Có 40+ tài liệu thuộc chủ đề "phép tính đơn biến"
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graph of f. (e) As you can see, vertically shifting the graph of f doesn’t change its slope.. EXERCISE 5.3 Given the graph of f , sketch the graph of f . Include a number line indicating both the sign of f and the direction of the graph of f. EXERCISE 5.4 Given the graph of f , sketch two...
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13 Then C (s), also written dC ds , is the function that gives the instantaneous rate of change of calories/mile with respect to speed.. 7 tells us that when the speed of the cyclist is 12 mph, the number of calories used per mile is increasing at a rate of 7 calories/mile per mph. The years 1665–1666 were the...
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The derivative determines a function only up to a constant. Although the derivative gives us information about the shape of the parabola, it doesn’t give us any information about the vertical positioning of the parabola. The derivative will not help us pick out any one member of the family of parabolas drawn.. The Graph of a Quadratic Function. The derivative...
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Beginning with the basic parabola y = x 2 and using the results of Section 3.4 on shifting, stretching, shrinking, and flipping, we can obtain any other parabola in which we are interested.. SOLUTION This is the graph of y = x 2 shifted to the right 4 units, shrunk vertically to half the height at each x -value, and...
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(If there are 10 seats left unsold, the airline will charge each passenger for the flight.) The plane seats 220 travelers and only round-trip tickets are sold on the charter flights.. (a) Let x = the number of unsold seats on the flight. Express the revenue received for this charter flight as a function of the number of unsold seats....
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EXAMPLE 7.4 Consider the function f (x. Find lim x → 3 x(x 2(x − 3). The argument given in Example 7.3 holds without alteration since we always worked with the condition x = 3. A single hole in the graph makes no difference in the limit. EXAMPLE 7.5 Let f (x. Find lim x → 3 f (x).. 3...
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(a) Sketch the graph of f and sketch the graph of f . graph of f (x. Figure 7.19. (c) Approaching the problem from a graphical viewpoint, we notice that the graph of f does not appear locally linear at x = 0. No matter how much the region around x = 0 is magnified the graph has a sharp...
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If f is continuous on the closed and bounded interval [a, b] and f (a. B, then somewhere in the interval f attains every value between A and B.. Figure 7.25. If a function f is continuous on a closed interval [a, b], then f takes on both a maximum (high) and a minimum (low) value on [a, b]. Figure...
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We’ll use the rate at which water is decreasing today to approximate the change in water level over the next two days.. the water is decreasing at a rate of 115 gallons/day so after two days we’d expect it to decrease by about 230 gallons.. This will help us see how a tangent line can be of use. 16.8 is...
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L 1 ± L 2 The limit of a sum/difference is the sum/difference of the limits.. L 1 · L 2 The limit of a product is the product of the limits (in particular, g(x) may be constant:. We can use principles (1) and (2) given above to prove two properties of derivatives, the Constant Multiple Rule and the Sum...
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Find the equation of the tangent line to f (x. For what value(s) of x is the slope of the tangent line to f (x. Exponential Functions. Look around you at quantities in the world that change. Or observe the early stages of the spread of infectious disease in a large susceptible population. the rate at which the disease spreads...
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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)...