Let be a twice-differentiable function with and

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

Problem

Let

g

be a twice-differentiable function with

g(4)=7

and

g'(4)=2

. A student uses the tangent line at

x=4

to approximate

g(5)

and reports a value of

9

. A later measurement shows the true value is

g(5)=9.6

. (a) Determine whether the student’s linear approximation at

x=4

is an overestimate or an underestimate of

g(5)

. Justify your conclusion numerically. (b) Using your answer to part (a), state the sign of

g''(c)

for some

c

(4,5)

guaranteed by the Mean Value Theorem for derivatives. Explain thoroughly. (c) Suppose that on the interval

[4,5]

the magnitude of

g''

never exceeds

M

. Derive an inequality involving

M

that must be satisfied by the observed error at

x=5

. Briefly discuss how this bound is consistent with the sign you found in part (b).