Estimate instantaneous rates with secants and tangents

Which gradient describes the instantaneous rate at one input?

Reduce the offset through all four settings. Compare the left, right and central gradients

What do the two one-sided gradients do as h decreases?

From a secant to a tangent

For a smooth curve, inspect secants closer to the target. Keep the offset nonzero when computing a quotient. Do not substitute h = 0 into a difference quotient

1. 1A 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
2. 2One-sided difference The offset may be positive or negative. Subtract the target output and divide by the same signed offset \(\displaystyle \frac{f(a+h)-f(a)}{h},\quad h\ne0\)

Exact data

\(\displaystyle P=(-1,4),\quad Q=(3,-2)\)

A supplied tangent passes through these exact labelled points. Calculate its exact gradient

Exact data

\(\displaystyle f(2)=5,\quad f(2.1)=5.63\)

Calculate the right-hand secant gradient as an estimate of the tangent gradient at x = 2

Exact data

\(\displaystyle f(2)=5,\quad f(1.9)=4.43\)

Calculate the left-hand secant gradient

Use both sides

A central difference joins equal offsets on opposite sides of the target. At input 1.9, the output is 4.43; at input 2.1, it is 5.63. Their output difference is 1.20 and the full input width is 0.2

1. 1Central difference Use equal offsets on opposite sides. The total width is twice the offset \(\displaystyle f\prime(a)\approx\frac{f(a+h)-f(a-h)}{2h},\quad h>0\)

Exact data

\(\displaystyle f(1.9)=4.43,\quad f(2.1)=5.63,\quad h=0.1\)

Calculate the central difference at x = 2 using the symmetric pair

The explorer gave an exact central difference of 2 for y = x² at x = 1. What can we conclude for another smooth curve?

Estimates

\(\displaystyle 3.0001,\quad3.000001\)

The two finest estimates agree when rounded. Write their common approximation to 2 decimal places

Choose and report an estimate

With exact smooth polynomial data, compare nearby estimates and round once at the end. Real measurements may be noisy, so ever-smaller intervals do not automatically improve a measured rate

1. 1Keep 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