Activity 5.6.4.
Consider the functions \(f(x) = 2-x^2\text{,}\) \(g(x) = 2-x^3\text{,}\) and \(h(x) = 2-x^4\text{,}\) all on the interval \([0,1]\text{.}\) For each of the questions that require a numerical answer in what follows, write your answer exactly in fraction form.
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(a)
On the three sets of axes in the provided figure, sketch a graph of each function on the interval \([0,1]\text{,}\) and compute \(L_1\) and \(R_1\) for each. What do you observe?
(b)
Compute \(M_1\) for each function to approximate \(\int_0^1 f(x) \,dx\text{,}\) \(\int_0^1 g(x) \,dx\text{,}\) and \(\int_0^1 h(x) \,dx\text{,}\) respectively.
(c)
Compute \(T_1\) for each of the three functions, and hence compute \(S_2\) for each of the three functions.
(d)
Evaluate each of the integrals \(\int_0^1 f(x) \,dx\text{,}\) \(\int_0^1 g(x) \,dx\text{,}\) and \(\int_0^1 h(x) \,dx\) exactly using the First FTC.
(e)
For each of the three functions \(f\text{,}\) \(g\text{,}\) and \(h\text{,}\) compare the results of \(L_1\text{,}\) \(R_1\text{,}\) \(M_1\text{,}\) \(T_1\text{,}\) and \(S_2\) to the true value of the corresponding definite integral. What patterns do you observe?
