Exponential function
A cloud server scales its traffic: every hour that passes, the number of requests is multiplied by a fixed factor, rather than having a constant amount added to it. A capacitor discharges and its voltage falls by being multiplied by the same factor in every equal interval of time. In both cases the variable $t$ is not in the base but in the exponent, and that completely changes the way the quantity grows or decays. This is the function you are going to formalise in this section.
The exponential function
Every function $f(x)=a^x$ passes through the fixed point $(0,1)$, because $a^0=1$, and through $(1,a)$, because $a^1=a$.
Among all the possible bases, one plays a special role: the number $e\approx2{,}71828\ldots$, an irrational number that appears naturally in continuous growth processes (compound interest with instantaneous compounding, radioactive decay, charging and discharging of capacitors). The function $f(x)=e^x$ is the reference exponential in science and engineering, and its base is always written like this, without approximating, unless the problem asks for a numerical value.
Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using…
Given the exponential function $f(x) = 3^x$, calculate the numerical values of $f(2)$, $f(-1)$, $f(0)$ and $f(\frac{1}{2})$. Express the results in exact form, using roots or fractions if necessary.
Find the algebraic expression for an exponential function of the form $f(x) = k \cdot a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.
Find the algebraic expression for an exponential function of the form $f(x) = k \cdot a^x$, knowing that its graph passes through the points $(0, 3)$ and $(2, 12)$.
Determine the domain of definition of the following function: $f(x) = \frac{x+2}{e^x - e^2}$
Determine the domain of definition of the following function:
$f(x) = \frac{x+2}{e^x - e^2}$
Growth and decay
The formal reason: when going from $x$ to $x+1$, the value is always multiplied by $a$, that is, $\dfrac{f(x+1)}{f(x)}=a$ for every $x$. If this constant factor is greater than $1$, each step increases the value; if it is less than $1$, each step reduces it.
Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base: $f(x) = 4^x$…
Classify the following exponential functions as increasing or decreasing, justifying your answer based on the value of their base:
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$f(x) = 4^x$
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$g(x) = \left(\frac{1}{3}\right)^x$
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$h(x) = 0,8^x$
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$i(x) = e^x$
Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0,5^x$,…
Match each of the graphs labelled A, B, C and D in the following figure with its corresponding exponential function. The functions are: $f(x) = 2^x$, $g(x) = 0,5^x$, $h(x) = 3^x$ and $i(x) = -2^x$.
The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$. Determine analytically the coordinates of the point…
The attached graph shows the functions $f(x) = e^x$ and $g(x) = e^{-x}$, together with the horizontal line $y = e$.
Determine analytically the coordinates of the point of intersection between the functions $f(x)$ and $g(x)$. Then, calculate the exact area of the triangle whose vertices are this point of intersection and the two points where the curves intersect the line $y = e$.
Domain, range and horizontal asymptote
When the graph is translated, the asymptote is translated with it, and it can be found rigorously by studying the limit. In $g(x)=a^{x-p}+q$:
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The exponent $x-p$ shifts the curve $p$ units horizontally.
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The final $+q$ shifts it $q$ units vertically.
If $a>1$, when $x\to-\infty$ the term $a^{x-p}\to0^+$, so $$\lim_{x\to-\infty} \left(a^{x-p}+q\right) = q.$$ The new horizontal asymptote is therefore $y=q$. (If $0 Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$. Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$. Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function $g(x) = 2^{x-3} + 1$. Determine the domain, range and the equation of the horizontal asymptote of $g(x)$.Determine the domain, range and the equation of the horizontal asymptote of the function $f(x) = 2^x - 5$.
Calculate analytically the coordinates of the x-intercept and y-intercept of the function $f(x) = 3^{x+1} - 9$.
Starting from the function $f(x) = 2^x$, explain in detail what geometric transformations must be applied to its graph to obtain the graph of the function…
Exponential equations
There are four basic techniques, from the simplest to the most general.
Solve the following exponential equation by equating the bases on both sides of the equation: $5^{2x-1} = 125$
Solve the following exponential equation by equating the bases on both sides of the equation:
$5^{2x-1} = 125$
Solve the following exponential equation by making an appropriate substitution: $9^x - 10 \cdot 3^x + 9 = 0$
Solve the following exponential equation by making an appropriate substitution:
$9^x - 10 \cdot 3^x + 9 = 0$
Solve the following exponential equation using logarithms: $4^{x-1} = 15$ Express the result in exact form using logarithms and also provide an approximation rounded to…
Solve the following exponential equation using logarithms:
$4^{x-1} = 15$
Express the result in exact form using logarithms and also provide an approximation rounded to two decimal places.
Models of growth and decay
Compound interest with many compounding periods per year tends, as the number of periods grows without limit, to continuously compounded interest: $C(t)=C_0\cdot e^{rt}$. It is the same idea as Newton's law of cooling or radioactive decay measured with the constant $k$ instead of the half-life: they all share the form $e^{\pm kt}$.
A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a…
A population of bacteria doubles every hour. If there are initially $150$ bacteria in the culture, write the function $P(t)$ that models the number of bacteria as a function of time $t$ (expressed in hours). How many bacteria will be in the culture after $4$ hours?
An amount of $5000$ euros is invested in a bank deposit offering an annual compound interest rate of $3\%$. Write the function that determines the accumulated capital…
An amount of $5000$ euros is invested in a bank deposit offering an annual compound interest rate of $3\%$.
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Write the function that determines the accumulated capital $C(t)$ after $t$ years.
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Calculate the total capital that will have accumulated after $10$ years.
According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is…
According to Newton's law of cooling, the temperature of an object at time $t$ (in minutes) is given by the expression $T(t) = T_a + (T_0 - T_a)e^{-kt}$, where $T_a$ is the ambient temperature and $T_0$ is the initial temperature of the object. A cup of coffee at $90^\circ\text{C}$ is left in a room kept at a constant temperature of $20^\circ\text{C}$. If after $10$ minutes the temperature of the coffee has dropped to $60^\circ\text{C}$:
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Calculate the exact value of the cooling constant $k$.
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Determine the total time elapsed from the beginning for the coffee to reach a temperature of $30^\circ\text{C}$.
Back to the beginning: If the server's traffic starts at $N_0$ requests and is multiplied by $k$ every hour, its evolution is $N(t) = N_0 \cdot k^t$; in the same way, if the capacitor's voltage falls by being halved every second, it follows the decreasing model $V(t) = V_0 \cdot (1/2)^t$.