2.3 Estimating Derivatives of a Function at a Point

Learning objectives

—CHA-2.D—Estimate derivatives• CHA-2.D Estimate derivatives.• CHA-2.D.1 The derivative at a point can be estimated from information given in tables or graphs.• CHA-2.D.2 Technology can be used to calculate or estimate the value of a derivative of a function at a point.• Enduring understanding CHA-2: Derivatives allow us to determine rates of change at an instant by applying limits to knowledge about rates of change over intervals.

To estimate $f'(a)$ from a table, use values close to $a$ on opposite sides when available. Their secant slope approximates the tangent slope because the interval is centered on the target input.

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

For the explicitly defined function $f(x)=x^3$ :

$x$	$f(x)$
$1.9$	$6.859$
$2.0$	$8.000$
$2.1$	$9.261$

Using the values equally spaced around $2$ , $f'(2)\approx\dfrac{9.261-6.859}{2.1-1.9}=\dfrac{2.402}{0.2}=12.01$ . The symbol $\approx$ is appropriate because a finite secant interval is being used in place of the limiting tangent slope.

From a graph, estimate the slope of the tangent at the target point using two readable points on the tangent line. With technology, evaluate a numerical derivative at the target input and report reasonable precision; the displayed result is still an estimate unless exact analysis is supplied.

Do not use the slope of a visibly distant secant when closer data are available, and do not read two arbitrary points on the curve as if they lay on the tangent. Table spacing, graph scale, and rounded values all affect the estimate.

Next Topic

—2.4 Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not ExistRead