Estimate a Derivative | Math Coach Amy

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Key Ideas Easy Medium Hard

\(\dfrac{f(a + h) - f(a - h)}{2h}\)

best estimate when both sides are equally close

What to Remember

A symmetric quotient is the usual best estimate

When the table has points equally spaced on both sides of a,

\(f'(a) \approx \dfrac{f(a + h) - f(a - h)}{2h}\)

Use the closest points

A one-sided quotient is the right tool when the table has values on only one side. Closer x-values beat farther ones.

A straight piece has an exact slope

On a line segment, the derivative is the rise over the run of that segment. At a corner, the left slope and the right slope disagree, so the derivative does not exist.

The denominator is the change in x

For points \(a - h\) and \(a + h\), the denominator is \(2h\), not h.

Level 1

Easy

Read a slope from a small symmetric table or a straight piece of a graph.

Example

The table gives position in meters at time t seconds. Estimate the velocity at \(t = 4\) with the symmetric difference quotient.

\(t\)	3	4	5
\(s(t)\)	10	14	20

\(\dfrac{s(5) - s(3)}{5 - 3}\)

\(\dfrac{20 - 10}{2} = 5\)

5 meters per second

Answer: 5 meters per second

Use the same position table. Estimate the velocity at \(t = 4\) with the forward quotient from 4 to 5.

\(t\)	3	4	5
\(s(t)\)	10	14	20

Show Answer

\(\dfrac{s(5) - s(4)}{5 - 4}\)

\(\dfrac{20 - 14}{1} = 6\)

6 meters per second

Answer: 6 meters per second

Use the same position table. Estimate the velocity at \(t = 4\) with the backward quotient from 3 to 4.

\(t\)	3	4	5
\(s(t)\)	10	14	20

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\(\dfrac{s(4) - s(3)}{4 - 3}\)

\(\dfrac{14 - 10}{1} = 4\)

4 meters per second

Answer: 4 meters per second

Estimate \(f'(2)\) with the symmetric difference quotient.

\(x\)	1.9	2.0	2.1
\(f(x)\)	4.62	5.00	5.44

Show Answer

\(\dfrac{f(2.1) - f(1.9)}{2.1 - 1.9}\)

\(\dfrac{5.44 - 4.62}{0.2} = \dfrac{0.82}{0.2}\)

\(4.1\)

Answer: \(4.1\)

The graph of f is made of straight pieces. Estimate \(f'(3)\).

Graph of f

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x = 3 lies on the segment from \((2, 0)\) to \((5, 3)\).

\(\dfrac{3 - 0}{5 - 2} = 1\)

Answer: \(1\)

Use the same graph. Estimate \(f'(0)\).

Graph of f

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x = 0 lies on the segment from \((-2, 4)\) to \((2, 0)\).

\(\dfrac{0 - 4}{2 - (-2)} = \dfrac{-4}{4} = -1\)

Answer: \(-1\)

Level 2

Medium

Choose the closest points, and notice a corner.

Example

The table gives values only at 3 and to the right of 3. Give the best estimate of \(f'(3)\).

\(x\)	3	3.2	3.5
\(f(x)\)	8	8.9	10.4

The closest available point is \(x = 3.2\).

\(\dfrac{f(3.2) - f(3)}{3.2 - 3} = \dfrac{8.9 - 8}{0.2} = 4.5\)

The point \(x = 3.5\) is farther, so it is a weaker estimate.

Answer: \(4.5\)

Which quotient is the best estimate of \(f'(2)\)?

\(x\)	1.9	2.0	2.1
\(f(x)\)	4.62	5.00	5.44

A. \(\dfrac{f(2.1) - f(1.9)}{0.2}\)

B. \(\dfrac{f(2.1) - f(2)}{0.1}\)

C. \(\dfrac{f(2) - f(1.9)}{0.1}\)

D. \(\dfrac{f(2.1) - f(1.9)}{2.1}\)

Show Answer

The points 1.9 and 2.1 are equally close to 2, one on each side.

The symmetric quotient uses denominator \(2.1 - 1.9 = 0.2\).

The one-sided quotients are reasonable, but they use less of the table.

Choice D divides by 2.1, which is not the change in x.

Answer: A

Use the graph. Find \(f'(2)\), or say it does not exist.

