The equation of the tangent

Calculus, grade 10 (5 units) Lesson 11 of 26Learning

The equation of the tangent

The tangent at a point on the graph

The tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={x_1}$ is a line through the point $\displaystyle (x_1,y_1)$ , and its slope is the derivative there:

\displaystyle m

\displaystyle =

\displaystyle f'(x_1)

A line of slope $\displaystyle m$ through $\displaystyle (x_1,y_1)$ :

\displaystyle y-y_1

\displaystyle =

\displaystyle m(x-x_1)

So the equation of the tangent is found in three steps:

$\displaystyle {y_1}={f(x_1)}$ – the $\displaystyle y$ -coordinate of the point;
$\displaystyle {m}={f'(x_1)}$ – the slope;
substitute in $\displaystyle {y-y_1}={m(x-x_1)}$ and bring it to the form $\displaystyle {y}={mx+b}$ .

Example:

\displaystyle {f(x)}={x^3-2x+1}

\displaystyle {x}={1}

The example below finds the equation of the tangent to the graph of $\displaystyle {f(x)}={x^3-2x+1}$ at the point where $\displaystyle {x}={1}$ .

$\displaystyle {y_1}={f(1)}={1-2+1}={0}$ : the point is $\displaystyle (1,0)$ .
$\displaystyle {f'(x)}={3x^2-2}$ , so $\displaystyle {m}={f'(1)}={3-2}={1}$ .
$\displaystyle {y-0}={1\cdot(x-1)}$ , that is $\displaystyle {y}={x-1}$ .

\displaystyle f(x)=x^{3}-2x+1

\displaystyle y=x-1

\displaystyle A=(1,0)

Near $\displaystyle (1,0)$ the tangent almost merges with the graph. Far from the point it may cut the graph again – here at $\displaystyle {x}={-2}$ , at the point $\displaystyle (-2,-3)$ .

A tangent of a given slope

When the slope of the tangent is given and not the point, the slope fixes the point: solve the equation $\displaystyle {f'(x)}={m}$ .
Parallel lines are lines with the same slope, so a tangent parallel to a given line is a tangent with the slope of that line.

The example below finds the tangent to the graph of $\displaystyle {f(x)}={x^2-6x+5}$ parallel to the line $\displaystyle {y}={2x+3}$ .

The slope of the line is 2, so solve $\displaystyle {f'(x)}={2}$ : $\displaystyle {2x-6}={2}$ , that is $\displaystyle {x}={4}$ .
$\displaystyle {y_1}={f(4)}={16-24+5}={-3}$ : the point is $\displaystyle (4,-3)$ .
$\displaystyle {y+3}={2(x-4)}$ , that is $\displaystyle {y}={2x-11}$ .

\displaystyle f(x)=x^{2}-6x+5

\displaystyle y=2x+3

\displaystyle y=2x-11

\displaystyle A=(4,-3)

A horizontal tangent, and a tangent and the axes

A horizontal tangent: its slope is 0, so it is at a point where $\displaystyle {f'(x)}={0}$ . Its equation is $\displaystyle {y}={y_1}$ . On a parabola – the tangent at the vertex.
Where the tangent meets the axes: the tangent is a straight line, so substitute $\displaystyle {x}={0}$ in it (where it meets the $\displaystyle y$ -axis) or $\displaystyle {y}={0}$ (where it meets the $\displaystyle x$ -axis).

For example, the tangent $\displaystyle {y}={2x-11}$ meets the $\displaystyle y$ -axis at $\displaystyle (0,-11)$ , and the $\displaystyle x$ -axis where $\displaystyle {2x-11}={0}$ , that is at $\displaystyle (5.5,0)$ .

Exercise 1

The function is $\displaystyle {f(x)}={x^2-3x+2}$ .
The point $\displaystyle (3,2)$ is on its graph.

Work out the slope of the tangent to the graph at the point $\displaystyle (3,2)$ .

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

/ – 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 the equation of the tangent on the board.

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

\displaystyle (3,2)

Exercise 2

The function is $\displaystyle {f(x)}={-x^3+3x^2}$ .
Write on the board the equation of the tangent to the graph of $\displaystyle f$ at the point where $\displaystyle {x}={1}$ .

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

\displaystyle (1,2)

Exercise 3

The function is $\displaystyle {f(x)}={x^3-3x^2}$ .
Its graph has two tangents of slope 9.

Write an equation and solve it: at which values of $\displaystyle x$ is the slope of the tangent 9?

Write the equations of both tangents on the board.

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

Exercise 4

The function is $\displaystyle {f(x)}={x^2+4x}$ .

At which value of $\displaystyle x$ is the tangent to the graph horizontal?
Write an equation and solve it.

Write the equation of the horizontal tangent on the board.

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

Exercise 5

The function is $\displaystyle {f(x)}={x^2-4}$ .

Write on the board the equation of the tangent to the graph at the point $\displaystyle (1,-3)$ .

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

\displaystyle (1,-3)

Find the $\displaystyle x$ -coordinate of the point where the tangent meets the $\displaystyle x$ -axis.

The tangent meets the $\displaystyle y$ -axis at $\displaystyle (0,-5)$ .
Work out the area of the triangle the tangent makes with the axes.

