Exercises

1. Suppose that the the velocity of an object traveling in one dimension is given by $v(t) = t+sin(t)$ for $0 \leq t \leq 6$. Let $s(t)$ be the position of the object at time $t$, assuming that $s(0)=0$.
   1. As explained in your text, the position of the object at $t=6$ is equal to the area under the velocity curve from $t=0$ to $t=6$. Use the midpoint rule with $20$ subdivision to approximate the position of the object at $t=6$.
   2. Let $R_n$ be the right endpoint rule and $L_n$ the left endpoint rule approximations with $n$ subdivisions to the position of the object at $t=6$. Explain why the following relation holds for any value of $n$.
      
      $$L_n < s(6) < R_n$$
      
      
      (Hint - look at the results of leftbox and rightbox commands.)
   3. Evaluate $R_{500}$ and $L_{500}$. Based on your results, do you think your approximation using the midpoint rule with $20$ subintervals was within $0.1$ of the exact value for the area? Explain why or why not.
2. Consider the function
   
   $$g(x) = \frac{x^2+4x+1}{x^2+2x}$$
   
   
   on the interval $[1,5]$.
   1. Use the error bound formula to find the smallest value of $n$ that guarantees that $M_n$ approximates the area to within $0.0005$. That is, find the smallest value of $n$ that guarantees that $\mid EM_n \mid < 0.0005$.

   2. The value of $n$ given by the error bound is usually conservative. That is, in practice the desired accuracy can be achieved with a smaller value of $n$. Given that
      
      $$\int_1^5 g(x) \, dx \approx 6.075665746$$
      
      
      find the smallest value of $n$ such that $\mid EM_n \mid < 0.0005$.