Combined investigations and families of functions

Exponential and logarithmic functions, grade 12 (5 units) Lesson 17 of 23Learning

Combined investigations and families of functions

Two horizontal asymptotes:

\displaystyle \frac{e^x}{e^x+1}

The example below investigates $\displaystyle {f(x)}={\frac{e^x}{e^x+1}}$ – a rational function of $\displaystyle e^x$ :

The domain: every $\displaystyle x$ , since $\displaystyle e^x+1>0$ .
$\displaystyle 0<e^x<e^x+1$ , so $\displaystyle 0<f(x)<1$ for every $\displaystyle x$ .
As $\displaystyle x\to-\infty$ , $\displaystyle e^x\to0$ and $\displaystyle f(x)\to0$ : the asymptote $\displaystyle {y}={0}$ on the left.
As $\displaystyle x\to\infty$ , $\displaystyle {f(x)}={\frac{1}{1+e^{-x}}\to1}$ : the asymptote $\displaystyle {y}={1}$ on the right.
$\displaystyle {f'(x)}={\frac{e^x(e^x+1)-e^x\cdot e^x}{(e^x+1)^2}}={\frac{e^x}{(e^x+1)^2}>0}$ : the function increases throughout, with no extremum points.
$\displaystyle {f''(x)}={\frac{e^x(1-e^x)}{(e^x+1)^3}}$ , and its sign is the sign of $\displaystyle 1-e^x$ .

\displaystyle f(x)=\frac{e^{x}}{e^{x}+1}

\displaystyle y=0

\displaystyle y=1

\displaystyle P=(0,0.5)

Where $\displaystyle x<0$ , $\displaystyle e^x<1$ and $\displaystyle f''(x)>0$ – concave up; where $\displaystyle x>0$ , $\displaystyle f''(x)<0$ – concave down.
An inflection point: $\displaystyle \left(0,\frac{1}{2}\right)$ – where the function increases fastest.

A vertical tangent at the end of the domain:

\displaystyle \sqrt{e^x-1}

The example below investigates $\displaystyle {f(x)}={\sqrt{e^x-1}}$ :

The domain: $\displaystyle e^x-1\geq0$ , that is $\displaystyle e^x\geq1$ and $\displaystyle x\geq0$ .
$\displaystyle {f(0)}={0}$ : the graph starts at the origin.
$\displaystyle {f'(x)}={\frac{e^x}{2\sqrt{e^x-1}}>0}$ where $\displaystyle x>0$ : the function increases throughout, and the end point $\displaystyle (0,0)$ is a minimum point.
As $\displaystyle x\to0^+$ , the denominator tends to 0 and the numerator to 1, so $\displaystyle f'(x)\to\infty$ : the graph leaves the origin upright, and the tangent there is the y-axis.
As $\displaystyle x\to\infty$ , $\displaystyle f(x)\to\infty$ : no horizontal asymptote.

\displaystyle f(x)=\sqrt{e^{x}-1}

Investigating

\displaystyle \frac{\ln x}{\sqrt{x}}

The example below investigates $\displaystyle {f(x)}={\frac{\ln x}{\sqrt{x}}}$ :

The domain: $\displaystyle x>0$ . The x-axis: $\displaystyle (1,0)$ .
As $\displaystyle x\to0^+$ , $\displaystyle \ln x\to-\infty$ and $\displaystyle \sqrt{x}\to0^+$ , so $\displaystyle f(x)\to-\infty$ : the vertical asymptote $\displaystyle {x}={0}$ .
As $\displaystyle x\to\infty$ , $\displaystyle \ln x$ grows more slowly than $\displaystyle \sqrt{x}$ , so $\displaystyle f(x)\to0$ : the horizontal asymptote $\displaystyle {y}={0}$ .
$\displaystyle {f'(x)}={\frac{\frac{1}{x}\cdot\sqrt{x}-\ln x\cdot\frac{1}{2\sqrt{x}}}{x}}={\frac{2-\ln x}{2x\sqrt{x}}}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={e^2}$ .
$\displaystyle f$ increases where $\displaystyle 0<x<e^2$ and decreases where $\displaystyle x>e^2$ : a maximum $\displaystyle \left(e^2,\frac{2}{e}\right)\approx(7.39,0.74)$ .

