Graphs of composite functions and of derivatives

Exponential and logarithmic functions, grade 12 (5 units) Lesson 16 of 23Learning

Graphs of composite functions and of derivatives

The graph of

\displaystyle e^{f(x)}

$\displaystyle e^{f(x)}$ increases exactly where $\displaystyle f$ increases, and decreases where $\displaystyle f$ decreases. The reason is in the derivative: $\displaystyle {\left(e^{f(x)}\right)'}={f'(x)\cdot e^{f(x)}}$ , and $\displaystyle e^{f(x)}>0$ , so the derivative has the sign of $\displaystyle f'(x)$ . So, without differentiating:

$\displaystyle e^{f(x)}$ has its extremum points at the same values of $\displaystyle x$ as $\displaystyle f$ , of the same type;
the value at an extremum point is $\displaystyle e$ to the power of the value of $\displaystyle f$ there;
$\displaystyle e^{f(x)}>0$ always; and where $\displaystyle f(x)\to-\infty$ , $\displaystyle e^{f(x)}\to0$ – the horizontal asymptote $\displaystyle {y}={0}$ .

For example, $\displaystyle {f(x)}={2x-x^2}$ (dashed) and $\displaystyle {g(x)}={e^{2x-x^2}}$ :

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

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

\displaystyle M=(1,2.718)

$\displaystyle f$ has the maximum $\displaystyle (1,1)$ , so $\displaystyle g$ has the maximum $\displaystyle {(1,e^1)}={(1,e)}$ .
As $\displaystyle x\to\pm\infty$ , $\displaystyle 2x-x^2\to-\infty$ , so $\displaystyle g(x)\to0$ : the asymptote $\displaystyle {y}={0}$ on both sides.

The graph of

\displaystyle \ln f(x)

$\displaystyle \ln f(x)$ is defined only where $\displaystyle f(x)>0$ , and there it increases and decreases with $\displaystyle f$ : $\displaystyle {\left(\ln f(x)\right)'}={\frac{f'(x)}{f(x)}}$ , and $\displaystyle f(x)>0$ .

The extremum points of $\displaystyle \ln f(x)$ are at the same values of $\displaystyle x$ as those of $\displaystyle f$ , inside the domain.
Where $\displaystyle f(x)\to0^+$ , $\displaystyle \ln f(x)\to-\infty$ – a vertical asymptote.
$\displaystyle {\ln f(x)}={0}$ where $\displaystyle {f(x)}={1}$ .

For example, $\displaystyle {g(x)}={\ln(4-x^2)}$ :

The domain: $\displaystyle 4-x^2>0$ , that is $\displaystyle -2<x<2$ ; the vertical asymptotes $\displaystyle {x}={-2}$ and $\displaystyle {x}={2}$ .
$\displaystyle 4-x^2$ has the maximum $\displaystyle (0,4)$ , so $\displaystyle g$ has the maximum $\displaystyle (0,\ln4)\approx(0,1.39)$ .

\displaystyle f(x)=4-x^{2}

\displaystyle g(x)=\ln(4-x^{2})

\displaystyle x=-2

\displaystyle x=2

The family

\displaystyle e^{a(x-p)^2+k}

In the functions $\displaystyle {y}={e^{a(x-p)^2+k}}$ , $\displaystyle a\neq0$ , the exponent is a quadratic function with the vertex $\displaystyle (p,k)$ . So:

where $\displaystyle a>0$ : a minimum point $\displaystyle \left(p,e^k\right)$ ; where $\displaystyle a<0$ : a maximum point $\displaystyle \left(p,e^k\right)$ .
Where $\displaystyle {p}={0}$ the function is even.
Where $\displaystyle a<0$ , the exponent tends to $\displaystyle -\infty$ on both sides: the asymptote $\displaystyle {y}={0}$ on both sides.
$\displaystyle e^k>0$ : no function of the family has an extremum point whose y-coordinate is 0 or negative.

For example, a function of the family with the maximum point $\displaystyle (1,e^2)$ : $\displaystyle a<0$ , $\displaystyle {p}={1}$ , $\displaystyle {k}={2}$ , such as $\displaystyle {y}={e^{-(x-1)^2+2}}$ . Any negative $\displaystyle a$ fits, so there are infinitely many such functions.

