5.1.2 Applications of Differentiation

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

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 

• 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  has derivative  so
    •  is decreasing for
    •  is stationary at
    •  is increasing for 

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

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

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 .

 Plot the graph of 
  Sketch the graph as part of the solution

 The GDC will display the  and  coordinates of the first minimum point

• 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