| 单词 | exponential function |
| 例句 | The sharp increase at the right smashes down the rest of the curve, so its true nature is a bit obscured, but from the lowest percentile up, this is roughly an exponential function. Seven secrets of dating from the experts at OkCupid 2014-09-28T04:00:00Z It is a characteristic of the exponential function that the final value is never reached, but 0.632 of the remainder to that value is achieved in every time, τ . College Physics for AP Courses 2015-08-12T00:00:00Z Graph the exponential function by making a table. College Algebra with Corequisite Support 2020-09-23T00:00:00Z Both geometric sequences and exponential functions have a constant ratio. College Algebra 2021-12-21T00:00:00Z This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. Intermediate Algebra 2020-05-06T00:00:00Z For the following exercises, use a graphing calculator to find the equation of an exponential function given the points on the curve. Precalculus 2021-12-01T00:00:00Z When we take derivatives of polynomials, exponential functions, sines, and cosines, we get polynomials, exponential functions, sines, and cosines. Calculus, Volume 3 2016-03-30T00:00:00Z Graph the exponential function by making a table. College Algebra with Corequisite Support 2020-09-23T00:00:00Z The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small. College Algebra 2021-12-21T00:00:00Z Explain how you can distinguish between exponential functions and polynomial functions. Intermediate Algebra 2020-05-06T00:00:00Z Figure 1.37 The graph shows an exponential function times the square root of an exponential function. Calculus, Vol 2 2016-03-30T00:00:00Z We encountered exponential functions with complex exponents earlier. Calculus, Volume 3 2016-03-30T00:00:00Z The exponential function whose base is e, is called the natural exponential function. College Algebra with Corequisite Support 2020-09-23T00:00:00Z To see the difference between an exponential function and a power function, we compare the functions y = x2 and y = 2 x. Calculus, Volume 1 2016-03-30T00:00:00Z As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. Intermediate Algebra 2020-05-06T00:00:00Z Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The solution methods we examine are different from those discussed earlier, and the solutions tend to involve trigonometric functions as well as exponential functions. Calculus, Volume 3 2016-03-30T00:00:00Z Recall that an exponential function is any equation written in the form such that and are positive numbers and Any positive number can be written as for some value of . College Algebra with Corequisite Support 2020-09-23T00:00:00Z Note that exponential functions satisfy the general laws of exponents. Calculus, Volume 1 2016-03-30T00:00:00Z This means an exponential function does have an inverse. Intermediate Algebra 2020-05-06T00:00:00Z Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. Algebra and Trigonometry 2015-02-13T00:00:00Z With two exponential functions, unless the exponents are equal, the functions are linearly independent. Calculus, Volume 3 2016-03-30T00:00:00Z If you have a savings account, you have experienced the use of an exponential function. College Algebra with Corequisite Support 2020-09-23T00:00:00Z As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances. Calculus, Volume 1 2016-03-30T00:00:00Z In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. Intermediate Algebra 2020-05-06T00:00:00Z In this section, we will learn techniques for solving exponential functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Like logarithms, or the zero, or exponential functions. The Hidden Depth in Math's Simple Puzzles 2018-01-17T05:00:00Z “Other SAT topics that are either new or more difficult include histograms, scatter-plot interpretation, unit analysis, inference from a sample, two-way tables, exponential functions or compound interest,” he said. Is SAT angst easing off? Here are some tips for dispelling pre-test terror. 2015-09-06T04:00:00Z However, exponential functions and logarithm functions can be expressed in terms of any desired base b. Calculus, Volume 1 2016-03-30T00:00:00Z Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where 0 < a < 1. Intermediate Algebra 2020-05-06T00:00:00Z Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown. Algebra and Trigonometry 2015-02-13T00:00:00Z This involves looking at the time when they held the Olympics and afterwards and trying to fit an exponential function to it and then we can compare the different rates. The numbers of the year 2012 2012-12-29T01:09:08Z One of my favorites involved the population of a town growing according to some exponential function of time. SchoolBook: Real-World Problems + Group Learning = A.P. Calculus Success 2012-07-30T18:52:28Z Since exponential functions are one-to-one, we can conclude that u · v = w. Calculus, Volume 1 2016-03-30T00:00:00Z This means the exponential function will have an inverse. Intermediate Algebra 2020-05-06T00:00:00Z TRY IT #1 Which of the following equations represent exponential functions? Algebra and Trigonometry 2015-02-13T00:00:00Z It was once said - a tad hyperbolically - that the greatest shortcoming of the human race was the inability to understand the exponential function. Big number 2011-07-20T21:43:41Z Power functions, exponential functions — what was the point of all that? Power Tools 2010-03-28T23:57:00Z Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions. Calculus, Volume 1 2016-03-30T00:00:00Z All of our exponential functions have had either an integer or a rational number as the base. Intermediate Algebra 2020-05-06T00:00:00Z To evaluate an exponential function with a form other than the basic form, it is important to follow the