Use the duck principle to show that each equation is true. a. $\sqrt[4]{125} \cdot \sqrt[4]{5} = 5$ b. $\sqrt[3]{18} \cdot \sqrt[3]{12} = 6$ c. $\dfrac{\sqrt[5]{128}}{\sqrt[5]{16}} = \sqrt[5]{8}$ d. $\sqrt[4]{100} = \sqrt{10}$ e. $\sqrt[3]{4} \cdot \sqrt[6]{3} = \sqrt[6]{48}$

An electron has a mass of $10 ^ { - 28 }$ grams and a proton has a mass of $10 ^ { - 24 }$ grams. How many times greater is the mass of a proton than the mass of an electron?

Was Hubble’s original estimate of the distance to the Andromeda galaxy correct? Explain.

A coil has an inductance of 0.15 H and a resistance of $33 \Omega$. The coil is connected to a 6.0 V ideal battery. When the current reaches half its maximum value: (a) At what rate is magnetic energy being stored in the inductor? (b) At what rate is energy being dissipated? (c) What is the total power that the battery supplies?

Imagine a society that produces military goods and consumer goods, which we’ll call “guns” and “butter.” a. Draw a production possibilities frontier for guns and butter. Using the concept of opportunity cost, explain why it most likely has a bowed-out shape. b. Show a point that is impossible for the economy to achieve. Show a point that is feasible but inefficient. c. Imagine that the society has two political parties, called the Hawks (who want a strong military) and the Doves (who want a smaller military). Show a point on your production possibilities frontier that the Hawks might choose and a point that the Doves might choose. d. Imagine that an aggressive neighboring country reduces the size of its military. As a result, both the Hawks and the Doves reduce their desired production of guns by the same amount. Which party would get the bigger “peace dividend,” measured by the increase in butter production? Explain.

State the amplitude, where appropriate, of: y = sin 4x

(a) Let $f :\mathbb{R}^3 \to \mathbb{R}, \mathbf{x}_0 \in \mathbb{R}^3$. If $\mathbf{v}$ is a unit vector in $\mathbb{R}^3$, show that the maximum value of the directional derivative of $f$ at $\mathbf{x}_0$ along $\mathbf{v}$ is $||\nabla f (\mathbf{x}_0)||$.

Zinc reacts with 100.0 mL of 6.00 M cold, aqueous sulfuric acid through single replacement. a. How many grams of zinc sulfate can be produced? b. How many liters of hydrogen gas could be released at STP?

suppose $y = f (x)$ is defined for all $x$. For each description, sketch a graph with the indicated property.

Discontinuous at $x=2$ but continuous elsewhere with $\lim _{x \rightarrow 0} f(x)=\frac{1}{2}$

Write the rule for a quadratic function with a vertex of (0,0) that passes through (-1,3).

Suggest an explanation for why aspirin has a sour taste.

a. For what values of a and b will

$$g(x)=\left\{\begin{array}{ll}{a x+b,} & {x \leq-1} \\ {a x^{3}+x+2 b,} & {x>-1}\end{array}\right.$$

be differentiable for all values of x? b. Discuss the geometry of the resulting graph of g.

Investigate the services and fees of two banks. Bank 1 has a branch in your neighborhood or town, and Bank 2 is an online bank. Make a multimedia presentation outlining their services and fees and present it to the class.

How did Stalin gain and maintain power in the USSR?