Exercises: Concept of derivative

The following table shows the number of registered users on a new mobile app during its first months after launch:

• Month 1: $100$ users

• Month 2: $250$ users

• Month 3: $500$ users

• Month 4: $850$ users

• Month 5: $1300$ users

Calculate the average rate of change (ARC) of the number of users between months $2$ and $4$.

Given the polynomial function $f(x) = 3x^2 - 2x$, calculate its average rate of change analytically in the interval $[1, 4]$.

Observe the graph of the function $f(x)$ and the secant line drawn between the points with x-coordinates $x=1$ and $x=3$. Calculate visually the average rate of change (ARC) in the interval $[1, 3]$ using the values projected on the axes.

Using the definition of derivative as the limit of the average rate of change, $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, calculate the derivative of the function $f(x) = x^2 + 3$ at the point $x=2$.

Knowing that the derivative of a function $f(x)$ at the point $x=4$ is $f'(4) = -2$, state with reasons whether the function is increasing or decreasing at that point. Briefly justify the geometric meaning of this value with respect to the tangent line to the curve at that point.

Calculate the derivative of the linear function $f(x) = 5x - 7$ at the point $x=10$ by applying the definition of derivative using limits. Does the result depend on the chosen point?

The position of a cyclist on a straight track is given by the function $s(t) = 3t^2 + 2t$ (where $s$ is measured in metres and $t$ in seconds). Calculate the average speed of the cyclist in the time interval between $t=1$ and $t=4$ seconds.

From the graph of the function $f(x)$ and its tangent line at the point with x-coordinate $x=2$, determine the value of the derivative $f'(2)$.

Lucía wants to calculate the slope of the tangent line to the curve $f(x) = 2x^2 + 1$ at the point with abscissa $x = 1$. On her exam, she writes the following procedure:

"I apply the derivative formula by directly substituting the increment with zero:

$f'(1) = \frac{f(1+0) - f(1)}{0} = \frac{3 - 3}{0} = 0$.

Since the derivative is equal to $0$, I conclude that the tangent line is completely horizontal."

1. a) Point out the two conceptual errors made by Lucía: an algebraic one regarding division by zero and the idea of a limit, and a geometric one in her final conclusion.
2. b) Correct her solution by calculating the correct value of $f'(1)$ using the limit of the difference quotient: $\lim_{h \to 0} \frac{f(1+h) - f(1)}{h}$.
3. c) From the value obtained, determine with reasoning whether the function is increasing or decreasing at $x = 1$ and find the equation of the tangent line at that point.

Given the quadratic function $f(x) = -x^2 + 6x$, whose graph is a parabola with vertex at the point $(3, 9)$:

1. a) Before calculating anything, estimate with reasoning the sign of the derivatives $f'(2)$ and $f'(4)$, based on whether the parabola goes up or down on each side of its vertex. Furthermore, predict what exact numerical value the derivative right at the vertex, $f'(3)$, should have, justifying your prediction based on the tangent line.
2. b) Verify your estimation analytically by calculating $f'(2)$ using the formal definition of derivative: $f'(2) = \lim_{h \to 0} \frac{f(2+h) - f(2)}{h}$.
3. c) Compare the result of the limit with your initial estimation and write a conclusion relating the sign of the slope to the behavior of the function at that point.

The total distance covered by a cyclist descending a mountain pass (in metres) as a function of time is given by the function $e(t) = 150 + 2t + 0,5t^2$ ($t$ in seconds). Calculate the average rate of change of the distance from $t = 20$ to $t = 30$ seconds and interpret the physical meaning of the result obtained (the average speed over that section).

Find the equation of the secant line to the curve $f(x) = -x^2 + 6x$ passing through the points of intersection of this curve with the x-axis. (Note: calculate these x-intercepts first).

Making strict use of the definition of derivative by calculating the limit, find the value of $f'(1)$ for the function $f(x) = 2x^2 - 3x + 1$.

Find the equation of the tangent line to the graph of the parabola $f(x) = x^2 - 4x + 5$ at the point with x-coordinate $x=3$. First calculate the gradient using the definition of derivative with limits.

An object is dropped from the top of a building. Its height in metres after $t$ seconds is given by the expression $h(t) = 45 - 5t^2$. Calculate the instantaneous speed of the object at exactly $t=2$ seconds using the definition of derivative.

Given the function $f(x) = -2x^2 + 12x - 5$, determine whether the function is increasing or decreasing at $x=4$, having first calculated the instantaneous rate of change at that point using the definition of limit.

Observe the graph of the function $f(x)$ and its tangent line drawn at the point with x-coordinate $x=4$. Write the equation of the tangent line and deduce the value of $f'(4)$. What happens to the growth of the function at that exact point?

Given the function $f(x) = x^2$, calculate the average rate of change in the intervals $[3; 3,1]$, $[3; 3,01]$ and $[3; 3,001]$. In light of the results obtained, deduce what numerical value the ARC tends to as the width of the interval approaches zero, and state what this value is called mathematically.

A high-speed train (AVE) departs from a station. During the first $40$ seconds of acceleration, the distance in meters traveled as a function of time $t$ (in seconds) is given by $s(t) = 0,75t^2$.

