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Find the rectangular points for (3,120 )
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Solution: x=3cos120 =-15 and y=3sin120 =26
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As a check, you can veri@ that (-l5)2
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+ 2ti2 = 32
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Three-dimensional coordinate systems are usually righthanded In Fig 1-4 imagine your right hand positioned with fingers extended in the +x clrrection closing naturally so that your fingers rotate into the direction of the +y axis while your thumb points in the direction of the +Z axis It is this rotation of x into y to produce z with the right hand that specifies a right-handed coordinate system Points in the three-dimensional system are specified with three numbers (x,y,2)
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Fig 1-4 For certain types of problems, locating a point in space is more convenient with a cylindrical coordinate system, as shown in Fig 1-5 Notice that this is also a right-handed ih coordinate system wt the central axis of the cylinder as the z-axis
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Fig 1-5
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A point is located by specifjring a radius measured out fiom the origin in the +x Qrection, an angle in the x-y plane measured from the x-axis, and a height above the x-y plane Thus the coordinates in the cylinhcal system are (r, 0,z) The relation of these coorlnates to x,y,z is given in Fig 1-5
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CHAPTER1
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Logarithms and Exponents
Logarithms and exponents are used to describe several physical phenomena The exponential h c t i o n y = a" is a unique one with the general shape shown in Fig 1-6
Fig 1-6
This exponential equation y = a X cannot be solved for x using normal algebraic techniques The solution to y = a" is one of the definitions of the logarithmic function:
y=aX
x=log,y
The language of exponents and logarithms is much the same In exponential functions we say "a is the base raised to the power x" In logarithm functions we say "x is the logarithm to the base a ofy" The laws for the manipulation of exponents and logarithms are similar The manipulative rules for exponents and logarithms are summarized in the box below
The term "log" is usually used to mean logarithms to the base 10, while "ln" is used to mean logarithms to the base e The terms "natural" (for base e) and "common" (for base 10) are fiequently used
LAWS OF EXPONENTS AND LOGARITHMS (a")Y = a y ylog, x = log, y
1-9 convert the exponential statement 100= 102 to a logarithmic statement
Solution: y = a x is the same statement as x = log, y so 100= 102 is 2 = logl, 100
MATHEMA'TTCALBACKGROUND
1 10 convert the exponential statement e 2 = 74 to a (natural) logarithmic statement
Solution:
2 =74
so l n 7 4 = 2
1 11 Convert log 2 = 0301
Solution: ~ o O ~ O = 2 *
to an exponential statement
1 12
Find log(2 1)(43)'6
Solution: On your hand calculator raise 43 to the 16 power and multiply this result by 21 Now take the log to obtain 134 Second Solution: Applying the laws for the manipulation of logarithms write:
log(2 l)(43)'6 = log 21 + log 4316 log 2 I + 16log 43 = 032 + 1 01= 133 = (Note the round-off error in this second solution) This second solution is rarely used for numbers It is, however, used in solving equations
1-13
Solve 4=1n2x
Solution: Apply a manipulative rule for logarithms: 4 = In 2 + In x or 331 = In x
Now switch to exponentials: x = e331 274 = A very convenient phrase to remember in working with logarithms is "a logarithm is an exponent" If the logarithm of something is a number or an expression, then that number or expression is the exponent of the base of the logarithm
Remember: A logarithm is an exponent!
Functions and Graphs
Functions can be viewed as a series of mathematical orders The typical h c t i o n is written starting with y, or f(x), read as "fof x," short for function of x The mathematical function y orf (x) = x 2 + 2x + 1 is a series of orders or operations to be performed on an as yet to be specified value of x This set of orders is: square x, add 2 times x, and add 1 The operations specified in the function can be performed on individual values of x or graphed to show a continuous "function" It is the graphing that is most encountered in calculus We'll look at a variety of algebraic functions eventually leading into the concept of the limit
1- 14 Perform the functions f ( x ) = x3 - 3 x + 7 on the number 2, or, find f ( 2 )
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