Profit The profit from the production and sale of a product is $P(x)=47 x-0.01 x^2-8000$, where $x$ represents the number of units produced and sold. Give

(a) the profit from the production and sale of $2000$ units

(b) the value of $P(5000)$

( c ) the meaning of $P(5000)$.

Let p(x) and q(x) be continuous functions, and let V be the subspace of $C(-\infty,+\infty)$ consisting of all twice differentiable functions. Define $L : V \rightarrow V$ by $L(y(x))=y^{\prime \prime}(x)+p(x) y^{\prime}(x)+q(x) y(x)$. (a) Show that L is a linear transformation. (b) Consider the special case where p(x)=0 and q(x)=1. Show that the function $\phi(x)=c_{1} \sin x+c_{2} \cos x$ is in the kernel of L for all real values of $c_{1}$ and $c_{2}$.

The numbers 0 and 1 have special properties that are shared by no other numbers. For each of the following, decide whether the property is true for 0, 1, both, or neither. Explain your reasoning. a. If you add __________ to a number n, you get n. b. If the product of two numbers is ________, then one or both numbers are 0. c. The square of __________ is equal to itself. d. If you multiply a number n by _______, you get n. e. If you multiply a number n by ___________, you get 0. f. The opposite of ____________ is equal to itself.

Use the said table to find the standard score and percentile of the following data values.

a. A data value $1$ standard deviation above the mean

b. A data value $0.5$ standard deviation above the mean

c. A data value $1.5$ standard deviations below the mean

Let R be the quasi-triangular region in the first quadrant bounded only by the graphs of $f(x)=\sin x, g(x)=\cos x,$ and the x-axis over the interval $\left[0, \frac{x}{2}\right]$ Use algebraic methods to find the coordinates of the point of intersection of the graphs of f and g in the interval $\left[0, \frac{\pi}{2}\right]$.

A company can produce 100 home computers per day. The setup cost for a production run is $1,000. The cost of holding a computer in inventory for one year is$300. Customers demand 2,000 home computers per month (assume that 1 month = 30 days and 360 days = 1 year). What is the optimal production run size? How many production runs must be made each year?

(a) Does the moment of inertia of a rigid body depend in any way on the center of mass of the body? Explain. (b) Can a moment of inertia have a negative value? If so, explain what this would mean.

The displacement of a harmonic oscillator is given by $y=9.4 \sin (15 t)$, where the units of $y$ are meters and $t$ is measured in seconds. What is its maximum velocity?

Starting with a segment $\overline{A B}$ and a ray $\overrightarrow{A P}$, explain in diagrams and words how to create a compass and straightedge construction of a rhombus $A B C D$, where $\overline{A B}$ is a side of the rhombus and point $D$ is on the ray $\overrightarrow{A P}$.

Consider the equation y = 10 + 5x in intercept form.

a. Factor the right side of the equation. b. Use the commutative property of addition to swap the terms inside the parentheses. c. Your result should look similar to the point-slope form of the equation. What's missing? What is the value of this missing piece? d. What point could you use to write the point-slope equation in 7c What is special about this point?

An ideal gas at initial volume $V_1$ and pressure $P_1$ expands quasi-statically and adiabatically to volume $V_2$ and pressure $P_2$. Calculate the work done by the gas directly by integrating $P d V$, and show that your result is the same as that given by Equation 18-39.

Use the data shown in the box plot.

Describe the variability of the data.

Use the data given in earlier problems to answer the following:
(a) What percentage of the data is within $1$ standard deviation of the mean?
(b) What percentage of the data is within $2$ standard deviations of the mean?
(c) What percentage of the data is within $3$ standard deviations of the mean?

Over the next few decades, what is likely to happen to the world's production of oil?