\displaystyle f(x)=\frac{\ln x}{\sqrt{x}}

\displaystyle x=0

\displaystyle M=(7.389,0.736)

A family with a parameter:

\displaystyle x^2-a\ln x

The example below investigates the family $\displaystyle {f(x)}={x^2-a\ln x}$ , with $\displaystyle a>0$ a parameter:

The domain: $\displaystyle x>0$ , for every $\displaystyle a$ .
$\displaystyle {f'(x)}={2x-\frac{a}{x}}={\frac{2x^2-a}{x}}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={\sqrt{\frac{a}{2}}}$ (the negative solution is not in the domain).
$\displaystyle {f''(x)}={2+\frac{a}{x^2}>0}$ : every function of the family is concave up, so the point is a minimum and there are no inflection points.

On the graph: $\displaystyle {a}={2}$ and $\displaystyle {a}={8}$ .

\displaystyle f(x)=x^{2}-2\ln x

\displaystyle g(x)=x^{2}-8\ln x

$\displaystyle {a}={2}$ : the minimum $\displaystyle (1,1)$ .
$\displaystyle {a}={8}$ : the minimum $\displaystyle (2,4-8\ln2)\approx(2,-1.55)$ .
The larger $\displaystyle a$ , the further right the minimum point: the parameter changes where the extremum is, not its type.

Exercise 1

The function $\displaystyle {f(x)}={\frac{e^x}{x^2}}$ is given, for $\displaystyle x\neq0$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{e^{x}}{x^{2}}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark the extremum point on the graph.
Add a row for each of the two asymptotes: the vertical one, and the horizontal one on the left.

\displaystyle f(x)=\frac{e^{x}}{x^{2}}

Exercise 2

The function $\displaystyle {f(x)}={\sqrt{x}\ln x}$ is given, for $\displaystyle x>0$ .

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\sqrt{x}\ln x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Find $\displaystyle f''(x)$ .

$\displaystyle f(x)=\sqrt{x}\ln x$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Find where the function is concave up: solve the inequality $\displaystyle f''(x)>0$ .

Mark the extremum point and the inflection point on the graph.

\displaystyle f(x)=\sqrt{x}\ln x

Exercise 3

The family $\displaystyle {f(x)}={e^x-ax}$ is given, with $\displaystyle a>0$ a parameter.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=e^{x}-ax$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Show that for every $\displaystyle a>0$ the function has one extremum point, where $\displaystyle {x}={\ln a}$ , and find its type.

Find the y-coordinate of the extremum point.

$\displaystyle f(x)=e^{x}-ax$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

The extremum point is on the x-axis.
Find $\displaystyle a$ .

Substitute the $\displaystyle a$ you found, and mark the extremum point on the graph.

\displaystyle f(x)=e^{x}-ex

Exercise 4

The function $\displaystyle {f(x)}={\frac{x}{\ln x}}$ is given.

Find the domain, and explain.

Find $\displaystyle f'(x)$ .

$\displaystyle f(x)=\frac{x}{\ln x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Solve the equation $\displaystyle {f'(x)}={0}$ .

Mark the extremum point on the graph, and add a row with the equation of the vertical asymptote.

\displaystyle f(x)=\frac{x}{\ln x}

Formulas: exponential and logarithmic functions, grade 12 (5 units)

