Investigating exponential functions

Exponential and logarithmic functions, grade 12 (5 units) Lesson 14 of 23Learning

Investigating exponential functions

The steps of an investigation

A function with $\displaystyle e^x$ is investigated in the same steps as a rational function or a root function in grade 11:

the domain;
the points where the graph meets the axes;
asymptotes – vertical ones, and horizontal ones where $\displaystyle x$ is far from 0;
$\displaystyle f'(x)$ : the extremum points and the intervals of increase and decrease;
$\displaystyle f''(x)$ : the intervals of concavity and the inflection points;
a sketch of the graph.

$\displaystyle e^{f(x)}>0$ for every $\displaystyle x$ , so the sign of $\displaystyle e^{f(x)}\cdot g(x)$ is the sign of $\displaystyle g(x)$ , and it is 0 only where $\displaystyle {g(x)}={0}$ . It pays to take $\displaystyle e^{f(x)}$ out as a common factor in $\displaystyle f'$ and in $\displaystyle f''$ .

\displaystyle e^x

grows faster than any power

As $\displaystyle x$ grows, $\displaystyle e^x$ overtakes every power of $\displaystyle x$ , even when the power is larger at first:

$\displaystyle {x}={5}$ : $\displaystyle {x^3}={125}$ , $\displaystyle e^x\approx148$ .
$\displaystyle {x}={10}$ : $\displaystyle {x^3}={1000}$ , $\displaystyle e^x\approx22{,}026$ .
$\displaystyle {x}={20}$ : $\displaystyle {x^3}={8000}$ , $\displaystyle e^x\approx485{,}165{,}195$ .

Two facts follow, which we use for asymptotes:

As $\displaystyle x$ grows without limit, $\displaystyle {x^ne^{-x}}={\frac{x^n}{e^x}}$ comes close to 0.
Where $\displaystyle x$ is negative and far from 0, $\displaystyle x^ne^x$ comes close to 0.

Investigating

\displaystyle x^2e^{-x}

The example below investigates the function $\displaystyle {f(x)}={x^2e^{-x}}$ :

The domain: every $\displaystyle x$ .
The axes: $\displaystyle {x^2e^{-x}}={0}$ only where $\displaystyle {x}={0}$ , and $\displaystyle f(x)\geq0$ for every $\displaystyle x$ .
Asymptote: as $\displaystyle x$ grows, $\displaystyle \frac{x^2}{e^x}$ comes close to 0 – the asymptote $\displaystyle {y}={0}$ on the right. Where $\displaystyle x$ is negative and far from 0, $\displaystyle f(x)$ grows without limit.
$\displaystyle {f'(x)}={2xe^{-x}-x^2e^{-x}}={e^{-x}(2x-x^2)}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={0}$ or $\displaystyle {x}={2}$ .
A minimum $\displaystyle (0,0)$ , a maximum $\displaystyle \left(2,\frac{4}{e^2}\right)\approx(2,0.541)$ .
$\displaystyle {f''(x)}={e^{-x}(2-2x)-e^{-x}(2x-x^2)}={e^{-x}(x^2-4x+2)}$ , and $\displaystyle {f''(x)}={0}$ where $\displaystyle {x}={2\pm\sqrt{2}}$ .

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

\displaystyle M=(2,0.541)

\displaystyle P=(0.586,0.191)

\displaystyle Q=(3.414,0.384)

The graph is concave up where $\displaystyle x<2-\sqrt{2}$ or $\displaystyle x>2+\sqrt{2}$ , and concave down where $\displaystyle 2-\sqrt{2}<x<2+\sqrt{2}$ .
Two inflection points: $\displaystyle (0.586,0.191)$ and $\displaystyle (3.414,0.384)$ , roughly.
The right-hand inflection point is where the graph stops bending down and begins to come close to the asymptote.

Concavity on the two sides of an asymptote

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

The domain: $\displaystyle x\neq0$ . The vertical asymptote: $\displaystyle {x}={0}$ .
Where $\displaystyle x$ is negative and far from 0, $\displaystyle e^x$ comes close to 0, and so does $\displaystyle \frac{e^x}{x}$ : the asymptote $\displaystyle {y}={0}$ on the left.
$\displaystyle {f'(x)}={\frac{e^x\cdot x-e^x}{x^2}}={\frac{e^x(x-1)}{x^2}}$ : a minimum $\displaystyle (1,e)$ .
$\displaystyle {f''(x)}={\frac{e^x(x^2-2x+2)}{x^3}}$ . $\displaystyle x^2-2x+2>0$ for every $\displaystyle x$ (its discriminant is negative), so the sign of $\displaystyle f''$ is the sign of $\displaystyle x^3$ .

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

\displaystyle x=0

\displaystyle M=(1,2.718)

Concave down where $\displaystyle x<0$ , and concave up where $\displaystyle x>0$ .
The concavity changes at $\displaystyle {x}={0}$ , but there is no inflection point there: $\displaystyle {x}={0}$ is not in the domain.

Exercise 1

The function $\displaystyle {f(x)}={xe^x}$ is given.

Find $\displaystyle f'(x)$ , and take out a common factor.

$\displaystyle f(x)=xe^{x}$

/ – 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

Find $\displaystyle f''(x)$ , and take out a common factor.

$\displaystyle f(x)=xe^{x}$

/ – 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 graph is concave up: solve the inequality $\displaystyle f''(x)>0$ .

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

\displaystyle f(x)=xe^{x}

Exercise 2

The function $\displaystyle {f(x)}={e^{-\frac{x^2}{2}}}$ is given.
$\displaystyle {f'(x)}={-xe^{-\frac{x^2}{2}}}$ .

Find $\displaystyle f''(x)$ , and take out a common factor.

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

/ – 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 and the inflection points on the graph.

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

Exercise 3

The function $\displaystyle {f(x)}={(x^2-3)e^x}$ is given.

Find $\displaystyle f'(x)$ , and take out a common factor.

$\displaystyle f(x)=(x^{2}-3)e^{x}$

/ – 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 $\displaystyle f''(x)$ , and take out a common factor.

$\displaystyle f(x)=(x^{2}-3)e^{x}$

/ – 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 on the graph the extremum points, the inflection points and the points where it meets the x-axis.

\displaystyle f(x)=(x^{2}-3)e^{x}

Exercise 4

The function $\displaystyle {f(x)}={(x-a)e^x}$ is given, where $\displaystyle a$ is a parameter.
The graph of the function has an inflection point where $\displaystyle {x}={0}$ .

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

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

/ – 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

Put $\displaystyle {x}={0}$ into $\displaystyle {f''(x)}={0}$ , and solve the equation: find $\displaystyle a$ .

Put in the $\displaystyle a$ you found. $\displaystyle {f'(x)}={e^x(x-1)}$ .
Calculate the y-coordinate of the extremum point, decide its kind, and explain.

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

/ – 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

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
$\displaystyle \left(\int_{a}^{x}f(t)\,dt\right)^{\prime}=f(x)$