The crate of mass m is pushed to the left until the linear elastic spring of constant k has compressed a distance d from its unstretched length $\ell_{0}$. The crate is released from rest, and friction between the crate and the horizontal surface is negligible.Using the work-energy principle, determine the value of k so that the crate is moving at 3 m/s the instant the spring becomes uncompressed. Use m = 20 kg and d = 0.75 m.

Determine the polynomial function whose graph passes through the point.

$$(0.072,0.203),(0.120,0.238),(0.148,0.284)$$

Find a matrix P that orthogonally diagonalizes A. Verify that $P^TAP$ gives the correct diagonal form.

$$A=\left[\begin{array}{rr} {8} & {15} \\ {15} & {-8} \end{array}\right]$$

Solve the system of linear equations.

$$\begin{align*} x_1 - 2x_2 &= 0 \\ 6x_1+2x_2 &= 0 \\ \end{align*}$$

Use a software program or a graphing utility to solve the system of linear equations.

$$\begin{align*} 3x+3y+12z &= 6\\ x+y+4z &= 2 \\ 2x+5y +20z &= 10 \\ -x +2y+8z &= 4 \end{align*}$$

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. $\lim _{x \rightarrow 0} \frac{x}{e^{x}}=1$

The Michaelis-Menten equation models the rate v of an enzymatic reaction as a function of the concentration [S] of a substrate S. In the case of the enzyme chymotrypsin the equation is $v=\frac{0.14[\mathrm{S}]}{0.015+[\mathrm{S}]}$ Use a graphing calculator or computer to graph v as a function of [S].

Solve the system of linear equations.

$$\begin{align*} 0.05x_1 - 0.03x_2 &= 0.21 \\ 0.07x_1+0.02x_2 &= 0.17 \\ \end{align*}$$

Sketch the image of the unit square [a square with vertices at (0, 0), (1, 0), (1, 1), and (0, 1)] under the specified transformation. T is the shear represented by T(x, y) = (x + 2y, y).

Use the bond dissociation enthalpy data in the table to calculate $\Delta H^{\circ}$ for (a) the monofluorination and (b) the monoiodination of methane. What is the main reason the values are so different?

Suppose that the population dynamics of a species obeys a modified version of the logistic differential equation having the following form: $\frac{d N}{d t}=r\left(1-\frac{N}{K}\right)^{2} N$ where $r \neq 0 \text { and } K>0$ Apply the local stability criterion to the equilibrium $\hat{N}=K.$ What do you think your answer means about the stability of this equilibrium?

Why is the copious production of $\Lambda^{0}$ and K particles very difficult to reconcile with their slow decay, without the concept of strangeness? How does strangeness provide a reconciliation?

Rationalize each denominator. Assume that all radicands represent positive real numbers. $\frac{5}{\sqrt{10}}$

Determine whether the matrix is orthogonally diagonalizable.

$$\left[\begin{array}{rrr} {3} & {2} & {-3} \\ {-2} & {-1} & {2} \\ {-3} & {2} & {3} \end{array}\right]$$