Consider the function

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

Problem

f(x)=e^{x^{2}}

, which is twice differentiable for all real

x

. You wish to approximate

f(0.08)

. Two potential base points for a tangent–line approximation are offered: • Point A:

a=0

• Point B:

b=0.10

(a) Derive the equation of the tangent line to

f

x=a

and use it to approximate

f(0.08)

. (b) Derive the equation of the tangent line to

f

x=b

and use it to approximate

f(0.08)

. (c) Compute

f''(x)

and determine a numerical upper bound

M_{A}

for

|f''(x)|

on the closed interval connecting

a

and

0.08

. Repeat to obtain

M_{B}

for the interval connecting

b

and

0.08

. Show all work leading to each bound. (d) Using the bounds found in part (c), obtain an upper estimate for the absolute error of each approximation in parts (a) and (b). Clearly state both error bounds. (e) Which base point,

a

b

, yields the smaller theoretical maximum error? Cite the quantitative results from parts (c) and (d) to justify your conclusion.

Related theory

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