Investigating exponential functions

Exponential and logarithmic functions, grade 12 (4 units) Lesson 13 of 20Learning

Investigating exponential functions

The steps of an investigation

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

the domain;
the points where the graph meets the axes;
the derivative, the extremum points and the intervals of increase and decrease;
the intervals where the function is positive and negative;
asymptotes;
a sketch of the graph.

\displaystyle e^{f(x)}

is always positive

$\displaystyle e^{f(x)}>0$ for every $\displaystyle x$ , so in a function of the form $\displaystyle e^{f(x)}\cdot g(x)$ :

$\displaystyle {e^{f(x)}\cdot g(x)}={0}$ only where $\displaystyle {g(x)}={0}$ ;
the sign of $\displaystyle e^{f(x)}\cdot g(x)$ is the sign of $\displaystyle g(x)$ .

So it pays to take $\displaystyle e^{f(x)}$ out as a common factor of the derivative: $\displaystyle {e^{2x}(1+2x)}={0}$ only where $\displaystyle {1+2x}={0}$ .

Investigating

\displaystyle xe^{2x}

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

The domain: every $\displaystyle x$ .
The axes: $\displaystyle {xe^{2x}}={0}$ only where $\displaystyle {x}={0}$ . The graph meets both axes at the origin, $\displaystyle (0,0)$ .
The derivative: $\displaystyle {f'(x)}={e^{2x}+2xe^{2x}}={e^{2x}(1+2x)}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={-\frac{1}{2}}$ .
Where $\displaystyle x<-\frac{1}{2}$ , $\displaystyle f'(x)<0$ – the function decreases; where $\displaystyle x>-\frac{1}{2}$ , $\displaystyle f'(x)>0$ – it increases.
A minimum point: $\displaystyle {f\left(-\frac{1}{2}\right)}={-\frac{1}{2}e^{-1}}={-\frac{1}{2e}}$ , that is $\displaystyle \left(-\frac{1}{2},-\frac{1}{2e}\right)\approx(-0.5,-0.18)$ . It is also the absolute minimum.
Positive and negative: the sign of $\displaystyle f$ is the sign of $\displaystyle x$ – positive where $\displaystyle x>0$ , negative where $\displaystyle x<0$ .

\displaystyle f(x)=xe^{2x}

\displaystyle M=(-0.5,-0.184)

Where $\displaystyle x$ is negative and far from 0, $\displaystyle xe^{2x}$ is close to 0: the line $\displaystyle {y}={0}$ is a horizontal asymptote on the left.

The graph of the derivative

\displaystyle f'(x)

On the same axes, the graph of $\displaystyle f$ (dashed) and the graph of the derivative $\displaystyle {g(x)}={f'(x)}={e^{2x}(1+2x)}$ :

\displaystyle f(x)=xe^{2x}

\displaystyle g(x)=e^{2x}(1+2x)

The graph of $\displaystyle f'$ meets the x-axis at $\displaystyle {x}={-\frac{1}{2}}$ , right below the minimum point of $\displaystyle f$ .
Left of $\displaystyle -\frac{1}{2}$ the graph of $\displaystyle f'$ is below the x-axis, and $\displaystyle f$ decreases; right of it the graph is above the axis, and $\displaystyle f$ increases.

A horizontal asymptote

$\displaystyle e^x$ has the horizontal asymptote $\displaystyle {y}={0}$ : where $\displaystyle x$ is negative and far from 0, $\displaystyle e^x$ is close to 0. A transformation of $\displaystyle e^x$ moves the asymptote:

$\displaystyle {f(x)}={e^x+2}$ : a vertical shift by 2, and the asymptote is $\displaystyle {y}={2}$ (where $\displaystyle x$ is negative and far from 0).
$\displaystyle {g(x)}={3-e^{-x}}$ : $\displaystyle e^{-x}$ is close to 0 where $\displaystyle x$ is positive and large, so the asymptote is $\displaystyle {y}={3}$ – on the right.
In the function $\displaystyle a\cdot e^{mx+b}+k$ the horizontal asymptote is $\displaystyle {y}={k}$ .

