Applications of Differentiation (DP IB Analysis & Approaches (AA)) : Revision Note

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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.

picture-1

b) Find the values of x for whichspace f left parenthesis x right parenthesis 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 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 Tips and Tricks

  • 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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Paul

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