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y = x + x++;

adds the original value of x to x and assigns this result to y The value of x is incremented after its value has been obtained However, the statement

P2(q)

You Try It: What is Sec 1 ( 2/ 3) What is Csc 1 ( 2)

y = x + ++x;

obtains the value of x, increments x, and then adds that value to the original value of x The result is assigned to y As the output shows, simply changing ++x to x++ changes the number series from even to odd

P1(q)

4:

x _ F/2

intimidate you Although having two operators back-to-back is a bit unsettling at first glance, the compiler keeps it all straight Just remember, this expression simply adds the value of x to the value of x incremented

P1(q)

In the terms relational operator and logical operator, relational refers to the relationships that values can have with one another, and logical refers to the ways in which true and false values can be connected together Since the relational operators produce true or false results, they often work with the logical operators For this reason they will be discussed together here The relational operators are as follows:

Operator == != > < >= <= Meaning Equal to Not equal to Greater than Less than Greater than or equal to Less than or equal to

P1(q)

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Summary of Key Facts About the Inverse Trigonometric Functions x ; Cos x = cos x, 0 x ; 2 2 Tan x = tan x, < x < ; Cot x = cot x, 0 < x < ; 2 2 Sec x = sec x, x [0, /2) ( /2, ]; Csc x = csc x, x [ /2, 0) (0, /2] Sin x = sin x, d 1 d 1 Sin 1 x = Cos 1 x = , 1 < x < 1; , 1 < x < 1; 2 dx dx 1 x 1 x2 d d 1 1 Cot 1 x = , < x < ; , < x < ; Tan 1 x = 2 dx dx 1+x 1 + x2 d d 1 1 , |x| > 1; , |x| > 1; Sec 1 x = Csc 1 x = 2 1 dx dx |x| x |x| x 2 1 du 1 u2 = Sin 1 u + C; du = Cos 1 u + C; 1 u2 du du = Cot 1 u + C; 1 + u2 du = Csc 1 u + C |u| u2 1

The logical operators are shown next:

The outcome of the relational and logical operators is a bool value In general, objects can be compared for equality or inequality using == and != However, the comparison operators, <, >, <=, or >=, can be applied only to those types that support an ordering relationship Therefore, all of the relational operators can be applied to all numeric types However, values of type bool can only be compared for equality or inequality since the true and false values are not ordered For example, true > false has no meaning in C#

Now, if we recall the components of vector p(q) and its derivatives with respect to q, as given in Eq (72), the curvature can be expressed as k = sgn(l ) x y - y x ( x 2 + y 2 )3 2

Part I:

Hypatia is viewing a ten-foot-long tapestry that is hung lengthwise on a wall The bottom end of the tapestry is two feet above her eye level At what distance should she stand from the tapestry in order to obtain the most favorable view

For the logical operators, the operands must be of type bool, and the result of a logical operation is of type bool The logical operators, &, |, ^, and !, support the basic logical operations AND, OR, XOR, and NOT, according to the following truth table:

where the argument q has been dropped for brevity If the cam pro le is given instead in polar coordinates r = r(q), then q can be considered as a parameter, and the position vector of any point of the curve can be expressed as p(q ) = r (q )e r (720)

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