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

\displaystyle y=2

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

\displaystyle y=3

Finding a parameter

The graph of the function $\displaystyle {f(x)}={(x^2+ax+1)e^{-x}}$ meets the x-axis at the point where $\displaystyle {x}={-1}$ . The example below finds $\displaystyle a$ and investigates the function:

$\displaystyle {f(-1)}={0}$ : $\displaystyle {(1-a+1)e^{1}}={0}$ , and $\displaystyle e\neq0$ , so $\displaystyle {2-a}={0}$ and $\displaystyle {a}={2}$ .
$\displaystyle {f(x)}={(x^2+2x+1)e^{-x}}={(x+1)^2e^{-x}}$ .
$\displaystyle {f'(x)}={2(x+1)e^{-x}-(x+1)^2e^{-x}}={e^{-x}(x+1)(1-x)}$ , and $\displaystyle {f'(x)}={0}$ where $\displaystyle {x}={-1}$ or $\displaystyle {x}={1}$ .
$\displaystyle f$ decreases where $\displaystyle x<-1$ and where $\displaystyle x>1$ , and increases where $\displaystyle -1<x<1$ .
A minimum $\displaystyle (-1,0)$ , a maximum $\displaystyle \left(1,\frac{4}{e}\right)\approx(1,1.47)$ .
$\displaystyle f(x)\geq0$ for every $\displaystyle x$ , and the graph meets the y-axis at $\displaystyle (0,1)$ .
The horizontal asymptote is $\displaystyle {y}={0}$ , where $\displaystyle x$ is positive and large.

\displaystyle f(x)=(x+1)^{2}e^{-x}

Exercise 1

The function $\displaystyle {f(x)}={(x-1)e^x}$ is given.

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

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

/ – 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

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

Find the interval of increase: solve the inequality $\displaystyle f'(x)>0$ .

Mark on the graph the extremum point and the points where it meets the axes.

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

Exercise 2

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

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

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

/ – 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

Find the intervals of increase and decrease.

Mark the extremum points on the graph.

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

State the kind of each extremum point, and write the equation of the horizontal asymptote.

Exercise 3

The function $\displaystyle {g(x)}={-2e^{x-1}+4}$ is given.

Write the equation of the horizontal asymptote, and say whether $\displaystyle g$ increases or decreases.

Find the point where the graph meets the x-axis: solve the equation $\displaystyle {g(x)}={0}$ .

Mark on the graph the points where it meets the axes.

\displaystyle g(x)=-2e^{x-1}+4

\displaystyle y=4

Exercise 4

The function $\displaystyle {f(x)}={(x+a)e^x}$ is given, $\displaystyle a$ a parameter.
The function has an extremum point where $\displaystyle {x}={2}$ .

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

Find $\displaystyle a$ .

Put in the $\displaystyle a$ you found, and calculate the y-coordinate of the extremum point.
State its kind, and explain.

/ – 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 (4 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
$\displaystyle \log_{a}(x^{b})$ $\displaystyle =$ $\displaystyle b\cdot\log_{a}x$
The natural logarithm – to the base $\displaystyle e$
$\displaystyle \ln x=\log_{e}x$

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 $\displaystyle T$
$\displaystyle q^{T}$ $\displaystyle =$ $\displaystyle \frac{1}{2}$

Derivatives

$\displaystyle (e^{x})^{\prime}$ $\displaystyle =$ $\displaystyle e^{x}$
$\displaystyle (\ln x)^{\prime}$ $\displaystyle =$ $\displaystyle \frac{1}{x}$
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 and the area

$\displaystyle \int e^{x}\,dx=e^{x}+C$
$\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C$ $\displaystyle m\neq0$
In any interval where $\displaystyle x\neq0$
$\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C$
$\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C$ $\displaystyle m\neq0$
$\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$