Let be twice-differentiable on with for all in that interval

MATHEMATICS · Single correct · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

Let

f

be twice-differentiable on

(1,3)

with

f''(x)>0

for all

x

in that interval. The tangent line to

f

x=2

is used to approximate

f(2.1)

. Which of the following describes the relationship between the approximation

T

and the actual value

f(2.1)

? (a) The information given is insufficient to determine whether

T

overestimates or underestimates

f(2.1)

. (b)

T

is an overestimate because

f''(2)>0

implies the slope is increasing, placing the tangent line above the curve to the right of

x=2

. (c)

T

is an underestimate because the graph of

f

is concave up on

(1,3)

. (d)

T

is an overestimate because the graph of

f

is concave up on

(1,3)