Mathematics

Problem

A continuous function

g

is shown on a coordinate grid. The curve is increasing on the interval

2\le x\le4

and passes through the point

P(3,5)

. Two additional labeled points on the curve are

A(2.5,4.0)

and

B(3.5,6.2)

. Near

x=3

the graph is concave up. Throughout this problem, assume

g

is differentiable on the open interval that contains the three points described. (a) Describe a procedure that uses the information given in the graph to obtain a numerical approximation for

g'(3)

. Your answer should identify the quantities that must be computed and how they are combined. (b) Without performing any calculations, explain why an approximation that relies on points closer to

x=3

is generally more accurate than an approximation that uses points farther from

x=3

. Reference specific geometric features of the graph in your explanation. (c) Decide whether the slope of the line through

A

and

B

is an overestimate or an underestimate of

g'(3)

. Justify your answer using the concavity information provided.

Intuitive explanation

The key to understanding the derivative here is that $g'(3)$ represents the instantaneous rate of change of $g$ at $x=3$ —the slope of the tangent line at point $P$ . Since you have two nearby points $A$ and $B$ on the curve, you can estimate this slope using the secant line between them, which approximates the tangent line when the points are close to $x=3$ . The concavity information tells you whether your estimate is an overestimate or underestimate of the true derivative.

Intuitive explanation

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