Applications of Differentiation (DP IB Applications & Interpretation (AI)): Revision Note

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  at that point into the curve's derivative function

• For example, if 

  • then 

  • and the gradient of when is 

  • and the gradient of when is 

• 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

Increasing & Decreasing Functions

What are increasing and decreasing functions?

• A function,, is increasing if

  • This means the value of the function (‘output’) increases as increases

• A function,, is decreasing if

  • This means the value of the function (‘output’) decreases as increases

• A function,, is stationary if

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

 (for increasing intervals) and/or

 (for decreasing intervals)

• Most functions are a combination of increasing, decreasing and stationary

  • a range of values of (interval) is given where a function satisfies each condition

  • e.g.  The function has derivative so

    •  is decreasing for

    •  is stationary at

    •  is increasing for 

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

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, , of the tangent to the function at is

• Therefore find the equation of the tangent to the function  at the point by substituting the gradient, , and point into , giving:

  •  

• (You could also substitute into but it is usually quicker to substitute into )

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

How do I find the equation of a normal?

• The gradient of the normal to the function at is

• Therefore find the equation of the normal to the function  at the point by using

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.

• A local minimum point, will be the lowest value of in the local vicinity of the value of 

  • The function may reach a lower value further afield

• Similarly, a local maximum point, will be the greatest value of in the local vicinity of the value of 

  • The function may reach a greater value further afield

• The graphs of many functions tend to infinity for large values of 
  (and/or minus infinity for large negative values of )

• 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 .

 STEP 1

 Plot the graph of 
  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  and  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