Có 40+ tài liệu thuộc chủ đề "toán học ứng dụng"
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(Use any means at your disposal.) (b) Find j (x. In part (a) the input of f is increased by 1. What is the effect on the graph?. In part (b) the output of f is increased by 1. In part (a) the input of f is increased by 2. In part (b) the output of f is increased by...
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Later in this text, when we are looking for the rate of change of a composite function, we will use the same approach of decomposition to decompose the function and find its rate of change from our knowledge of the rates of change of these simpler functions. For each of the functions given below, give possible formulas for f (x)...
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Your calculator probably won’t indicate the pinhole in the graph.. The zeros of the function f (x) are at x. Find the zeros of the following functions. The graph of y = f (x) is symmetric about the y-axis. Which of the following functions is equal to f (x)?. Below is the graph of y = 2 x . Using...
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change of − 1 minute/day. As we can see by looking at the graph in Figure 4.2, over small enough intervals, the data points either lie on a line or lie close to some line that can be fitted to the data. However, the line that fits the data best varies with the interval chosen. When looked at over the...
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A, B , C, D, and E are points on the graph of f . (b) What is the slope of L 1 ? (c) What is the equation of L 1. (e) What is the length of the line segment joining points B and D?. (f ) What is the slope of L 2 ? (g) What is the...
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6 this model correctly counts the $5 commission for the first six items in with the base rate, but then it gives an additional $25 for each of those first six items. This amounts to giving a $30 commission for each of the first six items sold. ($25/item)(7 items) instead of the correct. The slope of C(x) corresponds to the...
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The rock hits the ground in 4 seconds. What is its average velocity in the first second? In each consecutive second? Recall that the average velocity is given by. average velocity in the 1st second = s t = s(1) 1. 16 ft/sec average velocity in the 2nd second = s t = s(2) 2. 48 ft/sec average velocity in...
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Therefore the slope of the tangent line is 10/27. the equation of the tangent line is of the form y − y x − x 1. The tangent line goes through the point. Alternatively, the equation of the tangent line can be written. The limiting process enables us to get a handle on the slope of a curve and the...
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0 the graph of f has a horizontal tangent line.. Figure 5.15. Conclude that the derivative of x 2 + 3x is the sum of the derivatives of x 2 and 3x.. It is designed for your second reading of the chapter.. This means getting rid of the square roots. 192 CHAPTER 5 The Derivative Function. So the derivative of...
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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...