The rates of change in population for two cities are \( P^{\prime}(t)=45 \) for Alphaville and \( Q^{\prime}(t)=105 e^{0.04 t} \) for Betaburgh, where \( t \) is the number of years since 1990 , and \( P^{\prime} \) and \( Q^{\prime} \) are measured in people per year. In 1990, Alphaville had a population of 5500 , and Betaburgh had a population of 3500 . Answer parts a) through \( c \) ).

8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \).

USing first principle prove that \( y=\sin x \) then \( \frac{d y}{d x}=\cos x \)

By using Newton-Raphson Method, determine the root for \( x^{3}+2 x^{2}-1=0 \). Given \( x_{0}=0.5 \). Give the answer correct to four decimal places. Dengan mengomenan

Calcule \( \int \frac{x^{4} - 5x^{2} + 6}{x-2} \,dx \) aplicando o método de divisão longa.

7. Find \( \lim _{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \) 8. If \( f(x)=x^{2}+10 \sin x \), show that there is a number \( c \) such that \( f(x)=1000 \)