Applications of Differentiation (DP IB Maths: AA SL)

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,, 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 

 (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 

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