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
Worked Example
a)
Determine whether is increasing or decreasing at the points where and .
b)
Find the values of for which is an increasing function.
Tangents & Normals
What is a tangent?
- At any point on a curve (the graph of a (non-linear) function), the tangent is the straight line that passes through that point and has the same gradient as the curve at that point
How do I find the equation of a tangent?
- The equation of the tangent to the function at the point is
What is a normal?
- At any point on a curve (the graph of a (non-linear) function), the normal is the straight line that passes through that point and is perpendicular to the tangent at that point
How do I find the equation of a normal?
- The equation of the normal to the function at the point is
Exam Tip
- The equations of a tangent and a normal are not in the formula booklet
- However both can be derived from the equation of a straight line
- This is given in the formula booklet
Worked Example
The function is defined by
a)
Find an equation for the tangent to the curve at the point where , giving your answer in the form .
b)
Find an equation for the normal to the curve at the point where , giving your answer in the form , where , and are integers.