高中数学函数题目 在线等答案或中文翻译

高中数学函数题目 在线等答案或中文翻译
题目是英文版本的.求题解,需要详细过程.或者是题目的中文翻译,都可以
1. Sara was sitting on a swing, talking about the mathematics she had been doing at school and swinging gently to and fro. She noticed that the period of one complete swing was 3 seconds and that if she did nothing to keep going, the amplitude of her motion decreased from 1 metre to 50cm in 2 minutes. If Sara’s horizontal displacement is x metres after t second, find a function of the form x=Ae-ktcos nt to describe this motion. Find an expression for Sara’s horizontal velocity after t seconds.
2. Show that the area under the curve y=x2-1/12 from x=r-1/2 to x=r+1/2. By considering the area under this curve from x=1/2 to x=n+1/2, show that :1²+2²+3²+……….+n²=n(n+1)(2n+1)/6
3. Suppose that x has density f(x)=12x²(1-x) for 0Is x a density function? Find the mode, median and mean of x. sketch the graph of f and show the location of the mean, median and mode on the x-axis. What is the probability that a randomly selected value from this distribution is between 0.2and 0.4?