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Activity 5.2.3 .
Suppose that
\(f(t) = \frac{t}{1+t^2}\) and
\(F(x) = \int_0^x f(t) \, dt\text{.}\)
Figure 5.2.5. Axes for plotting \(f\) and \(F\text{.}\)
(a) On the axes at left in
Figure 5.2.5 , plot a graph of
\(f(t) = \frac{t}{1+t^2}\) on the interval
\(-10 \le t \le 10\text{.}\) Clearly label the vertical axes with appropriate scale.
(b) What is the key relationship between
\(F\) and
\(f\text{,}\) according to the Second FTC?
(c) Use the first derivative test to determine the intervals on which
\(F\) is increasing and decreasing.
(d) Use the second derivative test to determine the intervals on which
\(F\) is concave up and concave down. Note that
\(f'(t)\) can be simplified to be written in the form
\(f'(t) = \frac{1-t^2}{(1+t^2)^2}\text{.}\)
(e) Using technology appropriately, estimate the values of
\(F(5)\) and
\(F(10)\) through appropriate Riemann sums.
(f) Sketch an accurate graph of
\(y = F(x)\) on the righthand axes provided, and clearly label the vertical axes with appropriate scale.
