Show that

$$y = \sin \left( \tan ^ { - 1 } x + C \right)$$

is the general solution of

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

Then use the addition formula for the sine function to show that the general solution may be written

$$y = \frac { ( \cos C ) x + \sin C } { \sqrt { 1 + x ^ { 2 } } }$$

Use a(t) = - 32 feet per second per second as the acceleration due to gravity. (Neglect air resistance.) A ball is thrown vertically upward from a height of 6 feet with an initial velocity of 60 feet per second. How high will the ball go?

Marley has a vegetable garden in her yard. During a storm, heavy rain falls. The rain runs over the garden and some of the soil is washed away. Which term best describes this movement of soil from one place to another? F. deposition G. discharge H. erosion I. weathering

How does Franklin establish common ground with his audience?

What is the formula for hydrobromic acid?

Prove each statement by mathematical induction.

$$3 ^ { 2 n } - 1$$

is divisible by 8, for each integer

$$n \geq 0$$

.

What are the three main methods of charging objects?

Explain how to find the lengths of the longer leg and hypotenuse of a

$$30 ^ { \circ } - 60 ^ { \circ } - 90 ^ { \circ }$$

triangle if the shorter leg is 10 ft.

Solve the system of linear equations using any method. Explain why you chose the method.

$$\left. \begin{array} { l } { 8 x + 3 y = - 9 } \\ { - 8 x + y = 29 } \end{array} \right.$$

An isosceles triangle has two sides of length

$$9x + 3$$

. The perimeter of the triangle is

$$30x + 10$$

. a) Determine the ratio of the base to the perimeter, in simplified form. State the restriction on x. b) Explain why the restriction on x in part (a) is necessary in this situation.

a. Write the equation of the line that represents the linear approximation to the following functions at the given point a. b. Graph the function and the linear approximation at a. c. Use the linear approximation to estimate the given function value. d. Compute the percent error in your approximation, 100|approximation-exact|/|exact|, where the exact value is given by a calculator. f(x)=ln (1+x); a=0; f(0, 9)

Without solving the equation, determine the number and type of solutions.

$$2 x ^ { 2 } + 5 x + 4 = 0$$

The following while loop implements a way to multiply two numbers that was developed by the ancient Egyptians. [Pre-condition: A and B are positive integers, x=A, y=B, and product=0.]/ while

$$( y \neq 0 )$$

, r: := y mod 2, if r=0, then do

$$x : = 2 \cdot x$$

, y :=y div 2, end do, if r=1, then do product :=product +x, y:=y-1, end do, end while, [Post-condition: product=

$$A \cdot B$$

/ Prove the correctness of this loop with respect to its preand post-conditions by using the loop invariant

$$I ( n ) : \quad * x y +$$

product =

$$A \cdot B$$

."

The area of a square is 225 square centimeters. a. Make a diagram and determine the length of each side. b. Use the Pythagorean theorem to find the length of its diagonal.