Có 60+ tài liệu thuộc chủ đề "toán học ứng dụng"
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bc 2 = bcc = (bc) 2 e.g., for x = 0. Answers are provided at the end of the chapter.. The function f is called an exponential function if it can be written in the form f (t. This function is called exponential because the variable, t , is in the exponent. The domain of the exponential function is...
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Estimate the zeros of the function (1.04) t − 2. Suppose we put M 0 dollars in a bank at interest rate r per year compounded annually. (If the interest rate is 5%, then r = 0.05.) Assume the money is put in the bank at the beginning of the year and interest is compounded at the end of the...
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t = 0 corresponds to this year’s salary, t = 1 to the salary one year from now, and so on. Which person’s situation would you prefer to be your own?. Let us suppose that of the hundred thousand people living in this town, which is, of course, uncultured and behind the times, there are only three of your sort....
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10.1 ANALYSIS OF EXTREMA. What route must be taken to travel the distance between two cities in the shortest amount of time? When should a farmer harvest his crop in order to maximize his profit? What dimensions will minimize the amount of material required to construct a can of a given volume? Example 6.1 was such an optimization problem. If...
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x on [–1, 2]. Figure 10.12. Suppose x = c is an interior critical point of a continuous function f . How can we tell if, at x = c, f has a local maximum, local minimum, or neither?. One approach is to look at the sign of f to determine whether f changes from increasing to decreasing across x...
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10.3 PRINCIPLES IN ACTION. In this section we put the principles discussed in Sections 10.1 and 10.2 into action.. EXAMPLE 10.6 Below is a graph of f , the derivative of f . Figure 10.21. (a) Identify all critical points of f . Which of these critical points are also stationary points of f. (b) On a number line plot...
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(c) Is the slope of the revenue curve constant or does it vary with q? Interpret the slope of the revenue curve in terms of the economic model. Economists call the slope of the revenue curve the marginal revenue.. (d) Is the slope of the cost curve constant or does it vary with q? Interpret the slope of the cost...
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EXERCISE 11.4 Construct a polynomial equation with the specification given. (a) a third degree polynomial equation with roots at x. 2, x = 3, and x = 0 (b) a second degree polynomial equation with a double root at x. 1 (c) a second degree polynomial equation with no roots. (d) a fifth degree polynomial equation with roots only at...
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11.3 POLYNOMIAL FUNCTIONS AND THEIR GRAPHS. EXAMPLE 11.6 Graph f (x. This is the function from Example 11.2. The graph of f (x) does not cross the x -axis anywhere between these zeros. By determining the sign of f (x) for just one test value of x on the interval. 2, 0), we can determine the sign of f (x)...
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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...