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