order of operations. Algebra and Trigonometry 2015-02-13T00:00:00Z We have found that man is an exponential function where time enters as an exponent. Manhood of Humanity. But when you need mathematical dynamite, it’s time to unpack the exponential functions. Power Tools 2010-03-28T23:57:00Z For the following exercises, sketch the graph of the exponential function. Calculus, Volume 1 2016-03-30T00:00:00Z We will now look at an exponential function with an irrational number as the base. Intermediate Algebra 2020-05-06T00:00:00Z Q&A Do two points always determine a unique exponential function? Algebra and Trigonometry 2015-02-13T00:00:00Z The peculiar faculty belonging exclusively to humans which I designate as “time-binding” I have clearly defined as an exponential function of time in the following chapter. Manhood of Humanity. Just as every office worker needs both a stapler and a staple remover, every mathematician needs exponential functions and logarithms. Power Tools 2010-03-28T23:57:00Z The natural exponential function is y = ex and the natural logarithmic function is y = ln x = loge x. Calculus, Volume 1 2016-03-30T00:00:00Z Before we can look at this exponential function, we need to define the irrational number, e. Intermediate Algebra 2020-05-06T00:00:00Z But keep in mind that we also need to know that the graph is, in fact, an exponential function. Algebra and Trigonometry 2015-02-13T00:00:00Z For the purpose of the moment it is important to realize that a state is the governing center of an accumulation of human beings—of time-binding powers—increasing exponential functions of time. Manhood of Humanity. It is straightforward to show that properties of exponents hold for general exponential functions defined in this way. Calculus, Volume 1 2016-03-30T00:00:00Z Given an exponential function or logarithmic function in base a, we can make a change of base to convert this function to any base b > 0, b ≠ 1. Calculus, Volume 1 2016-03-30T00:00:00Z If you own a bank account, you have experienced the use of an exponential function. Intermediate Algebra 2020-05-06T00:00:00Z TRY IT #6 Find an equation for the exponential function graphed in Figure 6. Algebra and Trigonometry 2015-02-13T00:00:00Z The supreme factor is the use of the mind—the exponential function of time—the time-binding energy of man. Manhood of Humanity. One of the most prevalent applications of exponential functions involves growth and decay models. Calculus, Volume 1 2016-03-30T00:00:00Z The hyperbolic functions involve combinations of the exponential functions ex and e−x. Calculus, Volume 1 2016-03-30T00:00:00Z In the following exercises, graph each exponential function. Intermediate Algebra 2020-05-06T00:00:00Z For most real-world phenomena, however, e is used as the base for exponential functions. Algebra and Trigonometry 2015-02-13T00:00:00Z The time-binding energies as well as “life” follow the same type of exponential function. Manhood of Humanity. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. Calculus, Volume 1 2016-03-30T00:00:00Z As we discussed in Introduction to Functions and Graphs, exponential functions play an important role in modeling population growth and the decay of radioactive materials. Calculus, Volume 1 2016-03-30T00:00:00Z What happens to an exponential function as the values of x decreases? Intermediate Algebra 2020-05-06T00:00:00Z Explain why the values of an increasing exponential function will eventually overtake the values of an increasing linear function. Algebra and Trigonometry 2015-02-13T00:00:00Z Animal standards belong to a class of life whose capacity is not an exponential function of Time. Manhood of Humanity. The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. Calculus, Volume 1 2016-03-30T00:00:00Z Write the exponential function that relates the total population as a function of t. Calculus, Volume 1 2016-03-30T00:00:00Z Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function. Intermediate Algebra 2020-05-06T00:00:00Z Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Algebra and Trigonometry 2015-02-13T00:00:00Z The quantity, PRT of progress made in the Tth generation contains T as an exponent, and so the quantity, varying as time T passes, is called an exponential function of the time. Manhood of Humanity. General exponential functions are defined in terms of ex, and the corresponding inverse functions are general logarithms. Calculus, Volume 1 2016-03-30T00:00:00Z Figure 4.73 An exponential function grows at a faster rate than a power function. Calculus, Volume 1 2016-03-30T00:00:00Z We have seen that growth and decay are modeled by exponential functions. Intermediate Algebra 2020-05-06T00:00:00Z For the following exercises, identify whether the statement represents an exponential function. Algebra and Trigonometry 2015-02-13T00:00:00Z Nature made man an increasing exponential function of time, a time-binder, a power able to transform and direct basic powers. Manhood of Humanity. Exponential growth and exponential decay are two of the most common applications of exponential functions. Calculus, Volume 1 2016-03-30T00:00:00Z The exponential function ex grows faster than any power function xp, p > 0. Calculus, Volume 1 2016-03-30T00:00:00Z Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions. Intermediate Algebra 2020-05-06T00:00:00Z For the following exercises, find the formula for an exponential function that passes through the two points given. Algebra and Trigonometry 2015-02-13T00:00:00Z Hyperbolic functions are defined in terms of exponential functions. Calculus, Volume 1 2016-03-30T00:00:00Z All translations of the exponential function can be summarized by the general equation See Table 3. Algebra and Trigonometry 2015-02-13T00:00:00Z