From the graph of

\displaystyle f'

\displaystyle f

Below is the graph of the derivative $\displaystyle {g(x)}={f'(x)}={(x-1)e^x}$ – not of $\displaystyle f$ itself. The example below reads the properties of $\displaystyle f$ from it:

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

The graph of $\displaystyle f'$ is below the x-axis where $\displaystyle x<1$ and above it where $\displaystyle x>1$ : $\displaystyle f$ decreases and then increases.
$\displaystyle f'$ meets the x-axis at $\displaystyle {x}={1}$ and changes sign from $\displaystyle -$ to $\displaystyle +$ : $\displaystyle f$ has a minimum at $\displaystyle {x}={1}$ .
$\displaystyle f'$ decreases where $\displaystyle x<0$ , that is $\displaystyle f''<0$ – $\displaystyle f$ is concave down; $\displaystyle f'$ increases where $\displaystyle x>0$ – $\displaystyle f$ is concave up.
$\displaystyle f'$ has a minimum at $\displaystyle {x}={0}$ , so $\displaystyle f$ has an inflection point at $\displaystyle {x}={0}$ .

In short: a zero of $\displaystyle f'$ where it changes sign – an extremum point of $\displaystyle f$ ; an extremum point of $\displaystyle f'$ – an inflection point of $\displaystyle f$ .

From the graph of

\displaystyle f''

\displaystyle f

The graph of $\displaystyle f''$ shows the concavity of $\displaystyle f$ :

where the graph of $\displaystyle f''$ is above the x-axis, $\displaystyle f$ is concave up;
where it is below the axis, $\displaystyle f$ is concave down;
a zero of $\displaystyle f''$ where it changes sign is the x-coordinate of an inflection point of $\displaystyle f$ .

The graph of $\displaystyle f''$ does not show where $\displaystyle f$ increases or decreases: that takes $\displaystyle f'$ .

Exercise 1

The function $\displaystyle {g(x)}={\ln(x^2-6x+10)}$ is given.

Find the domain: solve the inequality.

Mark the extremum point of $\displaystyle g$ on the graph.

\displaystyle g(x)=\ln(x^{2}-6x+10)

Explain, without differentiating, why the extremum point of $\displaystyle g$ is there: find the vertex of $\displaystyle x^2-6x+10$ .

Exercise 2

The function $\displaystyle {h(x)}={e^{4x-x^2-3}}$ is given.

Write the exponent in vertex form.
Explain, without differentiating, the type of the extremum point of $\displaystyle h$ , and find its coordinates.

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

\displaystyle h(x)=e^{4x-x^{2}-3}

Exercise 3

The graph below shows the derivative of a function $\displaystyle f$ : $\displaystyle {f'(x)}={g(x)}={(x^2-1)e^{-x}}$ .

Find where $\displaystyle f$ increases: solve the inequality $\displaystyle f'(x)>0$ .

On the graph of $\displaystyle f'$ , mark the points whose $\displaystyle x$ is the x-coordinate of an extremum point of $\displaystyle f$ .

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

On the graph of $\displaystyle f'$ , mark the points whose $\displaystyle x$ is the x-coordinate of an inflection point of $\displaystyle f$ .

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

Find the type of each extremum point of $\displaystyle f$ , and give the reason from the graph of $\displaystyle f'$ .

Exercise 4

The family $\displaystyle {y}={e^{a(x-p)^2+k}}$ , $\displaystyle a\neq0$ , is given.

The points $\displaystyle A(1,1)$ and $\displaystyle B(0,e)$ are marked on the graph.
A function of the family has the minimum point $\displaystyle A$ , and its graph passes through $\displaystyle B$ .
Write the function in a row of your own.

\displaystyle A=(1,1)

\displaystyle B=(0,2.718)

Does the family have a function whose minimum point is $\displaystyle (0,-1)$ ? Give a reason.

Exercise 5

The graph below shows the second derivative of a function $\displaystyle f$ : $\displaystyle {f''(x)}={g(x)}={x^2-4}$ .

On the graph of $\displaystyle f''$ , mark the points whose $\displaystyle x$ is the x-coordinate of an inflection point of $\displaystyle f$ .

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

Find where $\displaystyle f$ is concave down.

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