91Ó°ÊÓ

If \(\int_{-x}^{0} v(t) d t=\int_{0}^{x} v(t) d t\) (equal areas left and right of zero), then \(v(x)\) is an ________ function. Take derivatives to prove it.

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=1 / y\)

Find \(c\) in \(S-I=c(\Delta x)^{4}\left[y^{\prime \prime}(1)-y^{\prime \prime \prime}(0)\right]\) by taking \(y=x^{4}\) and \(\Delta x=1\)

Example 2 said that \(\int_{2 x}^{3 x} d t / t\) does not really depend on \(x\) (or \(t !)\). Substitute \(x u\) for \(t\) and watch the limits on \(u\).

Compute the area of 208 rectangles under \(u(x)=\sqrt{x}\) from \(x=0\) to \(x=4\)

Find a function \(y(x)\) that solves the differential equation. \(d y / d x=x / y\)

A point \(P\) is chosen randomly along a semicircle (see figure: equal probability for equal arcs). What is the average distance \(y\) from the \(x\) axis? The radius is \(1 .\)

Find \(c\) in \(G-I \Leftrightarrow c(\Delta x)^{4}\left[y^{\prime \prime \prime}(1)-y^{\prime \prime \prime}(-1)\right]\) by taking \(y=x^{4}, \Delta x=2,\) and \(G=(-1 / \sqrt{3})^{4}+(1 / \sqrt{3})^{4}\)

Show that \(|1-5|<|1|+|-5|\). Always \(\left|v_{1}+v_{2}\right|<\left|v_{1}\right|+\left|v_{2}\right|\) unless

True or false, with reason: (a) All continuous functions have derivatives. (b) All continuous functions have antiderivatives. (c) All antiderivatives have derivatives. (d) \(A(x)=\int_{2 x}^{3 x} d t / t^{2}\) has \(d A / d x=0\).