Values of are recorded near and are shown below

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Values of

f(x)=\sqrt{x}

are recorded near

x=5

and are shown below.

x	4.80	4.90	4.95	5.00	5.05	5.10	5.20
f(x)	2.190890	2.213594	2.224859	2.236068	2.247222	2.258318	2.280351

Using only the information in the table, address the following tasks. (a) Compute the slope of the secant line through

(4.8, f(4.8))

and

(5.2, f(5.2))

. Interpret this value as an estimate of

f'(5)

. (b) Repeat the calculation using the points

(4.9, f(4.9))

and

(5.1, f(5.1))

. Record your result. (c) Repeat the calculation using the points

(4.95, f(4.95))

and

(5.05, f(5.05))

. Record your result. (d) Based on your answers to parts (a)–(c), select the estimate you consider most reliable for

f'(5)

. Justify your choice by discussing the trade-off between interval width and error sources.

Related theory

Derivative Foundations
Intuitive idea of derivatives
Limits and Derivatives

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