derivative at the point with x-coordinate $x = 2$.

Determine the slope of the tangent line to the graph of the function $f(x) = \sqrt{x+1}$ at the point where $x = 3$, applying the limit of the instantaneous rate of change:

The number of followers of a new YouTube channel evolves according to the function $S(t) = t^2 + 10t + 100$, where $t$ is the time in days since its creation.

1. a) Calculate the average rate of change of followers between day $t = 0$ and day $t = 10$.
2. b) What is the instantaneous growth of followers exactly on day $t = 5$? (Calculate $S'(5)$ using the definition).

Consider the rational function $f(x) = \frac{2}{x}$.

Prove, applying the definition of the derivative as a limit, that the value of the instantaneous rate of change at $x = 1$ is $f'(1) = -2$. Interpret the sign of the result obtained.

Observe the following graph of a function $f(x)$. Without performing calculations, order the following values from smallest to largest, justifying your answer based on the steepness of the lines:

• The average rate of change in the interval $[1, 3]$.

• The derivative of the function at the point $x = 1$.

• The derivative of the function at the point $x = 3$.

The temperature of a metal plate (in degrees Fahrenheit) varies with time $t$ (in minutes) following the model $T(t) = 20 + 4t - 0.1t^2$.

Calculate the instantaneous rate of change of the temperature at the instant $t = 10$ minutes. State the units of measurement of the result and explain whether the plate is heating up or cooling down at that moment.

The growth of followers of a new account on a well-known social network follows the function $S(t) = t^2 + 4t + 100$, where $t$ is the time in days since its creation and $S(t)$ is the number of followers.

1. a) Calculate the average rate of change (ARC) of the number of followers in the intervals $[0, 5]$ and $[5, 10]$.
2. b) Explain the meaning of the results obtained in the context of the account's growth. In which interval has the growth been faster?

Calculate the derivative of the function $f(x) = \dfrac{3}{x + 1}$ at the point with x-coordinate $x = 2$ strictly using the definition of the derivative via the limit:

Consider the function $f(x) = x^2 - 6x + 5$. Determine the point on the graph where the tangent line has a slope of $m = 2$. Once the point is found, write the full equation of this tangent line in point-slope form.

Given the piecewise function:

Calculate the values of $a$ and $b$ so that the function is differentiable at the point $x = 1$. Remember that for a function to be differentiable, it must first be continuous.

A skater moves along a track whose position (in feet) with respect to the start is given by the expression $e(t) = \sqrt{4t + 9}$, where $t$ is the time in seconds.

1. a) Calculate the average velocity of the skater between the instants $t = 0$ and $t = 4$ seconds.
2. b) Find the instantaneous velocity at the moment $t = 4$ seconds using the definition of the derivative.

Find the equation of the tangent line and the normal line to the curve $f(x) = x^3 - 3x^2 + 2$ at the point where the graph intersects the y-axis ($OY$).

Given the function $f(x) = \dfrac{x^2}{x - 1}$:

1. a) Calculate the average rate of change in the interval $[2, 4]$.
2. b) Determine if there is any point $c$ within the interval $(2, 4)$ such that the instantaneous derivative $f'(c)$ matches the ARC calculated in the previous section.

The temperature of an electronic component during the first hours of operation follows the function $T(t) = 20 + t \cdot e^{-t}$, where $T$ is the temperature in degrees Fahrenheit and $t$ is the time in hours.

Calculate the instantaneous rate of change of the temperature when exactly 1 hour has elapsed ($t = 1$). Is the temperature increasing, decreasing, or neither at that instant? Justify your answer based on the sign of the derivative.

Consider the piecewise function:

Determine the values of the real parameters $a$ and $b$ so that the function is differentiable at the point $x = 2$. To do this, you must first ensure continuity and, subsequently, the existence of finite and equal one-sided derivatives at that point using the definition of the derivative.

Calculate, making exclusive use of the definition of the derivative as the limit of the difference quotient, the value of the following limit:

Hint: Identify this limit with the derivative of a known function at a specific point and solve using standard differentiation rules to check your result.

A forest surveillance drone follows an approach trajectory whose height $h$ (in feet) with respect to time $t$ (in seconds) is given by the function:

You are asked to:

1. Calculate the Average Rate of Change (ARC) of the height in the interval $[0, 5]$. Explain its physical meaning.

2. Using the definition of the derivative at a point, find the instantaneous rate of ascent at the instant $t = 5$ seconds.

3. Check if there is any instant $t > 0$ at which the instantaneous velocity exactly matches the ARC calculated in the first part.

Given the function $f(x) = x \cdot |x - 3|$, study its differentiability at the point $x = 3$.

To do this:

1. Express the function as a piecewise function.

2. Calculate the one-sided derivatives $f'(3^-)$ and $f'(3^+)$ using the limit of the difference quotient.

3. Geometrically interpret the result obtained: does the graph have a corner point or a smooth tangent at $x=3$? Justify your answer.

Find the equations of the tangent lines to the curve $y = x^2 - 4x + 5$ that pass through the external point $A(0, 0)$ (the figure shows only one of them).

Note: Bear in mind that point A does not belong to the curve. You will need to set up the slope of the tangent at a generic point $(a, f(a))$ by equating it to the slope of the line joining that point to the origin.

Let the function be $f(x) = \frac{1}{x+1}$. Prove, applying the definition of the derivative, that its derivative function is $f'(x) = \frac{-1}{(x+1)^2}$ for any value of $x$ belonging to its domain.

From this result, determine the coordinates of the points on the graph where the tangent line is parallel to the angle bisector of the second quadrant ($y = -x$).

Consider the function $f(x) = \sqrt[3]{x^2}$.

1. Study the continuity of the function at the origin ($x=0$).

2. Analyze the differentiability at $x=0$ by calculating the limit of the difference quotient.

3. If the limit tends to infinity, what type of tangent line does the function present at that point? Describe the visual behavior of the graph in the vicinity of the origin.

In a biotechnology laboratory, the population of a colony of bacteria (in millions) is modeled by the function:

where $t$ is the time in hours. You are asked to:

1. Find the Instantaneous Rate of Change (IRC) of the population at $t = 1$ hour and at $t = 4$ hours.

2. Determine at which of these two instants the population is growing faster.

3. Is there any instant when the growth rate is zero? If so, calculate this value of $t$ and interpret what happens to the population at that moment.