Which function has the greater average rate of change over the interval $-5 \leq x \leq 0$ :the translation of the function $f(x)=\sqrt[3]{x}$ to the right 1 unit and up 2 units, or the function $r(x)=\sqrt[3]{x}+3$ ?

A hollow, conducting sphere with an outer radius of $0.250 \textrm{ m}$ and an inner radius of $0.200 \textrm{ m}$ has a uniform surface charge density of $+ 6.37 \times 10 ^ { - 6 } \mathrm { C } / \mathrm { m } ^ { 2 }$. A charge of $- 0.500 \mu C$ is now introduced at the center of the cavity inside the sphere.

(a) What is the new charge density on the outside of the sphere?

(b) Calculate the strength of the electric field just outside the sphere.

(c) What is the electric flux through a spherical surface just inside the inner surface of the sphere?

Translate into Latin: I am captured by the enemies of the brave Trojans.

A function f is given. Find the average rate of change of f between x = 0 and x = 2, and the average rate of change of f between x = 15 and x = 50. f(x) = 8 - 3x

Consider a conflict between two armies of x and y soldiers, respectively. During World War I, F. W. Lanchester assumed that if both armies are fighting a conventional battle within sight of one another, the rate at which soldiers in one army are put out of action (killed or wounded) is proportional to the amount of fire the other army can concentrate on them, which is in turn proportional to the number of soldiers in the opposing army. Thus Lanchester assumed that if there are no reinforcements and t represents time since the start of the battle, then x and y obey the differential equations

$$\begin{array}{l}{\frac{d x}{d t}=-a y} \\ {\frac{d y}{d t}=-b x}\end{array}$$

a, b > 0. (a) For two armies of strengths x and y fighting a conventional battle governed by Lanchester's differential equations, write a differential equation involving dy/dx and the constants of attrition a and b. (b) Solve the differential equation and hence show that the equation of the phase trajectory is $a y^{2}-b x^{2}=C$ for some constant C. This equation is called Lanchester's square law. The value of C depends on the initial sizes of the two armies.

The Lennon family used x dollars worth of electricity two years ago and then switched to balanced billing. a. Express the amount of their monthly balanced billing payment algebraically. b. Last year, they underpaid their bill and were charged c dollars at the end of the year to make up for the underpaid charges. Express the amount of this year's balanced billing monthly payment algebraically, if it was based on last year's usage.

Find the exact value of each expression. Do not use a calculator $\csc 45^{\circ} \tan 60^{\circ}$

At 3 P.M., a dry-bulb thermometer reading is $66^\circ F$. The wet-bulb reading is $66^\circ F$. What is the relative humidity? Explain.

How do the average rates of change for each pair of functions compare over the given interval? f(x) = -2x², g(x) = -4x², -4 ≤ x ≤ -2

A coast guard boat is 14.8 km east of a lighthouse. A disabled yacht is 7.5 km south of the lighthouse. a) How far is the coast guard boat from the yacht, to the nearest tenth of a kilometre? b) At what angle south of due west, to the nearest tenth of a degree, should the coast guard boat travel to reach the yacht?

Some diseases are thought to be caused by more than one gene.

Name each type of special quadrilateral that can meet the given condition. Make sketches to support your answers. two pairs of parallel sides

Calculate the average rate of change for each function over the given interval. g(x) = ∛x + 2 for -4 ≤ x ≤ 0

How do the key features of a polynomial function help you sketch its graph?