Applications of Differentiation (DP IB Maths: AA HL)

Revision Note

Author

Daniel I

Last updated

27 November 2023

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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 $f' (x)$ .

Vx0rvphg_rn-we-5-1-2-ib-ai-sl-finding-gradiens-parta

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

ksEAxyN__rn-we-5-1-2-ib-ai-sl-finding-gradiens-partb

3T8SSV-9_rn-we-5-1-2-ib-ai-sl-finding-gradiens-partc

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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.

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

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

Examiner 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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Previous:5.1.1 Introduction to DifferentiationNext:5.2.1 Differentiating Special Functions

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?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions