Extremum problems with exponential and logarithmic functions

Exponential and logarithmic functions, grade 12 (5 units) Lesson 18 of 23Learning

Extremum problems with exponential and logarithmic functions

The steps of a solution

An extremum problem is solved in the same steps as in grade 11; only the functions are new:

choose a variable, and write its interval from the problem;
write the quantity to make largest or smallest as a function of the variable;
differentiate, and solve $\displaystyle {f'(x)}={0}$ ;
check with the sign of $\displaystyle f'$ (or of $\displaystyle f''$ ) that the point is the maximum or minimum wanted;
on a closed interval, compare with the values at the ends too;
answer in words, with units.

A medicine’s concentration in the blood

The concentration of a medicine in the blood, $\displaystyle t$ hours after it was taken, is $\displaystyle {C(t)}={20te^{-0.5t}}$ milligrams per litre ( $\displaystyle t\geq0$ ).
When is the concentration highest, and what is it?

The example below finds the maximum of $\displaystyle C$ for $\displaystyle t\geq0$ :

$\displaystyle {C'(t)}={20e^{-0.5t}+20t\cdot(-0.5)e^{-0.5t}}={20e^{-0.5t}(1-0.5t)}$ .
$\displaystyle e^{-0.5t}>0$ , so $\displaystyle {C'(t)}={0}$ only where $\displaystyle {1-0.5t}={0}$ , that is $\displaystyle {t}={2}$ .
Where $\displaystyle 0\leq t<2$ , $\displaystyle C'(t)>0$ – the concentration rises; where $\displaystyle t>2$ , $\displaystyle C'(t)<0$ – it falls. So $\displaystyle {t}={2}$ is an absolute maximum.
$\displaystyle {C(2)}={40e^{-1}}={\frac{40}{e}\approx14.7}$ .

The highest concentration, about 14.7 milligrams per litre, comes two hours after the medicine is taken.

\displaystyle C(x)=20xe^{-0.5x}

\displaystyle M=(2,14.715)

A rectangle under the graph of

\displaystyle e^{-x^2}

Two vertices of a rectangle are on the x-axis, and the other two on the graph of $\displaystyle {y}={e^{-x^2}}$ .
What is the largest area of such a rectangle?

The example below names the vertices $\displaystyle (\pm x,0)$ and $\displaystyle \left(\pm x,e^{-x^2}\right)$ , $\displaystyle x>0$ – the graph is symmetric about the y-axis:

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

\displaystyle A=(-0.707,0)

\displaystyle B=(0.707,0)

\displaystyle C=(0.707,0.607)

\displaystyle D=(-0.707,0.607)

The base $\displaystyle 2x$ , the height $\displaystyle e^{-x^2}$ , the area $\displaystyle {S(x)}={2xe^{-x^2}}$ .
$\displaystyle {S'(x)}={2e^{-x^2}+2x\cdot(-2x)e^{-x^2}}={2e^{-x^2}(1-2x^2)}$ .
$\displaystyle {S'(x)}={0}$ where $\displaystyle {x^2}={\frac{1}{2}}$ , that is $\displaystyle {x}={\frac{1}{\sqrt{2}}\approx0.71}$ ( $\displaystyle x>0$ ).
$\displaystyle S'>0$ left of the point and $\displaystyle S'<0$ right of it: a maximum.
$\displaystyle {S\left(\frac{1}{\sqrt{2}}\right)}={\sqrt{2}e^{-\frac{1}{2}}\approx0.86}$ .

The triangle a tangent cuts from the axes

At the point $\displaystyle (t,e^{-t})$ , $\displaystyle t\geq0$ , a tangent is drawn to the graph of $\displaystyle {y}={e^{-x}}$ .
The tangent and the axes make a triangle.
What is the largest area of such a triangle?

The example below writes the area in terms of $\displaystyle t$ :

The slope of the tangent: $\displaystyle {\left(e^{-x}\right)'}={-e^{-x}}$ , and at the point $\displaystyle -e^{-t}$ . The tangent: $\displaystyle {y-e^{-t}}={-e^{-t}(x-t)}$ .
The y-axis ( $\displaystyle {x}={0}$ ): $\displaystyle {y}={e^{-t}(1+t)}$ . The x-axis ( $\displaystyle {y}={0}$ ): $\displaystyle {x}={t+1}$ .
The area: $\displaystyle {A(t)}={\frac{(t+1)\cdot e^{-t}(t+1)}{2}}={\frac{(t+1)^2e^{-t}}{2}}$ .

