Find the gradient of f at the indicated point. f(x, y)=

$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { - 1 / 2 }$$

; (3, 4)

What is the de Broglie wavelength of an electron traveling at $1.35 \times 10^{5} \mathrm{m} / \mathrm{s} ?$

What does the $\approx$ symbol mean?

(a) What is real GDP per capita unable to measure? (b) How does that limit economists' knowledge of individuals' standard of living? (c) Why do economists use it anyway?

What is the PDF of Z, the standard normal random variable?

Compute the dot product of the vectors and the cosine of the angle between them and determine if the vectors are orthogonal. $8 i-3 j+2 k,-8 i-3 j+k$

Plot the point in a coordinate plane. Describe the location of the point. (-5, -3)

Identifying Functions Think of a compound machine that you use every day, and identify the simple machines that make it up.

Find the dot product $\mathbf{v} \cdot \mathbf{w}$ and the angle between v and w. $\mathbf{v}=-2 \mathbf{i}+\mathbf{j}, \mathbf{w}=4 \mathbf{i}-3 \mathbf{j}$

Find the real solutions, if any, of each equation. Use the quadratic formula. $x^{2}+6 x+1=0$

What do we mean by fundamental risk, and why may such risk allow behavioral biases to persist for long periods of time?

Suppose that P(x) represents the percentage of income spent on health care in year x and I(x) represents income in year x. Determine a function H that represents total health care expenditures in year x.

Why is charge usually transferred by electrons? Which materials transfer electrons most easily? In what situations can positive charge move?

If ∂f/∂y is continuous in the rectangle D,show that there is a positive constant K such that |f(t,y1)−f(t,y2)|≤K|y1−y2|,(i) where (t,y1) and (t,y2)are any two points in D having the same t coordinate.This inequality is known as a Lipschitz20 condition. Hint:Hold t fixed and use the mean value theorem on f as a function of y only. Choose K to be the maximum value of|∂f/∂y| in D.