IB Mathematics Applications & Interpretation Topic SL…

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Curves are hard to work with, but near a single point a curve looks almost straight.
The tangent is the straight line that just touches the curve at a point and shares its gradient there.
The normal is the straight line through the same point at right angles to the tangent.

Finding The Equation Of A Tangent

Differentiate to get $f'(x)$, the gradient function.
Substitute the x-value $a$ to get the tangent gradient $m = f'(a)$.
Find the point $(a,\ f(a))$ the tangent passes through.
Put these into the line form $y - y_1 = m(x - x_1)$.

Example

To find the tangent to $y = x^3 - 2x$ at $x = 1$:

Point: $f(1) = 1 - 2 = -1$, so $(1, -1)$.
Gradient: $f'(x) = 3x^2 - 2$, so $f'(1) = 1$.
Line: $y + 1 = 1(x - 1)$, giving $y = x - 2$.

Example question

Find the equation of the tangent to $y = x^3 - 2x$ at $x = 2$.

Solution

Point: $f(2) = 8 - 4 = 4$, so $(2, 4)$.
Gradient: $f'(2) = 3(4) - 2 = 10$.
Line: $y - 4 = 10(x - 2)$, giving $y = 10x - 16$.

Finding The Equation Of A Normal

The normal is perpendicular to the tangent, and perpendicular gradients satisfy $m_1 m_2 = -1$.
So the normal gradient is $-\dfrac{1}{f'(a)}$, the negative reciprocal of the tangent gradient.
It passes through the same point $(a,\ f(a))$, then you use the same line form.

Example

To find the normal to $y = x^2$ at $x = 2$:

Point $(2, 4)$; tangent gradient $f'(2) = 4$.
Normal gradient $-\dfrac{1}{4}$.
Line: $y - 4 = -\dfrac{1}{4}(x - 2)$.

Example question

Find the equation of the normal to $y = x^2$ at the point $(1, 1)$.

Solution

Tangent gradient: $f'(x) = 2x$, so $f'(1) = 2$
Normal gradient: $-\dfrac{1}{2}$.
Line: $$y - 1 = -\dfrac{1}{2}(x - 1)$$

Tangent Versus Normal At A Glance

Line	Gradient	Passes through
Tangent	f'(a)	(a, f(a))
Normal	-1 / f'(a)	(a, f(a))

Example question

Find the tangent to $y = \dfrac{1}{x}$ at $x = 1$.

Solution

Rewrite $y = x^{-1}$, so $f'(x) = -x^{-2} = -\dfrac{1}{x^2}$.
At $x = 1$: point $(1, 1)$, gradient $-1$.
Line: $y - 1 = -1(x - 1)$, giving $y = -x + 2$.

Exam technique

There is no single tangent formula; you combine the line forms from SL 2.1 with the derivative.
The perpendicular condition $m_1 m_2 = -1$ is the key fact for the normal.
Keep full accuracy from the GDC, and your calculator can also produce a tangent line directly.
A horizontal tangent has gradient $0$, and its normal is the vertical line $x = a$.

Active recall

How do you find the gradient of the tangent to a curve at $x = a$?
What is the relationship between the gradients of a tangent and its normal?
Write the normal gradient in terms of $f'(a)$.
Find the tangent to $y = x^2$ at $x = 3$.

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Curves are hard to work with, but near a single point a curve looks almost straight.
The tangent is the straight line that just touches the curve at a point and shares its gradient there.
The normal is the straight line through the same point at right angles to the tangent.

Finding The Equation Of A Tangent

Differentiate to get $f'(x)$, the gradient function.
Substitute the x-value $a$ to get the tangent gradient $m = f'(a)$.
Find the point $(a,\ f(a))$ the tangent passes through.
Put these into the line form $y - y_1 = m(x - x_1)$.

Example

To find the tangent to $y = x^3 - 2x$ at $x = 1$:

Point: $f(1) = 1 - 2 = -1$, so $(1, -1)$.
Gradient: $f'(x) = 3x^2 - 2$, so $f'(1) = 1$.
Line: $y + 1 = 1(x - 1)$, giving $y = x - 2$.

Example question

Find the equation of the tangent to $y = x^3 - 2x$ at $x = 2$.

Solution

Point: $f(2) = 8 - 4 = 4$, so $(2, 4)$.
Gradient: $f'(2) = 3(4) - 2 = 10$.
Line: $y - 4 = 10(x - 2)$, giving $y = 10x - 16$.

Finding The Equation Of A Normal

The normal is perpendicular to the tangent, and perpendicular gradients satisfy $m_1 m_2 = -1$.
So the normal gradient is $-\dfrac{1}{f'(a)}$, the negative reciprocal of the tangent gradient.
It passes through the same point $(a,\ f(a))$, then you use the same line form.

Example

To find the normal to $y = x^2$ at $x = 2$:

Point $(2, 4)$; tangent gradient $f'(2) = 4$.
Normal gradient $-\dfrac{1}{4}$.
Line: $y - 4 = -\dfrac{1}{4}(x - 2)$.

Example question

Find the equation of the normal to $y = x^2$ at the point $(1, 1)$.

Solution

Tangent gradient: $f'(x) = 2x$, so $f'(1) = 2$
Normal gradient: $-\dfrac{1}{2}$.
Line: $$y - 1 = -\dfrac{1}{2}(x - 1)$$

Tangent Versus Normal At A Glance

Line	Gradient	Passes through
Tangent	f'(a)	(a, f(a))
Normal	-1 / f'(a)	(a, f(a))

Example question

Find the tangent to $y = \dfrac{1}{x}$ at $x = 1$.

Solution

Rewrite $y = x^{-1}$, so $f'(x) = -x^{-2} = -\dfrac{1}{x^2}$.
At $x = 1$: point $(1, 1)$, gradient $-1$.
Line: $y - 1 = -1(x - 1)$, giving $y = -x + 2$.

Exam technique

There is no single tangent formula; you combine the line forms from SL 2.1 with the derivative.
The perpendicular condition $m_1 m_2 = -1$ is the key fact for the normal.
Keep full accuracy from the GDC, and your calculator can also produce a tangent line directly.
A horizontal tangent has gradient $0$, and its normal is the vertical line $x = a$.

Active recall

How do you find the gradient of the tangent to a curve at $x = a$?
What is the relationship between the gradients of a tangent and its normal?
Write the normal gradient in terms of $f'(a)$.
Find the tangent to $y = x^2$ at $x = 3$.

IB Mathematics Applications & Interpretation SL 5.4—tangents and Normals Notes

IB Mathematics Applications & Interpretation SL 5.4—tangents and Normals Notes

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Finding The Equation Of A Tangent

Finding The Equation Of A Normal

Tangent Versus Normal At A Glance

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Finding The Equation Of A Tangent

Finding The Equation Of A Normal

Tangent Versus Normal At A Glance

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 5.4—tangents and Normals Notes

Number and Algebra

Functions

Geometry and Trigonometry

Statistics and Probability

Calculus

IB Mathematics Applications & Interpretation SL 5.4—tangents and Normals Notes

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Finding The Equation Of A Tangent

Finding The Equation Of A Normal

Tangent Versus Normal At A Glance

A Tangent Is The Best Straight-Line Copy Of A Curve At A Point

Finding The Equation Of A Tangent

Finding The Equation Of A Normal

Tangent Versus Normal At A Glance