Concepts: rate of change and derivative of a function at a point
In 1st Bachillerato you already built all the machinery: average rate of change, difference quotient, limit, derivative. In 2nd Bachillerato you do not define anything from scratch again, but you do need to have it at your fingertips: the university entrance exam (EBAU) mixes these concepts with parameters, piecewise functions and physical contexts, and there is no time to rediscover the idea in the exam.
Review: average rate of change
Calculate the average rate of change (ARC) of the function $f(x) = x^2 - 4x + 5$ on the interval $[1, 4]$. Explain graphically what the value obtained represents in…
Calculate the average rate of change (ARC) of the function $f(x) = x^2 - 4x + 5$ on the interval $[1, 4]$. Explain graphically what the value obtained represents in relation to the secant line that passes through the points $A$ and $B$ of the curve corresponding to the endpoints of that interval (they are marked in the figure).
A Year 13 student trains for the 100-metre sprint. The distance covered (in metres) as a function of time (in seconds) is given by the expression $s(t) = 0,5t^2 + 2t$.…
A Year 13 student trains for the 100-metre sprint. The distance covered (in metres) as a function of time (in seconds) is given by the expression $s(t) = 0,5t^2 + 2t$.
Calculate the average speed of the athlete between $t = 2$ s and $t = 6$ s. Which mathematical concept does this value correspond to?
Review: derivative at a point and one-sided derivatives
Given the function $f(x) = 3x^2 - 1$, use the definition of derivative at a point via the limit: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ to find the value of…
Given the function $f(x) = 3x^2 - 1$, use the definition of derivative at a point via the limit:
$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$to find the value of the derivative at the point with x-coordinate $x = 2$.
Consider the rational function $f(x) = \frac{2}{x}$. Show, applying the definition of derivative as a limit, that the value of the instantaneous rate of change at…
Consider the rational function $f(x) = \frac{2}{x}$.
Show, applying the definition of 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.
Tangent line and normal line
Find the equation of the tangent line and of the normal line to the curve $f(x) = x^3 - 3x^2 + 2$ at the point where the graph crosses the y-axis ($OY$).
Find the equation of the tangent line and of the normal line to the curve $f(x) = x^3 - 3x^2 + 2$ at the point where the graph crosses the y-axis ($OY$).
Determine the gradient 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…
Determine the gradient 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:
$$f'(3) = \lim_{x \to 3} \frac{f(x) - f(3)}{x - 3}$$Continuity, differentiability and instantaneous velocity
The number of subscribers 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. a)…
The number of subscribers 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.
- a) Calculate the average rate of change of subscribers between day $t = 0$ and day $t = 10$.
- b) What is the instantaneous growth of subscribers on exactly day $t = 5$? (Calculate $S'(5)$ using the definition).
The temperature of a metal plate (in degrees Celsius) varies with time $t$ (in minutes) following the model $T(t) = 20 + 4t - 0,1t^2$. Calculate the instantaneous rate…
The temperature of a metal plate (in degrees Celsius) 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 time $t = 10$ minutes. State the units of measurement of the result and explain whether the plate is heating up or cooling down at that moment.