For the following exercise, determine the point(s), if any, at which the function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. $f(x)=\frac{x}{x^{2}-x}$

Calculate the dot product $\mathbf{u} \cdot \mathbf{v}$. $u = 4i + 2j - 3k, v = -2i + j - 2k$

What determines the severity of a shock? Can you say that a certain voltage is hazardous without further information?

For the following exercise, determine the point(s), if any, at which the function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other. $f(x)=\frac{2}{x^{2}+1}$

In the following exercise, use direct substitution to show that the limit leads to the indeterminate form 0/0. Then, evaluate the limit. $\lim _{x \rightarrow 2} \frac{x-2}{x^{2}-2 x}$

In the following exercise, use the graph of the function y = f(x) shown here to find the values, if possible. Estimate when necessary. $\lim _{x \rightarrow 0^{-}} f(x)$

For the following exercise, consider the function $f(x)=-x^{2}+1$. Approximate the area of the region between the x-axis and the graph of f over the interval [-1, 1].

Why is the trigone of the urinary bladder important clinically?

Find the weights for the numerical differentiation formula. $f^{\prime}(0) \approx A_{0} f(0)+A_{1} f(h)+A_{2} f(2 h)+A_{3} f(3 h)$

How do sound vibrations of atoms differ from thermal motion?

For the following exercise, consider the function $f(x)=-x^{2}+1$. Sketch the graph of f over the interval [-1, 1].

Twin jet engines on an airplane are producing an average sound frequency of 4100 Hz with a beat frequency of 0.500 Hz. What are their individual frequencies?

For the following exercise, points P(4, 2) and Q(x, y) are on the graph of the function $f(x)=\sqrt{x}$. Use the value in the preceding exercise to find the equation of the tangent line at point P.

For each of the following rings R exhibit the elements of Spec R, the open sets U in Spec R, the sections $\mathcal{O}(U)$ of the structure sheaf for Spec R for each open U, and the stalks $\mathcal{O}_{P}$ at each point P $\in$ Spec R: (a) $\mathbb{Z} / 4 \mathbb{Z}$ (b) $\mathbb{Z} / 6 \mathbb{Z}$ (c) $\mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 3 \mathbb{Z}$ (d) $\mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z} \times \mathbb{Z} / 2 \mathbb{Z}$