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Find f if $$f^{\prime \prime}(x)=x^{-2}, f(1)=0, \quad f(2)=0$$

$$f^{\prime \prime}(x)=x^{-2}$$

$$f^{\prime}(x)=-x^{-1}+C_{1}$$

$$=-\frac{1}{x}+c_{1}$$

$$f(x)=-\ln (x)+c_{1} x+c_{2}$$

$$f(1)=-\ln (1)+c_{1} 1+c_{2}=c_{1}+c_{2}$$

$$c_{1}+c_{2}=0 \rightarrow c_{1}=-c_{2}$$

$$f(2)=-\ln (2)+c_{1}(2)+c_{2}$$

$$-\ln (2)+2 c_{1}+c_{2}=0$$

$$-\ln (2)-2 c_{2}+c_{2}=0$$

$$-\ln (2)-c_{2}=0$$

$$c 2=-\ln (2)=-0.69 \simeq 0.7$$

$$C 1=0.69 \simeq 0.7$$

$$c_{1}=\ln (2)$$
$$c_{2}=-\ln (2)$$

$$F(x)=-\ln (x)+\ln (2) x-\ln (2)$$

Find position function if $$a(t)=3 \cos (t)-2 \sin (t), s(0)=0, v(0)=4$$

$$a(t)=3 \cos (t)-2 \sin (t)$$

$$v(t)=3 \sin (t)+2 \cos (t)+c_{1}$$

$$v(0)=3 \sin (0)+2 \cos (0)+c_{1}$$

$$4=0+2(1)+c 1 \rightarrow c_{1}=4-2=2$$

$$v(t)=3 \sin (t)+2 \cos t+2$$

$$S(t)=-3 \cos (t)+2 \sin (t)+2 t+c^{2}$$

$$S(0)=-3 \cos (0)+2 \sin (0)+2(0)+C_{2}$$

$$0=-3(1)+2(0)+0+C_2 \rightarrow C_{2}=3$$

$$s(t)=-3 \cos t+2 \sin (t)+2 t+3$$

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