Có 60+ tài liệu thuộc chủ đề "khóa học tính toán"
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Find a polynomial to fit the graph below.. Each of the graphs on the following page is the graph of a polynomial P (x). (b) Despite the fact that you have just categorized each of the polynomials as being of either odd or even degree, none of the polynomials graphed are even functions and none are odd functions. (d) Determine...
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The graph will look like one of the following.. Figure 11.23. Where is the numerator of the simplified expression zero? (For a fraction to be zero, its numerator must be zero.). Figure out the sign of f (x). The sign of any function can change only across a zero or a point of discontinuity. Draw a “sign of f (x)”...
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12.1 WHAT DOES IT MEAN FOR F AND G TO BE INVERSE FUNCTIONS?. If f is a function whose action can be undone, we refer to the function that undoes the action of f as its inverse function and denote it by f − 1 . “f inverse” is not. EXAMPLE 12.1 Below are some simple functions together with their...
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This makes sense: The function f cubes its input, multiplies the result by 4, and then adds 2. 4x 3 +2 add 2 EXAMPLE 12.6 Let g(x. At first glance, it may seem that if g is the squaring function, its inverse must be the square root function. But we must be careful. g is not invertible because it is...
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Then t = log 10 6561. Let’s make sure the definition of log 10 x is clear by looking at some examples.. (a) log 10 100 is the exponent to which we must raise 10 in order to get 100.. (b) log 10 0.1 is the exponent to which we must raise 10 in order to get 0.l.. so log...
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log b x = log k x log k b. log 3 = 2 5 log 3. EXAMPLE 13.5 Solve for x if log 7 (4x. SOLUTION In order to solve for x we need to free the x from the logarithm. The first is to undo log 7 by exponentiating both sides of the equation with base 7.. Examples...
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ln x = 1 2 ln x. Find the equation of the straight line through the points (2, ln 2) and (3, ln 3).. Find the equation of the line through the points (2, ln 2) and (2. Take note of the values of r for which the latter is a good approximation of the former.. 13.4 GRAPHS OF LOGARITHMIC...
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b y = x ln b y = ln x y ln b = ln x y = ln x ln b log b x = ln x. ln b = 1 ln b ln x. We know that the derivative of a constant times f (x) is that constant times f (x), so. 1 ln b ln x =...
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Logarithms are useful when we’re trying to solve for a variable in the exponent.. If A = B and A and B are positive, then ln A = ln B.. ln 2, but ln 1 + ln ln 2.. y = log b M is equivalent to b y = M y ln b = ln M. y = ln...
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Recall that we have deposited $10,000 into a bank with a nominal annual interest rate of 100% and left it for one year. (When looked at in this context, it seems unreasonable that the limit would be 1.) Let’s experiment by returning to the numerical approach suggested by Example 15.1 and evaluating (1 + n 1 ) n for large...
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The families of functions we found as solutions to the differential equations in Example 15.6 are not simply more general solutions but in fact are the general solutions to each of the differential equations. By this we mean that any particular solution to the differential equation can be expressed in this form.. t Differential Equation Some Particular Solutions The General...
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Exploratory Problems for Chapter 15 511 (a) Let N = N (t ) be the number of moles of substance A at time t . (Note: The number of moles of substance B should be expressed in terms of the number of moles of substance A.). The rate at which N is changing is a function of N , the...
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(a) Which of the following are equal to e − x 2 ? Identify all correct answers.. (a) What is the domain of f ? The range?. (f ) Find the x -coordinates of the points of inflection.. 16.2 THE DERIVATIVE OF x n WHERE n IS ANY REAL NUMBER. We can use the Chain Rule to prove this fact.....
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Below is a graph of f (x) on the interval. on the domain. (a) Find f (x) in terms of g and its derivatives.. (a) On three separate sets of axes, draw the graphs of the function a(x. f (2x), labeling the x -intercepts, the y-intercept, and the x- and y-coordinates of the local extrema.. Your answers may be in...
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Use logarithmic differ- entiation to find dy dx . If you felt so inclined, you could come up with a “rule” for taking the derivative of functions of the form f (x) g(x) where f (x) is positive. You might call it “the Tower Rule” since you have a tower of functions, or you might think of a more descriptive...
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EXAMPLE 17.9 Suppose that x and y are functions of t (they vary with time) and x 2 + y 2 = 25. the coordinates of the bug at time t are given by (x, y) or (x(t. Suppose that dy dt. What is dx dt at this moment?. Let’s think about the question in terms of the bug. When...
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Calculating how many milligrams are in the body one week later requires simple addition.. The second question, how many milligrams will be in the body after several years of taking a 0.04 mg pill every morning, is certainly an important one, but at first glance it looks like it will be a lot of work. Several years is not a...
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For Problems 1 through 11, determine whether the series converges or diverges. Find the sum of the following. Determine whether each of the following geometric series converges or diverges. If the series converges, determine to what it converges.. Write each of the following series first as a repeating decimal and then as a fraction.. 572 CHAPTER 18 Geometric Sums, Geometric...
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The sum of money in the account is. The original million would stay in the bank generating interest. Let’s compute the “up-front value” of the prize money of 20 annual payments of $50,000.. This sum depends on the interest rate in the bank account. 15 Find P , the total amount of money in the account earmarked for you.. How...
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Note that the sum of the present values of his payments (pulled back to the present using an interest rate of 12%) should equal his loan.. What is the level of the drug in the patient’s body two weeks into treatment, immediately after taking the 28th pill?. Suppose that the half-life of the medica- tion in the patient’s bloodstream is...