Graph of f

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Left of 2, the segment from \((-2, 4)\) to \((2, 0)\) has slope

\(\dfrac{0 - 4}{2 - (-2)} = -1\)

Right of 2, the segment from \((2, 0)\) to \((5, 3)\) has slope

\(\dfrac{3 - 0}{5 - 2} = 1\)

The one-sided slopes are not equal. The graph has a corner.

The derivative does not exist.

Answer: does not exist

Use the graph. Estimate \(f'(-3)\).

Graph of f

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x = −3 lies on the segment from \((-4, 2)\) to \((-2, 4)\).

\(\dfrac{4 - 2}{-2 - (-4)} = \dfrac{2}{2} = 1\)

The hole is at \(x = -2\), not at \(x = -3\).

Answer: \(1\)

Values satisfy \(f(0) = 1\), \(f(2) = 5\), and \(f(4) = 13\). The average rate on \([0, 4]\) is 3. What is the forward estimate of \(f'(2)\) that uses \(x = 4\)?

Show Answer

Average rate on \([0, 4]\):

\(\dfrac{13 - 1}{4} = 3\)

Forward estimate at 2:

\(\dfrac{f(4) - f(2)}{4 - 2} = \dfrac{13 - 5}{2} = 4\)

Those are different numbers. The forward estimate is the one asked for.

Answer: \(4\)

Estimate \(f'(3)\) with \(f(2.9) = 7.6\) and \(f(3.1) = 8.4\).

Show Answer

\(\dfrac{f(3.1) - f(2.9)}{3.1 - 2.9}\)

\(\dfrac{8.4 - 7.6}{0.2} = \dfrac{0.8}{0.2} = 4\)

Answer: \(4\)

Level 3

Hard

The denominator, the distance, or a corner decides the estimate.

Example

Use the closest symmetric pair to estimate \(f'(2)\). Then explain why the pair 1.0 and 3.0 is weaker.

\(x\)	1.0	1.9	2.0	2.1	3.0
\(f(x)\)	0	4.6	5.0	5.6	9.0

The closest pair around 2 is 1.9 and 2.1.

\(\dfrac{f(2.1) - f(1.9)}{2.1 - 1.9} = \dfrac{5.6 - 4.6}{0.2} = 5\)

The pair 1.0 and 3.0 gives

\(\dfrac{9.0 - 0}{3.0 - 1.0} = \dfrac{9}{2} = 4.5\)

Those x-values are farther from 2, so the secant is a weaker stand-in for the tangent.

Answer: \(5\)

A student estimates \(f'(2)\) from \(f(1.9) = 4.6\) and \(f(2.1) = 5.6\) by computing \(\dfrac{5.6 - 4.6}{0.1} = 10\). What is the correct symmetric estimate?

Show Answer

The change in x is \(2.1 - 1.9 = 0.2\), not 0.1.

\(\dfrac{5.6 - 4.6}{0.2} = 5\)

Answer: \(5\)

Use the graph. The left-hand slope at \(x = 2\) is \(-1\) and the right-hand slope is 1. A student averages them and says \(f'(2) = 0\). Is the derivative 0?

Show Answer

A derivative exists only when the one-sided derivatives are equal.

\(-1 \neq 1\)

Averaging the two slopes does not create a derivative.

The derivative does not exist at the corner.

Answer: no

The position of a particle, in meters, satisfies \(s(4.9) = 12\), \(s(5) = 13\), and \(s(5.1) = 12.4\). Estimate the velocity at \(t = 5\), and say whether the estimate is positive or negative.

Show Answer

\(\dfrac{s(5.1) - s(4.9)}{5.1 - 4.9} = \dfrac{12.4 - 12}{0.2} = \dfrac{0.4}{0.2} = 2\)

The estimate is positive, so the particle is moving in the positive direction at that instant.

Answer: 2 meters per second, positive

For the table of f below, compute the symmetric estimate of \(f'(2)\) and the value \(f(2)\). Which one is the derivative estimate?

\(x\)	1.9	2.0	2.1
\(f(x)\)	4.62	5.00	5.44

Show Answer

\(f(2) = 5.00\) is a function value.

\(\dfrac{5.44 - 4.62}{0.2} = 4.1\) is the derivative estimate.

Answer: \(4.1\)

Use the graph. At \(x = -2\), the open circle is at height 4 and the closed dot is at height 1. Can \(f'(-2)\) exist?

Graph of f

Show Answer

Both drawn pieces approach height 4, so the limit of f is 4.

\(f(-2) = 1\), so f is not continuous at \(x = -2\).

A derivative can exist only where the function is continuous.

The derivative does not exist.

Answer: no

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