(iv) Find the equation of the tangent to the curne \( y=x^{3}-x \) at \( x=2 \)

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) \( [n=4] \quad \int_{-1}^{1} \sqrt{\ln (2+x)} d x \)

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) \( [n=4] \quad \int_{-1}^{1} \sqrt{\ln (2+x)} d x \)

14. A mediodía, un barco A está a 150 km al oeste del barco B . El barco A navega hacia el este a \( 35 \mathrm{~km} / \mathrm{h} \) y el barco B navega hacia el norte a \( 25 \mathrm{~km} / \mathrm{h} \). ¿Qué tan rápido cambia la distancia entre los barcos a las \( 16: 00 \) ?

\( \begin{array}{ll}\text { (b) }[n=5] & \int_{0}^{1} \sin x^{2} d x \\ \text { (c) }[n=4] & \int_{0}^{2} x\left(e^{x}\right) d x\end{array} \)

\( \int e ^ { - \alpha t } \cos \beta t + \sin \beta t \)