Page \( \langle 5\rangle \) 。 5. (a) Take a \( \ln ( \) natural log) of both sides like we did in our class and (b) find \( \frac{d y}{d x} \) of part (a) and write your final answer in simplest form. \( y=(x+4)^{\left(5 x^{2}+1\right)} \). NOT allowed to use Chegg or other such websites. NO Desmos, Calculato and/or any other graphing website and/or utilities.

\( ( e ^ { x } - \sin y ) d x + ( \cos y ) d y \)

asymptotes of \( f \).

Proporre un esempio di funzione \( g(x) \) tale che \[ \lim _{x \rightarrow+\infty} \frac{g(x)}{3 \log ^{2} x+1}=+\infty \]

MAS182 - Applied Mathematics 13. Final Examination - S1 2021 The water content \( W(t) \) (measured in grams) in the cheese is governed by the differential equation \[ \frac{d W}{d t}=-(W-100)^{2} \] with initial condition \( W(0)=200 \mathrm{~g} \) and time is measured in hours.

5. [12 marks] Using calculus and showing all your working, sketch the function \[ f(x)=\frac{x}{x^{2}-4} \] noting the domain of \( f(x) \), any intercepts or asymptotes, symmetry, regions where the function is increasing or decreasing and any relative extrema. Note: You don't have to consider the second derivative and concavity.