Applications of Differentiation (DP IB Maths: AI HL)

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Daniel I

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Daniel I

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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 bold italic x at that point into the curve's derivative function
  • For example, if f open parentheses x close parentheses equals x squared plus 3 x minus 4
    • then f apostrophe open parentheses x close parentheses equals 2 x plus 3
    • and the gradient of y equals f open parentheses x close parentheses when x equals 1 is  f apostrophe open parentheses 1 close parentheses equals 2 open parentheses 1 close parentheses plus 3 equals 5
    • and the gradient of y equals f open parentheses x close parentheses when x equals negative 2 is  f apostrophe open parentheses negative 2 close parentheses equals 2 open parentheses negative 2 close parentheses plus 3 equals negative 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 fraction numerator d over denominator d x end fraction open parentheses space box enclose space space space space space space space space end enclose space close parentheses subscript x equals box enclose blank end enclose end subscript

Worked example

A function is defined by f open parentheses x close parentheses equals x cubed plus 6 x squared plus 5 x minus 12.

(a) Find f apostrophe open parentheses x close parentheses.

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

(b) Hence show that the gradient of y equals f open parentheses x close parentheses when x equals 1 is 20.

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

(c) Find the gradient of y equals f open parentheses x close parentheses when x equals negative 2.5.

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,space f left parenthesis x right parenthesis, is increasing ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold greater than bold 0
    • This means the value of the function (‘output’) increases asspace x increases
  • A function,space f left parenthesis x right parenthesis, is decreasing ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold less than bold 0
    • This means the value of the function (‘output’) decreases asspace x increases
  • A function,space f left parenthesis x right parenthesis, is stationary ifbold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold equals bold 0

Incr Decr Illustr 1

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
bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold greater than bold 0 (for increasing intervals) and/or
bold space bold italic f bold apostrophe bold left parenthesis bold italic x bold right parenthesis bold less than bold 0 (for decreasing intervals)

  • Most functions are a combination of increasing, decreasing and stationary
    • a range of values ofspace x (interval) is given where a function satisfies each condition
    • e.g.  The functionspace f begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px x to the power of size 16px 2 has derivativespace f to the power of size 16px apostrophe begin mathsize 16px style stretchy left parenthesis x stretchy right parenthesis end style size 16px equals size 16px 2 size 16px x so
      • space f left parenthesis x right parenthesis is decreasing for x less than 0
      • space f left parenthesis x right parenthesis is stationary at x equals 0
      • space f left parenthesis x right parenthesis is increasing for x greater than 0

Worked example

space f stretchy left parenthesis x stretchy right parenthesis equals x squared minus x minus 2

a)
Determine whetherspace f left parenthesis x right parenthesis is increasing or decreasing at the points where x equals 0 and x equals 3.

oT8b2gMg_5-1-2-ib-sl-ai-as-we1-soltn-a

b)
Find the values of x for whichspace f left parenthesis x right parenthesis is an increasing function.
mHWByvDO_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 only that point
  • 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 equals f open parentheses x close parentheses at left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis is bold italic f bold apostrophe stretchy left parenthesis bold italic x subscript 1 stretchy right parenthesis
  • Therefore find the equation of the tangent to the function y equals f left parenthesis x right parenthesis at the point left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis by substituting the gradient, f apostrophe open parentheses x subscript 1 close parentheses, and point left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis into y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses, giving:
    •  Error converting from MathML to accessible text.
  • (You could also substitute into y equals m x plus c but it is usually quicker to substitute into y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses)

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 equals f open parentheses x close parenthesesat left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis is
  • Therefore find the equation of the normal to the function y equals f left parenthesis x right parenthesis at the point left parenthesis x subscript 1 comma blank y subscript 1 right parenthesis by using Error converting from MathML to accessible text.

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 straight f left parenthesis x right parenthesis is defined by

 straight f stretchy left parenthesis x stretchy right parenthesis equals 2 x to the power of 4 plus 3 over x squared blank x not equal to 0

a)
Find an equation for the tangent to the curve y equals straight f left parenthesis x right parenthesis at the point where x equals 1, giving your answer in the form y equals m x plus c.

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

b)
Find an equation for the normal at the point where x equals 1, giving your answer in the form a x plus b y plus d equals 0, where a, b and d are integers.

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

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Local Minimum & Maximum Points

What are local minimum and maximum points?

  • Local minimum and maximum points are two types of stationary point
    • The gradient function (derivative) at such points equals zero
      i.e. space f apostrophe left parenthesis x right parenthesis equals 0
  • A local minimum point, left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis spacewill be the lowest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value of x
    • The function may reach a lower value further afield
  • Similarly, a local maximum point, left parenthesis x comma space f left parenthesis x right parenthesis right parenthesis spacewill be the greatest value ofspace f left parenthesis x right parenthesis in the local vicinity of the value of x
    • The function may reach a greater value further afield
  • The graphs of many functions tend to infinity for large values of x
    (and/or minus infinity for large negative values of x)
  • The nature of a stationary point refers to whether it is a local minimum or local maximum point

How do I find the coordinates and nature of stationary points?

  • The instructions below describe how to find local minimum and maximum points using a GDC on the graph of the function y equals f left parenthesis x right parenthesis.
 STEP 1
 Plot the graph of y equals f left parenthesis x right parenthesis
  
Sketch the graph as part of the solution

 STEP 2
 Use the options from the graphing screen to “solve for minimum”
 The GDC will display the x and y coordinates of the first minimum point
 Scroll onwards to see there are anymore minimum points
 Note down the coordinates and the type of stationary point

 STEP 3
 Repeat STEP 2 but use “solve for maximum” on your GDC
 
  • In STEP 2 the nature of the stationary point should be easy to tell from the graph
    • a local minimum changes the function from decreasing to increasing
      • the gradient changes from negative to positive
    • a local maximum changes the function from increasing to decreasing
      • the gradient changes from positive to negative

Stationary Points incr decr min max

Worked example

Find the stationary points ofspace f stretchy left parenthesis x stretchy right parenthesis equals x stretchy left parenthesis x squared minus 27 stretchy right parenthesis, and state their nature.

5-1-2-ib-sl-ai-only-we2-soltn

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Daniel I

Author: Daniel I

Expertise: Maths

Daniel has taught maths for over 10 years in a variety of settings, covering GCSE, IGCSE, A-level and IB. The more he taught maths, the more he appreciated its beauty. He loves breaking tricky topics down into a way they can be easily understood by students, and creating resources that help to do this.