Instantaneous rate estimates
Calculate a supplied tangent slope exactly or estimate it using nearby secants. Central differences compare equal offsets
\[D_h=\frac{f(a+h)-f(a-h)}{2h}\]
- A tangent gives an instantaneous rate A secant compares two distinct points. A tangent describes the local direction at one point. Nearby secant gradients can estimate its gradient
- One-sided difference The offset may be positive or negative. Subtract the target output and divide by the same signed offset \(\displaystyle \frac{f(a+h)-f(a)}{h},\quad h\ne0\)
- Central difference Use equal offsets on opposite sides. The total width is twice the offset \(\displaystyle f\prime(a)\approx\frac{f(a+h)-f(a-h)}{2h},\quad h>0\)
- Keep exact and approximate claims separate A quotient calculated exactly from two points is an exact secant gradient. It remains an approximation to the tangent gradient unless equality is established separately. Agreement of rounded estimates is not a proof
Worked example
Estimate the tangent gradient using a central difference
\[f(x)=x^3,\quad a=1,\quad h=0.1\]
- \(\displaystyle a-h=0.9,\quad a+h=1.1\)Choose symmetric neighbours
- \(\displaystyle f(0.9)=0.729,\quad f(1.1)=1.331\)Evaluate exactly before rounding
- \(\displaystyle D_{0.1}=\frac{1.331-0.729}{1.1-0.9}=\frac{0.602}{0.2}=3.01\)Use the full separation 2h
- \(\displaystyle f\prime(1)\approx3.01\)This is an approximation to the tangent gradient. The exact quotient is not asserted to be the exact derivative
Common mistakes
Half the required width
✕\(\displaystyle \frac{f(a+h)-f(a-h)}{h}\)✓\(\displaystyle \frac{f(a+h)-f(a-h)}{2h}\)
The central interval spans both equal offsets, so its full width is twice the offsetTreating a rounded estimate as exact
✕Matching rounded secants prove an exact derivative✓Matching rounded secants support an approximation
Rounding hides small differences. Exact equality needs separate justificationUsing zero offset
✕Set the offset to zero in the quotient to obtain the tangent✓Use nonzero offsets approaching zero
Direct substitution gives a zero denominatorDifficulty levels
introCalculate an exact supplied tangent slope or one nearby secant estimate
standardCompare one-sided or two-sided tables and report stable one-decimal estimates
challengeCalculate central differences from rules and tables, including near-stationary cases
examChoose symmetric neighbours, refine central differences and justify approximate conclusions
Questions about this topic
Can a negative offset be used?+
Yes for a one-sided quotient. Keep the numerator and denominator in the same subtraction order
Is a central difference always exact?+
No. It happens to equal the derivative for quadratic functions, but generally has an error for cubic and more complicated curves
Why not use extremely small offsets for measured data?+
Rounding or measurement error can dominate a very small output change. Inspect the quality and precision of the data
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