A function from its derivative and a point

Integral calculus, grade 11 (5 units) Lesson 15 of 20Learning

A function from its derivative and a point

A point picks an antiderivative

The derivative fixes the function up to a constant $\displaystyle C$ , and one point on the graph fixes $\displaystyle C$ . In the example below $\displaystyle {f'(x)}={\frac{1}{x^2}+\cos x}$ , $\displaystyle x\neq0$ , and the graph of $\displaystyle f$ passes through the point $\displaystyle \left(\frac{\pi}{2},3\right)$ :

$\displaystyle {f(x)}={\int\left(\frac{1}{x^2}+\cos x\right)dx}={-\frac{1}{x}+\sin x+C}$ .
$\displaystyle {f\left(\frac{\pi}{2}\right)}={3}$ : $\displaystyle {-\frac{2}{\pi}+1+C}={3}$ , so $\displaystyle {C}={2+\frac{2}{\pi}}$ .
$\displaystyle {f(x)}={-\frac{1}{x}+\sin x+2+\frac{2}{\pi}}$ .

A point from a tangent

Sometimes the point is not given directly, but hidden in a tangent. In the example below $\displaystyle {f'(x)}={3x^2-6x}$ , and the line $\displaystyle {y}={-3x+5}$ is tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={1}$ :

The point of contact, on the tangent: $\displaystyle {y}={-3+5}={2}$ , so $\displaystyle {f(1)}={2}$ .
$\displaystyle {f(x)}={x^3-3x^2+C}$ , and $\displaystyle {1-3+C}={2}$ , so $\displaystyle {C}={4}$ .
$\displaystyle {f(x)}={x^3-3x^2+4}$ .
The check: $\displaystyle {f'(1)}={3-6}={-3}$ – the slope of the tangent.

From the second derivative

From the second derivative to the function takes two integrations, so there are two constants, and two conditions are needed. In the example below $\displaystyle {f''(x)}={6x}$ , and $\displaystyle f$ has an extremum point $\displaystyle (1,-1)$ :

$\displaystyle {f'(x)}={3x^2+C_1}$ . At an extremum point $\displaystyle {f'(1)}={0}$ , so $\displaystyle {3+C_1}={0}$ and $\displaystyle {C_1}={-3}$ .
$\displaystyle {f(x)}={x^3-3x+C_2}$ . $\displaystyle {f(1)}={-1}$ , so $\displaystyle {1-3+C_2}={-1}$ and $\displaystyle {C_2}={1}$ .
$\displaystyle {f(x)}={x^3-3x+1}$ . $\displaystyle {f''(1)}={6>0}$ : the extremum is a minimum.

Exercise 1

$\displaystyle {f'(x)}={\frac{2}{\sqrt{x}}}$ , and $\displaystyle x>0$ .
The graph of $\displaystyle f$ passes through the point $\displaystyle A(4,10)$ .

Find all the antiderivatives of $\displaystyle f'$ .

/ – 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 $\displaystyle C$ .

Write $\displaystyle f$ on the board.

\displaystyle A=(4,10)

Exercise 2

$\displaystyle {f'(x)}={\sin2x}$ .
The line $\displaystyle {y}={1}$ is tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={\frac{\pi}{2}}$ .

Show that the slope of $\displaystyle f$ at $\displaystyle {x}={\frac{\pi}{2}}$ fits the tangent.

/ – 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 write it on the board.

\displaystyle y=1

Exercise 3

$\displaystyle {f''(x)}={6x-2}$ .
$\displaystyle f$ has a minimum point $\displaystyle (1,0)$ .

Find $\displaystyle f'(x)$ .

Find $\displaystyle f(x)$ , and write it on the board.

\displaystyle A=(1,0)

Exercise 4

A body moves along a line with velocity $\displaystyle {v(t)}={6t-3t^2}$ metres a second.
$\displaystyle t$ is the time in seconds, and at $\displaystyle {t}={0}$ the body is at the starting point: $\displaystyle {s(0)}={0}$ .

Find the position $\displaystyle s(t)$ .

/ – 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

When does the body return to the starting point?

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
$\displaystyle s(b)-s(a)=\int_{a}^{b}v(t)\,dt$