A twice–differentiable function satisfies the following conditions on the closed…

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

Problem

A twice–differentiable function

h

satisfies the following conditions on the closed interval

[3.5,4.5]

: •

h(4)=7.2

•

h'(4)=1.5

•

h''(x)<0

for all

x

[3.5,4.5]

Assume

|h''(x)|

remains bounded by

2

throughout the interval. (a) Write the equation of the tangent line to the graph of

h

x=4

. (b) Use the line from part (a) to approximate

h(4.3)

. Clearly indicate all steps. (c) Determine whether the approximation found in part (b) is guaranteed to be an underestimate or an overestimate of

h(4.3)

. Explain your reasoning, citing specific information given about

h''

. (d) Explain why

|h(4.3)-\text{approximation}|

cannot exceed

0.3

when

|h''(x)|\le 2

[3.5,4.5]

. Support your answer with appropriate reasoning (formal error-bound computations are not required).