Related questions with answers

Determine whether each statement is true or false. If the statement is true, prove it. If the statement is false, give a counterexample. Assume that the functions $f$ , $g$ , and $h$ take on only positive values.

\lg(2n)^2=\Theta(\lg n^2)

A piece of tungsten of mass 50 g is placed over a flame for some time. The metal is then quickly transferred to a well-insulated aluminium calorimeter of mass 120 g containing 300 g of water at

22 ^ { \circ } \mathrm { C }

. After some time the temperature of the water reaches a maximum value of

31 ^ { \circ } \mathrm { C }

. a. State what is meant by the internal energy of a piece of tungsten. b. Calculate the temperature of the flame. You may use these specific heat capacities: water

4.2 \times 10 ^ { 3 } \mathrm { Jkg } ^ { - 1 } \mathrm { K } ^ { - 1 }

, tungsten

1.3 \times 10 ^ { 2 } \mathrm { Jkg } ^ { - 1 } \mathrm { K } ^ { - 1 }

and aluminum

9.0 \times 10 ^ { 2 } \mathrm { Jkg } ^ { - 1 } \mathrm { K } ^ { - 1 }

. c. State and explain whether the actual flaame temperature is higher or lower than your answer to b.

A function commonly used in communications textbooks for the tail probabilities of Gaussian random variables is the complementary error function, defined as $\operatorname{erfc}(z)=\frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-x^{2}} d x$ . Show that $Q(z)=\frac{1}{2} \operatorname{erfc}\left(\frac{z}{\sqrt{2}}\right)$ .