Mathematics

Mathematics

Problem

On the interval

[0,1]

, which of the following statements is true about the lengths of the curves

y=x^{2}

and

y=e^{x}-1

? a) The curve

y=x^{2}

is longer because

2x>e^{x}

for every

x

in

[0,1]

. b) The curve

y=x^{2}

is longer because

\displaystyle\int_{0}^{1}2x\,dx=1

is greater than

\displaystyle\int_{0}^{1}e^{x}\,dx\approx0.718

. c) The curve

y=e^{x}-1

is longer because

e^{x}>2x

for all

x

in

[0,1]

. d) The curve

y=x^{2}

is longer because

x^{2}>e^{x}-1

for every

x

in

[0,1]

.

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