Growth, maxima and minima
A sprinter does not always run at the same speed: there are stretches where they accelerate more and stretches where they accelerate less. The average rate of change is the tool for comparing, with a single number, how quickly a function changes over different stretches.
Average rate of change
Calculate the average rate of change of the function $f(x) = 4x - 7$ in the intervals $[-2, 1]$ and $[1, 5]$. In light of the results, what relationship exists between…
Calculate the average rate of change of the function $f(x) = 4x - 7$ in the intervals $[-2, 1]$ and $[1, 5]$. In light of the results, what relationship exists between the average rate of change and the gradient of a linear function?
Given the quadratic function $g(x) = x^2 - 6x + 8$, find its average rate of change in the interval $[2, 4]$. What does the obtained result indicate about the value of…
Given the quadratic function $g(x) = x^2 - 6x + 8$, find its average rate of change in the interval $[2, 4]$. What does the obtained result indicate about the value of the function at the endpoints of the interval?
The temperature $T$ (in $^\circ\text{C}$) of a patient from admission to A&E is given by the formula $T(t) = 38,5 + 0,2t - 0,1t^2$, where $t$ is the time elapsed in…
The temperature $T$ (in $^\circ\text{C}$) of a patient from admission to A&E is given by the formula $T(t) = 38,5 + 0,2t - 0,1t^2$, where $t$ is the time elapsed in hours. Calculate the average rate of change of the temperature in the first two hours ($t \in [0, 2]$).
Three ways of measuring change
A tank contains $40$ litres at minute $2$ and $60$ at minute $6$:
| Measure | Calculation | Interpretation |
|---|---|---|
| Absolute change | $60-40=20$ L | How much it has increased |
| Relative change | $(60-40)/40=0{,}5=50\,\%$ | Increase relative to the initial value, which must be non-zero |
| Average rate of change | $(60-40)/(6-2)=5$ L/min | Change per unit of time |
On a graph, locate both points and draw the segment joining them. Its gradient is $5$, but that does not prove that five litres flowed in each minute. Enter the functions $y=5x+30$ and $y=40+1{,}25(x-2)^2$ into the graphing tool, restrict your view to $[2,6]$ and check that they share the end points and the ARC even though they change in different ways between them.
If you start from zero litres, the absolute change and the ARC still make sense, but the percentage increase relative to zero is not defined.
Increase and decrease
Without plotting its graph, determine the intervals of increase and decrease of the quadratic function $f(x) = 2x^2 + 8x - 3$.
Without plotting its graph, determine the intervals of increase and decrease of the quadratic function $f(x) = 2x^2 + 8x - 3$.
If we know that the average rate of change of a continuous function in the interval $[1, 4]$ is negative, can we state with absolute certainty that the function is…
If we know that the average rate of change of a continuous function in the interval $[1, 4]$ is negative, can we state with absolute certainty that the function is strictly decreasing throughout that interval? Reason your answer theoretically or using a graphical example.
A projectile is launched vertically upwards such that its height in metres is given by the function $h(t) = -5t^2 + 40t + 10$, where $t$ is the time in seconds.…
A projectile is launched vertically upwards such that its height in metres is given by the function $h(t) = -5t^2 + 40t + 10$, where $t$ is the time in seconds. Calculate the average rate of change of the height in the intervals $[0, 2]$ and $[6, 8]$. Physically interpret the meaning and the sign of the results obtained.
The average production cost of $x$ decametres of a special cable is given by the function $C(x) = \frac{400}{x} + 5x$ (in euros). Calculate the average rate of change of…
The average production cost of $x$ decametres of a special cable is given by the function $C(x) = \frac{400}{x} + 5x$ (in euros). Calculate the average rate of change of the cost between $x = 10$ and $x = 20$. Does the average cost increase or decrease in that production interval?
Maxima and minima
For a parabola $y = ax^2+bx+c$, with $a\ne0$ and domain $\mathbb R$, the vertex is its only extremum: an absolute maximum if $a<0$ (opens downwards), an absolute minimum if $a>0$ (opens upwards). Its $x$-coordinate is $xv = -\dfrac{b}{2a}$.
Observe the graph of the function $f(x)$ and determine its intervals of increase and decrease. Also write the coordinates of its relative maximums and minimums.
Observe the graph of the function $f(x)$ and determine its intervals of increase and decrease. Also write the coordinates of its relative maximums and minimums.
The following table shows the daily profits (in hundreds of euros) of a small shop over one week: | Day | M | T | W | T | F | S | S | |---|---|---|---|---|---|---|---| |…
The following table shows the daily profits (in hundreds of euros) of a small shop over one week:
| Day | M | T | W | T | F | S | S |
|---|---|---|---|---|---|---|---|
| Profit | 4 | 8 | 5 | 2 | 6 | 9 | 3 |
Identify on which days the relative maximums and minimums of the profits are reached during that week, justifying your answer based on increase and decrease.
Find the coordinates of the vertex of the parabola with equation $y = -3x^2 + 12x - 5$. State whether that vertex corresponds to an absolute maximum or an absolute…
Find the coordinates of the vertex of the parabola with equation $y = -3x^2 + 12x - 5$. State whether that vertex corresponds to an absolute maximum or an absolute minimum of the function.
The attached graph shows the concentration of a medication in a patient's bloodstream (in mg/L) as a function of the time elapsed (in hours) since it was administered.…
The attached graph shows the concentration of a medication in a patient's bloodstream (in mg/L) as a function of the time elapsed (in hours) since it was administered. Determine the intervals where the concentration increases and decreases, and state the time at which the maximum concentration is reached and its approximate value.
Optimisation: finding the best value
A bicycle factory estimates that its monthly profits, in thousands of euros, follow the function $B(x) = -0,5x^2 + 50x - 200$, where $x$ is the quantity of bicycles…
A bicycle factory estimates that its monthly profits, in thousands of euros, follow the function $B(x) = -0,5x^2 + 50x - 200$, where $x$ is the quantity of bicycles produced in tens of units. How many bicycles must they manufacture to obtain the maximum possible profit? How much will this profit be?
Find the values of the parameters $a$ and $b$ so that the quadratic function $f(x) = ax^2 + bx + 3$ reaches a relative maximum at the point $(2, 11)$.
Find the values of the parameters $a$ and $b$ so that the quadratic function $f(x) = ax^2 + bx + 3$ reaches a relative maximum at the point $(2, 11)$.
A farmer has $240$ metres of fencing to enclose a rectangular field adjacent to a straight river. If the side bordering the river does not need fencing, express the area…
A farmer has $240$ metres of fencing to enclose a rectangular field adjacent to a straight river. If the side bordering the river does not need fencing, express the area of the field as a function of its width $x$. What should the dimensions of the field be so that the enclosed area is as large as possible?
A bus company charges $30$ euros per ticket and, at that price, manages to sell $400$ tickets daily. A market study indicates that for each euro the ticket price…
A bus company charges $30$ euros per ticket and, at that price, manages to sell $400$ tickets daily. A market study indicates that for each euro the ticket price increases, $10$ fewer tickets will be sold. Express the company's daily revenue as a function of the price increase in euros, $x$. What ticket price maximises revenue and how many tickets will be sold at that price?