Applications of Differentiation (DP IB Maths: AA HL)

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Daniel I

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Maths

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Finding Gradients

How do I find the gradient of a curve at a point?

The gradient of a curve at a point is the gradient of the tangent to the curve at that point
Find the gradient of a curve at a point by substituting the value of $x$ at that point into the curve's derivative function
For example, if $f (x) = x^{2} + 3 x - 4$
- then $f' (x) = 2 x + 3$
- and the gradient of $y = f (x)$ when $x = 1$ is $f' (1) = 2 (1) + 3 = 5$
- and the gradient of $y = f (x)$ when $x = - 2$ is $f' (- 2) = 2 (- 2) + 3 = - 1$
Although your GDC won't find a derivative function for you, it is possible to use your GDC to evaluate the derivative of a function at a point, using $\frac{d}{d x} {()}_{x =}$

Worked example

A function is defined by $f (x) = x^{3} + 6 x^{2} + 5 x - 12$ .

(a) Find $f' (x)$ .

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(b) Hence show that the gradient of $y = f (x)$ when $x = 1$ is 20.

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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

picture-1

Find the values of

x

for which

f (x)

is an increasing function.

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Tangents & Normals

What is a tangent?

At any point on the graph of a (non-linear) function, the tangent is the straight line that touches the graph at a point without crossing through it
Its gradient is given by the derivative function

Grad Tang Norm Illustr 2

How do I find the equation of a tangent?

To find the equation of a straight line, a point and the gradient are needed
The gradient, $m$ , of the tangent to the function $y = f (x)$ at ${(x}_{1}, y_{1})$ is $f' (x_{1})$
Therefore find the equation of the tangent to the function $y = f (x)$ at the point ${(x}_{1}, y_{1})$ by substituting the gradient, $f' (x_{1})$ , and point ${(x}_{1}, y_{1})$ into $y - y_{1} = m (x - x_{1})$ , giving:
- $y - y_{1} = f^{'} (x_{1}) (x - x_{1})$
(You could also substitute into $y = m x + c$ but it is usually quicker to substitute into $y - y_{1} = m (x - x_{1})$ )

What is a normal?

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

Grad Tang Norm Illustr 3

How do I find the equation of a normal?

The gradient of the normal to the function $y = f (x)$ at ${(x}_{1}, y_{1})$ is $\frac{- 1}{f' (x_{1})}$
Therefore find the equation of the normal to the function $y = f (x)$ at the point ${(x}_{1}, y_{1})$ by using $y - y_{1} = \frac{- 1}{f^{'} (x_{1})} (x - x_{1})$

Exam Tip

You are not given the formula for the equation of a tangent or the equation of a normal
But both can be derived from the equations of a straight line which are 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 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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Applications of Differentiation (DP IB Maths: AA HL)

Revision Note

How do I find the gradient of a curve at a point?

What are increasing and decreasing functions?

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

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?

Author: Daniel I