A twice–differentiable function is shown on a set of labeled axes (units of 1 on each…

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

A twice–differentiable function

g

is shown on a set of labeled axes (units of 1 on each axis). At

x=0,2,4,6,8

, the curve intersects the grid points

(0,1)

(2,4)

(4,2)

(6,5)

, and

(8,3)

respectively, and the graph is smooth between these points. • From

x=0

x=2

, the curve is concave up and increases steadily so that the slope of the curve rises from approximately

0.5

x=0

to approximately

2.5

x=2

. • From

x=2

x=4

, the curve descends; the slope at

x=3

is visually horizontal, and at

x=4

the tangent appears to have slope about

-2

. • From

x=4

x=6

, the curve climbs again, becoming nearly vertical at

x=5

(slope near

+\text{large}

) and flattening to slope

1

x=6

. • From

x=6

x=8

, the curve decreases smoothly, and the tangent at

x=7

appears to have slope about

-1.5

. Answer the following questions about the derivative of

g

. (a) Describe the sign of

g'(x)

on each of the following open intervals and give a brief reason for each. (i)

(0,2)

(ii)

(2,4)

(iii)

(4,6)

(iv)

(6,8)

(b) Rank the magnitudes

|g'(2)|, |g'(4)|,

and

|g'(5)|

from least to greatest and justify your ranking using features you observe in the graph. (c) At

x=3

the tangent line is horizontal. Does

x=3

correspond to a local maximum, a local minimum, or neither for

g

? Explain your reasoning without computing any derivatives.