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YOU TRY IT A trust is established in your name which pays t + 10 dollars
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per year for every year in perpetuity, where t is time measured in years (here the present corresponds to time t = 0) Assume a constant interest rate of 4% What is the total value, in today's dollars, of all the money that will ever be earned by your trust account
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1 If possible, use l H opital s Rule to evaluate each of the following limits In each case, check carefully that the hypotheses of l H opital s Rule apply a lim b lim c lim d lim e lim cos x 1 4 2 x 0 x + x e 2x 1 2x x2 + x6 x 0 cos x 3 x 0 x [ln x]2 2 x 1 (x 1) (x 2)4 x 2 sin(x 2) (x 2) ex 1 x2 1
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2 If possible, use l H opital s Rule to evaluate each of the following limits In each case, check carefully that the hypotheses of l H opital s Rule apply x3 x 2 x + e 1 x x /2 ln x 2 x + x e 2x x + ln[x/(x + 1)] sin x 2x x + e ex 2 x 1/x ln |x| 2x x e
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c lim d lim
e lim
f lim
5 I N D E T E R M I N A T E F O R M S
3 If possible, use some algebraic manipulations, plus l H opital s Rule, to evaluate each of the following limits In each case, check carefully that the hypotheses of l H opital s Rule apply a lim x 2 e x
x +
b lim x 2 sin[1/x 2 ]
x +
c lim ln[x/(x + 1)]
x +
1 +1
d lim ln x e x
x +
e lim e 2x x 4
x
f lim x e 1/x
4 Evaluate each of the following improper integrals In each case, be sure to write the integral as an appropriate limit
x 4/5 dx (x 3) 6/5 dx 1 dx (x + 1)1/5 x dx (x 2)(x + 1) x+5 dx (x 2)1/3 sin x dx x
2 6
4 8
5 Evaluate each of the following improper integrals In each case, be sure to write the integral as an appropriate limit
e 2x dx
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x 2 e 2x dx x 2 ln x dx dx 1 + x2 dx x2 1 dx x 2 + x
chapter
Transcendental Functions
Polynomials are the simplest functions that we know, and they are easy to understand It only requires the most rudimentary understanding of multiplication and addition to calculate the values of a polynomials But many of the most important functions that arise in serious scientific work are transcendental functions A transcendental function is one that cannot be expressed as a polynomial, a root of a polynomial, or the quotient of polynomials x Examples of transcendental functions are sin x, tan x, log x, and e There are a great many more In this chapter we study properties of some of the most fundamental transcendental functions
CHAPTER OBJECTIVES
In this chapter, you will learn
Logarithms Logarithms to different bases Exponential functions Exponential functions with different bases Calculus with logarithmic and exponential functions Exponential growth and decay Inverse trigonometric functions
CALCULUS DeMYSTiFieD
60 Introductory Remarks
There are two types of functions: polynomial (and functions manufactured from polynomials) and transcendental A polynomial of degree k is a function of the form p(x) = a0 + a1 x + a2 x 2 + + ak x k Such a polynomial has precisely k roots, and there are algorithms that enable us to solve for those roots For most purposes, polynomials are the most accessible and easy-to-understand functions But there are other functions that are important in mathematics and physics These are the transcendental functions Among these more sophisticated types of functions are sine, cosine, the other trigonometric functions, and also the logarithm and the exponential The present chapter is devoted to the study of transcendental functions
61 Logarithm Basics
A convenient way to think about the logarithm function is as the inverse to the exponential function Proceeding intuitively, let us consider the function f (x) = 3x To operate with this f, we choose an x and take 3 to the power x For example, f (4) = 34 = 3 3 3 3 = 81 f ( 2) = 3 2 = 1 9
f (0) = 30 = 1 The inverse of the function f is the function g which assigns to x the power to which you need to raise 3 to obtain x For instance, g (9) = 2 because f (2) = 9 g (1/27) = 3 because f ( 3) = 1/27 g (1) = 0 because f (0) = 1 We usually call the function g the logarithm to the base 3 and we write g (x) = log3 x Logarithms to other bases are defined similarly
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