Chapter 5: Problem 60
Suppose \(\int_{1}^{x} f(t) d t=x^{2}-2 x+1 .\) Find \(f(x)\)
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Archimedes (287-212 B.C.), inventor, military engineer, physicist, and the greatest mathematician of classical times, discovered that the area under a parabolic arch like the one shown here is always two- thirds the base times the height. (a) Find the area under the parabolic arch \( y=6-x-x^2, -3 \leq x \leq 2 \) (b) Find the height of the arch. (c) Show that the area is two-thirds the base times the height.
In Exercises \(19-30,\) evaluate the integral using antiderivatives, as in Example \(4 .\) $$\int _ { 0 } ^ { 1 } e ^ { x } d x$$
Multiple Choice At \(x=\pi,\) the linearization of \(f(x)=\int_{\pi}^{x} \cos ^{3} t d t\) (A) \(y=-1\) (B) \(y=-x \quad\) (C) \(y=\pi\) (D) \(y=x-\pi \quad\) (E) \(y=\pi-x\)
Consider the integral \(\int_{-1}^{1} \sin \left(x^{2}\right) d x\) $$\begin{array}{l}{\text { (a) Find } f^{\prime \prime} \text { for } f(x)=\sin \left(x^{2}\right)} \\ {\text { (b) Graph } y=f^{\prime \prime}(x) \text { in the viewing window }[-1,1] \text { by }[-3,3] \text { . }} \\\ {\text { (c) Explain why the graph in part (b) suggests that }\left|f^{\prime \prime}(x)\right| \leq 3} \\ {\text { for }-1 \leq x \leq 1 .} \\ {\text { (d) Show that the error estimate for the Trapezoidal Rule in this case becomes }}\end{array} $$ $$\left|E_{T}\right| \leq \frac{h^{2}}{2}$$ $$\begin{array}{l}{\text { (e) Show that the Trapezoidal Rule error will be less than or equal to } 0.01 \text { if } h \leq 0.1 .} \\ {\text { (f) How large must } n \text { be for } h \leq 0.1 ?}\end{array}$$
Rectangular Approximation Methods Show that if \(f\) is a nonnegative function on the interval \([a, b]\) and the line \(x=(a+b) / 2\) is a line of symmetry of the graph of \(y=f(x)\) then \(L R A M_{n} f=\operatorname{RRAM}_{n} f\) for every positive integer \(n .\)
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