Let for . Derivatives of at are given…

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

h(x)=\sqrt{1+x}

for

x>-1

. Derivatives of

h

x=0

are given below.

h(0)=1,\qquad h'(0)=\dfrac12,\qquad h''(0)=-\dfrac14,\qquad h'''(0)=\dfrac38,\qquad h^{(4)}(0)=-\dfrac{15}{16}

(a) Construct the polynomial

P_4(x)

of degree 4 whose value and first four derivatives at

x=0

agree with those of

h

. Write

P_4

in standard form, collecting like powers of

x

. (b) Use

P_4

to estimate

\sqrt{1.5}

. Give your answer to 5 decimal places. (c) The next (fifth–degree) term in the series for

h

x=0

has coefficient

\dfrac{105}{32}

. Use this information to obtain an upper bound for the magnitude of the error in the estimate found in part (b). Express the bound as an inequality. (d) Determine whether the estimate in part (b) is an underestimate or an overestimate of the true value of

\sqrt{1.5}

. Give a brief justification that refers to the sign of the first neglected term.

Related theory

Browse the problem library