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

\displaystyle y=-e^{-1}x+2e^{-1}

\displaystyle P=(1,0.368)

$\displaystyle {A'(t)}={\frac{2(t+1)e^{-t}-(t+1)^2e^{-t}}{2}}={\frac{e^{-t}(t+1)(1-t)}{2}}$ , and $\displaystyle {A'(t)}={0}$ where $\displaystyle {t}={1}$ ( $\displaystyle t\geq0$ ).
$\displaystyle A'>0$ where $\displaystyle 0\leq t<1$ and $\displaystyle A'<0$ where $\displaystyle t>1$ : a maximum.
$\displaystyle {A(1)}={2e^{-1}}={\frac{2}{e}\approx0.74}$ . On the graph: the tangent at the point $\displaystyle (1,e^{-1})$ .

The absolute extremum on a closed interval

On a closed interval, the largest and the smallest value of a function are at an extremum point or at an end of the interval.

The example below finds the extreme values of $\displaystyle {f(x)}={\frac{\ln x}{x}}$ on the interval $\displaystyle 1\leq x\leq e^2$ :

$\displaystyle {f'(x)}={\frac{1-\ln x}{x^2}}={0}$ where $\displaystyle {x}={e}$ , inside the interval.
The values: $\displaystyle {f(1)}={0}$ , $\displaystyle {f(e)}={\frac{1}{e}\approx0.37}$ , $\displaystyle {f(e^2)}={\frac{2}{e^2}\approx0.27}$ .
The absolute maximum $\displaystyle \frac{1}{e}$ , at $\displaystyle {x}={e}$ ; the absolute minimum $\displaystyle 0$ , at the end $\displaystyle {x}={1}$ .

Exercise 1

A shop sells a product at a price of $\displaystyle x$ shekels.
The shop’s daily revenue from the product is $\displaystyle {R(x)}={100xe^{-0.1x}}$ shekels.
At what price is the daily revenue highest?

Find $\displaystyle R'(x)$ .

$\displaystyle R(x)=100xe^{-0.1x}$

/ – 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 $\displaystyle {R'(x)}={0}$ .

Calculate the highest revenue, rounded to the agora.

$\displaystyle R(x)=100xe^{-0.1x}$

/ – 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

Exercise 2

For every $\displaystyle x>0$ there is a point on the graph of $\displaystyle {y}={\ln x}$ and a point on the line $\displaystyle {y}={x}$ , one above the other.
The vertical distance between the two points is $\displaystyle {d(x)}={x-\ln x}$ .
Find the smallest vertical distance.

\displaystyle f(x)=\ln x

\displaystyle g(x)=x

Find $\displaystyle d'(x)$ .

$\displaystyle d(x)=x-\ln x$

/ – 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 $\displaystyle {d'(x)}={0}$ .

Calculate the smallest distance.

$\displaystyle d(x)=x-\ln x$

/ – 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

Exercise 3

One vertex of a rectangle is at the origin, and two of its sides are on the axes.
The vertex opposite the origin is on the graph of $\displaystyle {y}={3-\ln x}$ , for $\displaystyle 0<x<e^3$ .
The area of the rectangle is $\displaystyle {S(x)}={x(3-\ln x)}$ .

\displaystyle f(x)=3-\ln x

Find $\displaystyle S'(x)$ .

$\displaystyle S(x)=x(3-\ln x)$

/ – 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 $\displaystyle {S'(x)}={0}$ .

Calculate the largest area.

$\displaystyle S(x)=x(3-\ln x)$

/ – 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

Explain with the sign of $\displaystyle S'$ that this is indeed a maximum.

Exercise 4

The function $\displaystyle {f(x)}={x^2e^{-x}}$ is given, on the interval $\displaystyle -1\leq x\leq3$ .
Find the largest and the smallest value of the function on the interval.

Find $\displaystyle f'(x)$ .

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

/ – 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

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

Calculate the values of the function at the ends of the interval and at the extremum point other than $\displaystyle {x}={0}$ ( $\displaystyle {f(0)}={0}$ ).

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

/ – 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

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

/ – 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

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

/ – 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

Write the largest and the smallest value, and where each one is reached.

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