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Powers

A product of powers with the same base
$\displaystyle a^{x}\cdot a^{y}$ $\displaystyle =$ $\displaystyle a^{x+y}$
A quotient of powers with the same base
$\displaystyle \frac{a^{x}}{a^{y}}$ $\displaystyle =$ $\displaystyle a^{x-y}$
A power of a power
$\displaystyle (a^{x})^{y}$ $\displaystyle =$ $\displaystyle a^{x\cdot y}$
A power of a product
$\displaystyle (a\cdot b)^{x}$ $\displaystyle =$ $\displaystyle a^{x}\cdot b^{x}$
A power of a quotient
$\displaystyle \left(\frac{a}{b}\right)^{x}$ $\displaystyle =$ $\displaystyle \frac{a^{x}}{b^{x}}$
A negative exponent
$\displaystyle a^{-x}$ $\displaystyle =$ $\displaystyle \frac{1}{a^{x}}$
The exponent 0
$\displaystyle a^{0}$ $\displaystyle =$ $\displaystyle 1$ $\displaystyle a\neq0$
A rational exponent
$\displaystyle a^{\frac{m}{n}}$ $\displaystyle =$ $\displaystyle \sqrt[n]{a^{m}}$ $\displaystyle a>0$

Logarithms (within the domain)

The definition of the logarithm
$\displaystyle \log_{a}x=b\iff a^{b}=x$ $\displaystyle a>0,\ a\neq1,\ x>0$
$\displaystyle \log_{a}(a^{b})$ $\displaystyle =$ $\displaystyle b$
$\displaystyle a^{\log_{a}x}$ $\displaystyle =$ $\displaystyle x$
The logarithm of a product
$\displaystyle \log_{a}x+\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}(x\cdot y)$
The logarithm of a quotient
$\displaystyle \log_{a}x-\log_{a}y$ $\displaystyle =$ $\displaystyle \log_{a}\left(\frac{x}{y}\right)$
The logarithm of a power
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
Change of base
$\displaystyle \log_{c}x$ $\displaystyle =$ $\displaystyle \frac{\log_{a}x}{\log_{a}c}$
$\displaystyle \log_{a}b$ $\displaystyle =$ $\displaystyle \frac{1}{\log_{b}a}$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

The exponential and the logarithmic function

Every exponential as a power of $\displaystyle e$
$\displaystyle a^{x}$ $\displaystyle =$ $\displaystyle e^{x\ln a}$
$\displaystyle a^x$ and $\displaystyle \log_a x$ – inverse functions
$\displaystyle y=a^{x}\iff x=\log_{a}y$ $\displaystyle y>0$
The number $\displaystyle e$
$\displaystyle e\approx2.718$

Growth and decay

The amount after $\displaystyle t$ units of time
$\displaystyle f(t)=f(0)\cdot q^{t}$
The growth / decay factor – $\displaystyle p$ the percent per unit of time
$\displaystyle q=1\pm\frac{p}{100}$
The half-life $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$
The doubling time $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle 2$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (a^{x})^{\prime}$ $\displaystyle =$ $\displaystyle a^{x}\cdot\ln a$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
$\displaystyle (\log_{a}x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x\cdot\ln a}$
The derivative of $\displaystyle e^{f(x)}$
$\displaystyle \left[e^{f(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle e^{f(x)}\cdot f^{\prime}(x)$
The derivative of $\displaystyle \ln(f(x))$
$\displaystyle \left[\ln(f(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)}{f(x)}$
The derivative of a product
$\displaystyle \left[f(x)\cdot g(x)\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x)$
The derivative of a quotient
$\displaystyle \left[\frac{f(x)}{g(x)}\right]^{\prime}$ $\displaystyle =$ $\displaystyle \frac{f^{\prime}(x)\cdot g(x)-f(x)\cdot g^{\prime}(x)}{\left[g(x)\right]^{2}}$
The derivative of a composite function
$\displaystyle \left[f(u(x))\right]^{\prime}$ $\displaystyle =$ $\displaystyle f^{\prime}(u)\cdot u^{\prime}(x)$
The tangent’s equation at the point where $\displaystyle {x}={a}$
$\displaystyle y-f(a)=f^{\prime}(a)(x-a)$

Integrals, the area and accumulation

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int a^{x}\,dx=\frac{a^{x}}{\ln a}+C$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
Where $\displaystyle f(x)\neq0$
$\displaystyle \int\frac{f^{\prime}(x)}{f(x)}\,dx=\ln|f(x)|+C$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The accumulation function’s derivative
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$