Exercises

1. Consider the function $\cos(x)$ on the interval $[0,1]$. Use the leftbox and rightbox commands to display two rectangular approximations to the integral
   
   $$\int_0^1 \cos(x) \, dx$$
   
   
   You may use the default number of subintervals. From looking at your graphs, which do you think will give a better appoximation to the integral the right endpoint rule or the left endpoint rule? Explain your answer. If you have a hard time seeing a difference between the two methods, you might try decreasing the number of subintervals to two.

2. Consider the function $f(x) = x^3-3x^2+7x-2$ on the interval $[0,7]$. Use the command leftsum to approximate the definite integral
   
   $$\int_{0}^{7} f(x) \, dx = \frac{1659}{4}$$
   
   
   to two decimal places. Looking at the graph of $f$, can you explain why the value given by the leftsum command is always less than the value of the integral? If you used the rightsum command to approximate this same integral, do you think your approximations would be smaller than the value of the integral, larger than the integral, or could it be larger or smaller depending on the number of subintervals you use? Explain your answer.

3. Consider the function
   
   $$g(x) = \frac{4x^2+2x+3}{4x^2-1}$$
   
   
   on the interval $[1,4]$.
   1. Use the error bound formula to find the smallest value of $n$ that guarantees that $M_n$ approximates the area to within $0.00001$. That is, find the smallest value of $n$ that guarantees that $\mid EM_n \mid < 0.001$.

   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
      
      $$\int_1^4 g(x) \, dx \approx 4.608428$$
      
      
      find the smallest value of $n$ such that $\mid EM_n \mid < 0.001$ and compare it to the value you obtained in the previous exercise.