The graphs of a function, its derivative and its accumulation function

Integral calculus, grade 11 (5 units) Lesson 9 of 20Learning

The graphs of a function, its derivative and its accumulation function

Three graphs

By the fundamental theorem, the accumulation function $\displaystyle F$ is an antiderivative of $\displaystyle f$ , so $\displaystyle {F'}={f}$ , and $\displaystyle {F''}={f'}$ . So each of the three graphs tells about the others:

The graph of $\displaystyle f'$	The graph of $\displaystyle f$	The graph of $\displaystyle F$
	positive	rising
	negative	falling
	changes sign	an extremum point
positive	rising	concave up
negative	falling	concave down
changes sign	an extremum point	an inflection point

From the graph of

\displaystyle f

to the graph of

\displaystyle F

The example below reads the graph of $\displaystyle {F(x)}={\int_0^x f(t)\,dt}$ from the graph of $\displaystyle {f(x)}={x^2-2x-3}={(x+1)(x-3)}$ :

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

$\displaystyle f>0$ where $\displaystyle x<-1$ or $\displaystyle x>3$ : there $\displaystyle F$ rises. $\displaystyle f<0$ where $\displaystyle -1<x<3$ : there $\displaystyle F$ falls.
At $\displaystyle {x}={-1}$ $\displaystyle f$ changes sign from + to −: a maximum of $\displaystyle F$ . At $\displaystyle {x}={3}$ from − to +: a minimum.
$\displaystyle f$ falls where $\displaystyle x<1$ and rises where $\displaystyle x>1$ : $\displaystyle F$ is concave down and then concave up, and its inflection point is at $\displaystyle {x}={1}$ , the minimum of $\displaystyle f$ .
$\displaystyle {F(0)}={0}$ : the graph of $\displaystyle F$ passes through the origin.

The formula gives the heights of the points: $\displaystyle {F(x)}={\frac{x^3}{3}-x^2-3x}$ , so $\displaystyle {F(-1)}={\frac{5}{3}}$ and $\displaystyle {F(3)}={-9}$ .

\displaystyle F(x)=\frac{x^{3}}{3}-x^{2}-3x

Another lower limit

The accumulation function of the same $\displaystyle f$ from $\displaystyle {x}={1}$ is $\displaystyle {G(x)}={\int_1^x f(t)\,dt}={F(x)-F(1)}$ :

$\displaystyle {F(1)}={\frac{1}{3}-1-3}={-\frac{11}{3}}$ , so $\displaystyle {G(x)}={\frac{x^3}{3}-x^2-3x+\frac{11}{3}}$ .
The graph of $\displaystyle G$ is the graph of $\displaystyle F$ shifted $\displaystyle \frac{11}{3}$ up: it passes through $\displaystyle (1,0)$ and not through the origin.
The extremum and inflection points are at the same $\displaystyle x$ .

Exercise 1

The board below shows the graph of $\displaystyle {f(x)}={x^3-3x}$ .
$\displaystyle {F(x)}={\int_0^x f(t)\,dt}={\frac{x^4}{4}-\frac{3x^2}{2}}$ .

Mark the extremum points of $\displaystyle F$ on the board – at their height, $\displaystyle F(x)$ .

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

Mark the inflection points of $\displaystyle F$ on the board.

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

Exercise 2

The board below shows the graph of the accumulation function $\displaystyle {F(x)}={\int_0^x f(t)\,dt}$ .
$\displaystyle f$ is a quadratic function.
$\displaystyle F$ has a maximum at $\displaystyle (0,0)$ and a minimum at $\displaystyle (2,-4)$ .

Explain why $\displaystyle {f(x)}={a\cdot x(x-2)}$ , with $\displaystyle a>0$ .

Calculate $\displaystyle F(2)$ as an expression in $\displaystyle a$ , and find $\displaystyle a$ .

/ – 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

Write $\displaystyle f$ on the board.

\displaystyle F(x)=\,?

Exercise 3

The board below shows the graph of the derivative $\displaystyle f'$ of a function $\displaystyle f$ .

\displaystyle y=\,?