/ – 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 6

The function is $\displaystyle {f(x)}={x^2+bx}$ ( $\displaystyle b$ a parameter).
The tangent to the graph at the point where $\displaystyle {x}={1}$ is parallel to the line $\displaystyle {y}={5x}$ .

$\displaystyle {f'(x)}={2x+b}$ .
Write an equation in $\displaystyle b$ and solve it.

Write the equation of the tangent.

Formulas: calculus, grade 10 (5 units)

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Algebra

The square of a sum and of a difference
$\displaystyle (a\pm b)^2$ $\displaystyle =$ $\displaystyle a^2\pm2ab+b^2$
The difference of two squares
$\displaystyle a^2-b^2$ $\displaystyle =$ $\displaystyle (a-b)(a+b)$
The cube of a sum and of a difference
$\displaystyle (a\pm b)^3$ $\displaystyle =$ $\displaystyle a^3\pm3a^2b+3ab^2\pm b^3$
The sum and the difference of two cubes
$\displaystyle a^3\pm b^3$ $\displaystyle =$ $\displaystyle (a\pm b)(a^2\mp ab+b^2)$
The quadratic formula for $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Factoring a trinomial – $\displaystyle x_1$ , $\displaystyle x_2$ the roots of $\displaystyle {ax^2+bx+c}={0}$
$\displaystyle ax^2+bx+c$ $\displaystyle =$ $\displaystyle a(x-x_1)(x-x_2)$

The properties of a function

An even function – the graph is symmetric about the $\displaystyle y$ -axis
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle f(x)$
An odd function – the graph is symmetric about the origin
$\displaystyle f(-x)$ $\displaystyle =$ $\displaystyle -f(x)$
The root of a square
$\displaystyle \sqrt{x^2}$ $\displaystyle =$ $\displaystyle |x|$
The absolute value
$\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}$

Transformations of a graph

A shift of $\displaystyle k$ units up ( $\displaystyle k<0$ – down)
$\displaystyle y=f(x)+k$
A shift of $\displaystyle p$ units right ( $\displaystyle p<0$ – left)
$\displaystyle y=f(x-p)$
A vertical stretch ( $\displaystyle |a|>1$ ) or squeeze ( $\displaystyle |a|<1$ )
$\displaystyle y=a\cdot f(x)$
A horizontal squeeze by $\displaystyle a$ ( $\displaystyle a>1$ ) or stretch ( $\displaystyle 0<a<1$ )
$\displaystyle y=f(ax)$
A reflection in the $\displaystyle x$ -axis
$\displaystyle y=-f(x)$
A reflection in the $\displaystyle y$ -axis
$\displaystyle y=f(-x)$

The derivative

The average rate of change from $\displaystyle x_1$ to $\displaystyle x_2$ – the slope of the secant
$\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}$
The derivative at a point – the slope of the tangent
$\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}$
The derivative of a power
$\displaystyle (x^n)'$ $\displaystyle =$ $\displaystyle nx^{n-1}$
The derivative of a constant
$\displaystyle (c)'$ $\displaystyle =$ $\displaystyle 0$
The derivative of a constant multiple
$\displaystyle (c\cdot f(x))'$ $\displaystyle =$ $\displaystyle c\cdot f'(x)$
The derivative of a sum and of a difference
$\displaystyle (f(x)\pm g(x))'$ $\displaystyle =$ $\displaystyle f'(x)\pm g'(x)$
The instantaneous speed – the derivative of the distance with respect to time
$\displaystyle v(t)$ $\displaystyle =$ $\displaystyle s'(t)$

Rules of differentiation

The derivative of a product of functions
$\displaystyle [f(x)\cdot g(x)]'$ $\displaystyle =$ $\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)$
The derivative of a quotient of functions
$\displaystyle \left[\frac{f(x)}{g(x)}\right]'$ $\displaystyle =$ $\displaystyle \frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2}$
The derivative of a composite function – the outer derivative times the inner derivative
$\displaystyle [f(u(x))]'$ $\displaystyle =$ $\displaystyle f'(u)\cdot u'(x)$
The derivative of $\displaystyle \frac{1}{x}$
$\displaystyle \left(\frac{1}{x}\right)'$ $\displaystyle =$ $\displaystyle -\frac{1}{x^2}$
The derivative of $\displaystyle \sqrt{x}$
$\displaystyle (\sqrt{x})'$ $\displaystyle =$ $\displaystyle \frac{1}{2\sqrt{x}}$ $\displaystyle x>0$
The derivative of the root of a function
$\displaystyle (\sqrt{f(x)})'$ $\displaystyle =$ $\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}$ $\displaystyle f(x)>0$

The tangent to a graph

The slope of the tangent at the point $\displaystyle (x_1,y_1)$ on the graph
$\displaystyle m=f'(x_1)$
The equation of the tangent
$\displaystyle y-y_1=m(x-x_1)$

Rising, falling and extremum points

The function rises
$\displaystyle f'(x)>0$
The function falls
$\displaystyle f'(x)<0$
A point that may be an extremum
$\displaystyle f'(x)=0$

Asymptotes

A vertical asymptote – $\displaystyle a$ a zero of the denominator that is not a zero of the numerator
$\displaystyle x=a$
A horizontal asymptote – $\displaystyle b$ the value the function approaches as $\displaystyle x$ grows or falls without end
$\displaystyle y=b$