5.1.2 Applications of Differentiation

Increasing & Decreasing Functions

What are increasing and decreasing functions?

A function, $f (x)$ , is increasing if $f' (x) > 0$
- This means the value of the function (‘output’) increases as $x$ increases
A function, $f (x)$ , is decreasing if $f' (x) < 0$
- This means the value of the function (‘output’) decreases as $x$ increases
A function, $f (x)$ , is stationary if $f' (x) = 0$

How do I find where functions are increasing, decreasing or stationary?

To identify the intervals on which a function is increasing or decreasing

STEP 1

Find the derivative f'(x)

STEP 2

Solve the inequalities

$f' (x) > 0$ (for increasing intervals) and/or

$f' (x) < 0$ (for decreasing intervals)

Most functions are a combination of increasing, decreasing and stationary
- a range of values of $x$ (interval) is given where a function satisfies each condition
- e.g. The function $f (x) = x^{2}$ has derivative $f^{'} (x) = 2 x$ so
  - $f (x)$ is decreasing for $x < 0$
  - $f (x)$ is stationary at $x = 0$
  - $f (x)$ is increasing for $x > 0$

Worked Example

$f (x) = x^{2} - x - 2$

Determine whether

f (x)

is increasing or decreasing at the points where

x = 0

and

x = 3

5-1-2-ib-sl-ai-as-we1-soltn-a

Find the values of

x

for which

f (x)

is an increasing function.

Zvjge3OX_5-1-2-ib-sl-ai-as-we1-soltn-b

Tangents & Normals

What is a tangent?

At any point on a curve (the graph of a (non-linear) function), the tangent is the straight line that passes through that point and has the same gradient as the curve at that point

Grad Tang Norm Illustr 2

How do I find the equation of a tangent?

The equation of the tangent to the function $y = f (x)$ at the point ${(x}_{1}, y_{1})$ is

$y - y_{1} = f^{'} (x_{1}) (x - x_{1})$

What is a normal?

At any point on a curve (the graph of a (non-linear) function), the normal is the straight line that passes through that point and is perpendicular to the tangent at that point

Grad Tang Norm Illustr 3

How do I find the equation of a normal?

The equation of the normal to the function $y = f (x)$ at the point ${(x}_{1}, y_{1})$ is

$y - y_{1} = \frac{- 1}{f' (x_{1})} (x - x_{1})$

Exam Tip

The equations of a tangent and a normal are not in the formula booklet
- However both can be derived from the equation of a straight line $y - y_{1} = m (x - x_{1})$
- This is given in the formula booklet

Worked Example

The function $f (x)$ is defined by

$f (x) = 2 x^{4} + \frac{3}{x^{2}} x \neq 0$

Find an equation for the tangent to the curve

y = f (x)

at the point where

x = 1

, giving your answer in the form

y = m x + c

5-1-2-ib-sl-ai-aa-we2-soltn-a

Find an equation for the normal to the curve

y = f (x)

at the point where

x = 1

, giving your answer in the form

a x + b y + d = 0

, where

a

b

and

d

are integers.

5-1-2-ib-sl-ai-aa-we2-soltn-b

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Revision Notes

5.1.2 Applications of Differentiation

Increasing & Decreasing Functions

What are increasing and decreasing functions?

How do I find where functions are increasing, decreasing or stationary?

Worked Example

Tangents & Normals

What is a tangent?

How do I find the equation of a tangent?

What is a normal?

How do I find the equation of a normal?

Exam Tip

Worked Example

Author: Paul

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