The exponential function, y = ex, is its own derivative and its own integral. Calculus, Volume 1 2016-03-30T00:00:00Z Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. Intermediate Algebra 2020-05-06T00:00:00Z Use this fact to rewrite the formula for an exponential function that uses the number as a base. Algebra and Trigonometry 2015-02-13T00:00:00Z Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown. Algebra and Trigonometry 2015-02-13T00:00:00Z We will also investigate logarithmic functions, which are closely related to exponential functions. Algebra and Trigonometry 2015-02-13T00:00:00Z Figure 5.37 The graph shows an exponential function times the square root of an exponential function. Calculus, Volume 1 2016-03-30T00:00:00Z By graphing a few exponential functions, we will be able to see their unique properties. Intermediate Algebra 2020-05-06T00:00:00Z As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Algebra and Trigonometry 2015-02-13T00:00:00Z In this section, we will take a look at exponential functions, which model this kind of rapid growth. Algebra and Trigonometry 2015-02-13T00:00:00Z Three kinds of functions that are often useful in mathematical models are linear functions, exponential functions, and logarithmic functions. Algebra and Trigonometry 2015-02-13T00:00:00Z As mentioned at the beginning of this section, exponential functions are used in many real-life applications. Calculus, Volume 1 2016-03-30T00:00:00Z If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions. Intermediate Algebra 2020-05-06T00:00:00Z In fact, for any exponential function with the form is the constant ratio of the function. Algebra and Trigonometry 2015-02-13T00:00:00Z Describe how exponential functions and geometric sequences are similar. Algebra and Trigonometry 2015-02-13T00:00:00Z This is called an exponential function, not a power function. Algebra and Trigonometry 2015-02-13T00:00:00Z Let’s look at an example in which integration of an exponential function solves a common business application. Calculus, Volume 1 2016-03-30T00:00:00Z This is the shape we expect from an exponential function where a > 1. Intermediate Algebra 2020-05-06T00:00:00Z What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph? Algebra and Trigonometry 2015-02-13T00:00:00Z In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. Algebra and Trigonometry 2015-02-13T00:00:00Z The general form of the exponential function is where is any nonzero number, is a positive real number not equal to 1. Algebra and Trigonometry 2015-02-13T00:00:00Z Substitution is often used to evaluate integrals involving exponential functions or logarithms. Calculus, Volume 1 2016-03-30T00:00:00Z We know exponential functions and logarithmic function are very interrelated. Intermediate Algebra 2020-05-06T00:00:00Z The exponential function is one-to-one, so its inverse, is also a function. Algebra and Trigonometry 2015-02-13T00:00:00Z In more general terms, we have an exponential function, in which a constant base is raised to a variable exponent. Algebra and Trigonometry 2015-02-13T00:00:00Z Sometimes we are given information about an exponential function without knowing the function explicitly. Algebra and Trigonometry 2015-02-13T00:00:00Z We already examined exponential functions and logarithms in earlier chapters. Calculus, Volume 1 2016-03-30T00:00:00Z TRY IT #5 Given the two points and find the equation of the exponential function that passes through these two points. Algebra and Trigonometry 2015-02-13T00:00:00Z Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. Algebra and Trigonometry 2015-02-13T00:00:00Z No. Because the base of an exponential function is always positive, no power of that base can ever be negative. Algebra and Trigonometry 2015-02-13T00:00:00Z Use the regression feature to find an exponential function that best fits the data in the table. Algebra and Trigonometry 2015-02-13T00:00:00Z For example, we did not study how to treat exponential functions with exponents that are irrational. Calculus, Volume 1 2016-03-30T00:00:00Z Any exponential function with the form can be rewritten as an equivalent exponential function with the form where See Example 8. Algebra and Trigonometry 2015-02-13T00:00:00Z The inverse of every logarithmic function is an exponential function and vice-versa. Algebra and Trigonometry 2015-02-13T00:00:00Z For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied. Algebra and Trigonometry 2015-02-13T00:00:00Z Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. Algebra and Trigonometry 2015-02-13T00:00:00Z We then examine logarithms with bases other than e as inverse functions of exponential functions. Calculus, Volume 1 2016-03-30T00:00:00Z No. Recall that the range of an exponential function is always positive. Algebra and Trigonometry 2015-02-13T00:00:00Z We have seen that any exponential function can be written as a logarithmic function and vice versa. Algebra and Trigonometry 2015-02-13T00:00:00Z No. Factorials get large very quickly—faster than even exponential functions! Algebra and Trigonometry 2015-02-13T00:00:00Z Use the regression feature to find an exponential function that best fits the data in the table. Algebra and Trigonometry 2015-02-13T00:00:00Z This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of r. Calculus, Volume 1 2016-03-30T00:00:00Z Let’s take a look at the sequence This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. Algebra and Trigonometry 2015-02-13T00:00:00Z Knowing the behavior of exponential functions in general allows us to recognize when to use exponential regression, so let’s review exponential growth and decay. Algebra and Trigonometry 2015-02-13T00:00:00Z |
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