1. a) Explain the conceptual and physical difference between the Average Rate of Change of the distance over the time interval $[0, 20]$ and the Instantaneous Rate of Change at the exact instant $t = 20\text{ s}$.
2. b) Calculate the average speed of the train during those first $20$ seconds of the journey.
3. c) Apply the formal definition of derivative using limits to calculate the exact instantaneous speed at $20$ seconds: $s'(20) = \lim_{h \to 0} \frac{s(20+h) - s(20)}{h}$.
4. d) Express the obtained instantaneous speed in kilometers per hour ($\text{km/h}$) and interpret what this value indicates about the train's pace of travel at that instant.

Consider the quadratic function $f(x) = x^2 + bx + 1$, where $b$ is an unknown real parameter. It is known that the tangent line to the graph of $f$ at the point with abscissa $x = 3$ has the equation $y = 2x - 8$.

1. a) Without calculating limits, deduce directly from the equation of the tangent line the numerical value of the tangency point $f(3)$ and the value of the derivative $f'(3)$. Justify which properties you rely on.
2. b) State the formal definition of derivative $f'(3) = \lim_{h \to 0} \frac{f(3+h) - f(3)}{h}$ leaving the parameter $b$ indicated, solve the limit algebraically, and equate the result to the value obtained in the previous section to solve for $b$.
3. c) Check whether the value of $b$ you obtained actually makes the value of $f(3)$ coincide with the ordinate calculated in section a).

The number of subscribers to a video channel is given by the expression $S(x) = -0,1x^2 + 50x - 1000$, where $x$ is the number of days elapsed since its launch. Calculate $S'(200)$ using strictly the definition of derivative via limits, and interpret what this value indicates about the rate of growth of the channel at that moment.

Find the equation of the tangent line to the curve $f(x) = \frac{6}{x}$ at the point with x-coordinate $x=3$. To do so, first calculate the gradient by applying the definition of derivative as the limit of the average rate of change.

Find the exact point on the parabola $f(x) = x^2 - 4x + 7$ where the tangent line to the curve is parallel to the line with equation $y = 2x - 5$. Write down the full coordinates $(x, y)$ of this point.

Determine the values of the parameters $a$ and $b$ of the polynomial function $f(x) = ax^2 + bx$ knowing that its graph passes through the point $(1, 2)$ and that the slope of its tangent line at $x=1$ is equal to $3$.

Calculate the derivative of the radical function $f(x) = \sqrt{x+5}$ at the point $x=4$ using the definition of derivative via limits. (Hint: to evaluate the resulting limit, multiply and divide by the conjugate expression).

The vertex of a parabola corresponds to the point where its tangent line is horizontal (gradient equal to zero). Using the definition of derivative via limits, find the $x$-coordinate of the vertex of the parabola $f(x) = -2x^2 + 12x - 9$ by setting its derivative to zero.

The volume of a cube as a function of the length of its edge $x$ is given by $V(x) = x^3$. Calculate the instantaneous rate of change of the volume with respect to the edge when $x=5$ cm, using the definition of limit. If the error in measuring the edge is very small, how does it proportionally affect the calculation of the volume?

From the graph of the function $f(x)$ and the three tangent lines drawn at the points $A$, $B$ and $C$, order the values of the derivatives $f'(x_A)$, $f'(x_B)$ and $f'(x_C)$ from smallest to largest. Justify your answer based on the geometric gradient of each tangent line.

In a clinical trial, the concentration of a drug in the bloodstream (in $\text{mg/l}$) varies with time $t$ (in hours) according to the model $C(t) = 6t - t^2$ ($0 \le t \le 6$). The medical team needs to know exactly at what rate the concentration is changing $4$ hours after administration ($t = 4$). To do so, two analysts propose different working methods:

• Method A (Numerical approximation): Calculate the Average Rate of Change over three progressively smaller intervals: $[4; 4,1]$, $[4; 4,01]$, and $[4; 4,001]$, to observe toward what value the rate stabilizes.

• Method B (Formal algebraic limit): Calculate the exact derivative using the definition: $C'(4) = \lim_{h \to 0} \frac{C(4+h) - C(4)}{h}$.

1. a) Apply Method A by calculating the three average rates of change and indicate what value they appear to approach.
2. b) Apply Method B by expanding and simplifying the algebraic limit to find the exact value of $C'(4)$.
3. c) Compare both methods: analyze the advantages and disadvantages of each (accuracy, computation time, and rounding error risks) and justify which one is mathematically rigorous. Furthermore, explain what clinical meaning the sign of the result has for patient monitoring.

A forest surveillance drone descends following a vertical trajectory modeled by the function $h(x) = 0,25x^2 - 3x + 11$, where $x$ is the horizontal distance traveled in meters ($0 \le x \le 8$) and $h(x)$ is the height in meters. At a point with abscissa $x = a$, the drone activates a laser sensor that emits in the exact direction of its instantaneous motion (that is, along the tangent line to its trajectory). To calibrate the reading, the sensor needs to be activated at the instant when that tangent line has a slope exactly equal to $-1$.

1. a) Apply the formal definition of derivative using limits to deduce an expression for the slope $h'(a) = \lim_{t \to 0} \frac{h(a+t) - h(a)}{t}$ as a function of the generic abscissa $a$.
2. b) Set up an equation from the obtained expression to determine the exact coordinates $(a, h(a))$ where the sensor should be turned on.
3. c) Write the equation of the line followed by the sensor's light beam.
4. d) If the ground is at level $h = 0$, calculate at what horizontal distance $x$ the light beam will touch the ground surface, and reason whether the drone is in a descent phase at that point or has already begun to climb.