Mark the points on the $\displaystyle x$ -axis at whose $\displaystyle x$ the function $\displaystyle f$ has extremum points.

What kind is each extremum point of $\displaystyle f$ ? At what $\displaystyle x$ does $\displaystyle f$ have an inflection point? Explain.

Formulas: integral calculus, grade 11 (5 units)

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Basic integrals

The indefinite integral – all the antiderivatives
$\displaystyle \int f(x)\,dx=F(x)+C$ $\displaystyle F^{\prime}(x)=f(x)$
A power
$\displaystyle \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$
A constant
$\displaystyle \int k\,dx=kx+C$
$\displaystyle \int\frac{1}{x^{2}}\,dx=-\frac{1}{x}+C$
One over a root
$\displaystyle \int\frac{1}{\sqrt{x}}\,dx=2\sqrt{x}+C$
A root
$\displaystyle \int\sqrt{x}\,dx=\frac{2}{3}x\sqrt{x}+C$

The rules of integration

A constant factor
$\displaystyle \int k\cdot f(x)\,dx$ $\displaystyle =$ $\displaystyle k\int f(x)\,dx$
A sum and a difference
$\displaystyle \int\left(f(x)\pm g(x)\right)dx$ $\displaystyle =$ $\displaystyle \int f(x)\,dx\pm\int g(x)\,dx$
A linear inner function
$\displaystyle \int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ $\displaystyle m\neq0$
A composite function times the inner derivative
$\displaystyle \int f(u(x))\cdot u^{\prime}(x)\,dx=F(u(x))+C$
A power of a function times its derivative
$\displaystyle \int\left(u(x)\right)^{n}\cdot u^{\prime}(x)\,dx=\frac{\left(u(x)\right)^{n+1}}{n+1}+C$ $\displaystyle n\neq-1$

Trigonometric functions (in radians)

$\displaystyle \int\cos x\,dx=\sin x+C$
$\displaystyle \int\sin x\,dx=-\cos x+C$
$\displaystyle \int\frac{1}{\cos^{2}x}\,dx=\tan x+C$
From $\displaystyle {\cos2x}={1-2\sin^2x}$
$\displaystyle \sin^{2}x$ $\displaystyle =$ $\displaystyle \frac{1-\cos2x}{2}$
From $\displaystyle {\cos2x}={2\cos^2x-1}$
$\displaystyle \cos^{2}x$ $\displaystyle =$ $\displaystyle \frac{1+\cos2x}{2}$
From $\displaystyle {\sin2x}={2\sin x\cos x}$
$\displaystyle \sin x\cos x$ $\displaystyle =$ $\displaystyle \frac{1}{2}\sin2x$
From $\displaystyle {\sin^2x+\cos^2x}={1}$
$\displaystyle \tan^{2}x$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}x}-1$

The accumulation function and the definite integral

The accumulation function of $\displaystyle f$ from $\displaystyle a$
$\displaystyle F_{a}(x)=\int_{a}^{x}f(t)\,dt$
The fundamental theorem
$\displaystyle F_{a}^{\prime}(x)=f(x)$
The definite integral, $\displaystyle F$ an antiderivative of $\displaystyle f$
$\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)$
The same limit
$\displaystyle \int_{a}^{a}f(x)\,dx=0$
Swapping the limits
$\displaystyle \int_{b}^{a}f(x)\,dx$ $\displaystyle =$ $\displaystyle -\int_{a}^{b}f(x)\,dx$
Joining intervals
$\displaystyle \int_{a}^{b}f(x)\,dx+\int_{b}^{c}f(x)\,dx$ $\displaystyle =$ $\displaystyle \int_{a}^{c}f(x)\,dx$

Areas and accumulation

An area above the $\displaystyle x$ -axis
$\displaystyle S=\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\geq0$
An area below the $\displaystyle x$ -axis
$\displaystyle S=-\int_{a}^{b}f(x)\,dx$ $\displaystyle f(x)\leq0$
The area between two graphs
$\displaystyle S=\int_{a}^{b}\left(f(x)-g(x)\right)dx$ $\displaystyle f(x)\geq g(x)$
The displacement from the velocity
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$