Indicate whether the statement is true or false. If $R(w)=w^{3}$ then R'(-2) is negative.

Determine the x-coordinate of the mass center of the tapered steel rod of length L where the diameter at the large end is twice the diameter at the small end.

Find the derivative. Assume that a, b, c, and k are constant s. $w=\frac{3 y+y^{2}}{5+y}$

Determine the force T required to hold the uniform bar of mass m and length L in an arbitrary angular position $\theta$. Plot your result over the range $0 \leq \theta \leq 90^{\circ}$, and state the value of T for $\theta=40^{\circ}$.

Show that

$$\mathbf{x}^{1}=\left(\begin{array}{c} {1+\sqrt{1+\varepsilon}} \\ {1} \end{array}\right) \quad \text { and } \quad \mathbf{x}^{2}=\left(\begin{array}{c} {1-\sqrt{1+\varepsilon}} \\ {1} \end{array}\right)$$

are eigenvectors of the matrix

$$\left(\begin{array}{cc} {1} & {\varepsilon} \\ {1} & {-1} \end{array}\right)$$

with eigenvalues $\sqrt{1+\varepsilon} \text { and }-\sqrt{1+\varepsilon}$ respectively.

Find the mass of the surface lamina S of density $\rho$. $S: 2 y+6 x+z=18$, first octant, $\rho(x, y, z)=2 x$

Evaluate $\lim _{x \rightarrow 0} \frac{x}{|x-1|-|x+1|}$.

Express the set of all real numbers x satisfying the given conditions as an interval or a union of intervals. $x \leq-1$

A pipe flows full with water. Is it possible for the volume flow rate into the pipe to be different than the flow rate out of the pipe? Explain.

Suppose $\sum_{k=N+1}^{\infty} a_{k}=S$ and $a_{1}+a_{2}+\cdots+a_{N}=K$. Prove that $\sum_{k=1}^{\infty} a_{k}$ converges and its sum is S+K.

An engineer is designing a horizontal, rectangular conduit that will be part of a system that allows fish to bypass a dam. Inside the conduit, a flow of water at 40$^{\circ}$F will be divided into two streams by a flat, rectangular metal plate. Calculate the viscous drag force on this plate, assuming boundary-layer flow with free-stream velocity of 10 ft /s and plate dimensions of L = 6 ft and W = 4.0 ft .

On a clear day at a certain location, a 100-V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of $0.500 \times 10^{-4}$ T. Compute the energy densities of the electric field.

A solid particle falls through a viscous fluid. The falling velocity V, is believed to be a function of the fluid density pf , the particle density Pp, the fluid viscosity $\mu$, the particle diameter D, and the acceleration due to gravity g: By dimensional analysis, develop the $\mu$ -groups for this problem. Express your answer in the form $V=f\left(\rho_{f}, \rho_{p}, \mu, D, g\right)$, $\frac{V}{\sqrt{g D}}=f\left(\pi_{1}, \pi_{2}\right)$.

A 3-m-long rectangular weir is to be constructed in a 3-m-wide rectangular channel. (a). The maximum flow in the channel will be 4 m3/s. What should be the height P of the weir to yield a depth of water of 2 m in the channel upstream of the weir?