Tutorial 8 Numerical Methods 1. Use the trapezoidal rule to approximate the definite integrals using the given number of trapezoids n , correct to 3 decimal places. (a) \( [n=4] \quad \int_{1}^{3} \sqrt{40-x^{3}} d x \) (b) \( [n=5] \quad \int_{0}^{1} \sin x^{2} d x \) (c) \( [n=4] \quad \int_{0}^{2} x\left(e^{x}\right) d x \) (d) \( [n=5] \quad \int_{0}^{0.5} 2 e^{-x^{2}} d x \) (e) \( [n=4] \quad \int_{-1}^{1} \sqrt{\ln (2+x)} d x \)

(b) Prove that \( \left|x_{n}-x_{n+1}^{\infty}\right|=\frac{1}{2^{n-1}} \) for all \( n \in \mathbb{N} \)

1) \( y=\cos ^{-1}(\sin x) \Rightarrow y^{\prime}=? \) A) \( -\frac{\cos x}{|\cos x|} \) B) \( \frac{\cos x}{|\cos x|} \) C) \( -\frac{\cos x}{\sqrt{1-x^{2}}} \) D) \( \frac{\cos x}{\sqrt{1-x^{2}}} \) E) \( -\frac{\sin x}{\sqrt{1-x^{2}}} \)

24) \( \lim _{x \rightarrow 4} \frac{2-\sqrt{x}}{x^{2}-16}= \) ? \( \begin{array}{lllll}\text { A) }-\infty & \text { B) } \frac{1}{32} & \text { C) } 0 & \text { D) }-\frac{1}{32} & \text { E) Limit yoktur }\end{array} \)

\( F(x)=\int_{1}^{\pi} \frac{\cos t}{t} d t \) siendo \( x \geq 1 \)

25) \( \lim _{x \rightarrow 0} \frac{|x-2|}{x-2}= \) ? \( \begin{array}{lllll}\text { A) }-\infty & \text { B) Limit yoktur } & \text { C) } 1 & \text { D) } \infty & \text { E) }-1\end{array} \)