From shape to rate
Read gradient direction and magnitude separately from function height. Use exact coordinates and rules to support the graph
\[g(x)=f\prime(x)\]
- Height is not rate The ordinate f(x) tells you the function value. The ordinate of the gradient function tells you the tangent slope of f at the same input \(\displaystyle g(x)=f\prime(x)\)
- Read the sign of the gradient A positive gradient means the original curve is increasing; a negative gradient means decreasing. The original curve may lie above or below the axis in either case
- Check both sides of a zero Zero gradient means stationary. A positive-to-negative sign change gives a local maximum; negative-to-positive gives a local minimum. No sign change means a stationary point without a turn
- A gradient graph does not fix height Vertically shifted functions have identical gradients. A supplied function value selects the vertical position of the original curve
Worked example
A curve has the gradient function shown. Describe its stationary inputs and changes of direction
\[g(x)=(x-1)(x+1)=x^2-1\]
- \(\displaystyle g(x)=0\Longleftrightarrow x=-1\text{ or }x=1\)The zeros of g are stationary inputs of f.
- \(\displaystyle g(-2)=3>0,\quad g(0)=-1<0,\quad g(2)=3>0\)Use a sign witness in each interval
- \(\displaystyle +\ \longrightarrow\ -\quad\text{at }x=-1\)The original curve changes from increasing to decreasing: a local maximum
- \(\displaystyle -\ \longrightarrow\ +\quad\text{at }x=1\)The original curve changes from decreasing to increasing: a local minimum. Their heights are not determined by the gradient graph alone
Common mistakes
Using height as gradient
✕Above the axis means increasing✓A positive tangent gradient means increasing
A positive-height curve may be fallingEvery stationary point is a turn
✕A zero gradient always gives a maximum or minimum✓Check the gradient sign on both sides
A repeated gradient zero may have no sign changeIgnoring the supplied point
✕A gradient graph uniquely fixes the original curve✓Use a function value to determine its vertical position
Vertical translations preserve gradientsDifficulty levels
introRead tangent signs and exact interior rates of empirical straight segments
standardMatch quadratic curves and linear gradient graphs, using scale and a supplied point
challengeMatch cubic curves and distinguish stationary crossings from turns
examMatch quartic/cubic pairs and reconstruct a cubic choice from gradient and height information
Questions about this topic
Does zero gradient mean zero function height?+
No. It means the tangent is horizontal. The point may lie anywhere vertically
Can a rate be assigned at a corner of a piecewise-linear model?+
If the adjacent segment slopes differ, there is no single tangent gradient at the corner. The generated empirical questions use times strictly inside a segment
Do I need a differentiation rule here?+
The exact gradient information is supplied or can be read from a labelled tangent. Use it to relate the graphs; polynomial differentiation is developed separately
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 12: Rates and derivatives
In this chapter
Calculate and interpret an average rate of change12A Average and instantaneous rates: Relate graph direction and instantaneous rateEstimate instantaneous rates with secants and tangentsDerive polynomial gradients from first principlesDifferentiate polynomialsApply polynomial derivatives12C Polynomial differentiation: Relate graph direction and instantaneous rate