Question Approximate the \( x \)-value of the local minimum value of the function given below. Use Newton's method with the specified initial approximation \( x_{0} \) to find \( x_{2} \). Round your answer to the nearest thousandth. \[ f(x)=\frac{5}{6} x^{3}+\frac{5}{2} x^{2}-5 x+2, \quad x_{0}=3 \] Provide your answer below: \[ \] \( x_{2} \approx \square \)

Decide whether the following statement is true or false. The cube function is odd and is increasing on the interval \( (-\infty, \infty) \). Choose the correct answer below. False True

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

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.

1) \( f(x)=\sqrt{\frac{2}{\sqrt[x]{e 6 x+1}}} \)

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.