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Stewart J. Calculus.. early transcendentals (5ed.)(1300s)_MCet_.pdf |
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miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the y-intercept represent? (e) Why does a linear function give a suitable model in this situation?...
; 20. A study by the U. S. Office of Science and Technology in 1972
estimated the cost (in 1972 dollars) to reduce automobile emissions by certain percentages:
Reduction in emissions (%) 50 55 60 65 70 Cost per car (in $) 45 55 62 70 80 Reduction in emissions (%) 75 80 85 90 95 Cost per car (in $) 90 100 200 375 600...
0. To obtain the graph of f x a distance c units upward f x a distance c units downward f x a distance c units to the right f x a distance c units to the left...
; 27. Use a graphing calculator with exponential regression capaUse a graphing calculator with exponential regression capability to model the U.S. population since 1900. Use the model to estimate the population in 1925 and to predict the population in the years 2010 and 2020....
E X A M P L E 4 Find the inverse function of f x
S O L U T I O N According to (5) we first write...
(a) Linear function (c) Exponential function (e) Polynomial of degree 5 functions. (a) f x x (c) h x x3 (a) y (c) y (e) (g) y sin x ex 1x sx...
Take Cases We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value. Work Backward Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x 5 7, we suppose that x is a number that satisfies 3x 5 7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x 4. Since each of these steps can be reversed, we have solved the problem. Establish Subgoals In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled). If we can first reach these subgoals, then we may be able to build on them to reach our final goal. Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true. Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle....
data: Unknown: hypotenuse h Given quantities: perimeter P, area 25 m 2
|||| Draw a diagram...
It appears that as we shorten the time period, the average velocity is becoming closer to 49 m s. The instantaneous velocity when t 5 is defined to be the limiting value of these average velocities over shorter and shorter time periods that start at t 5. Thus, the (instantaneous) velocity after 5 s is 49 m s You may have the feeling that the calculations used in solving this problem are very similar to those used earlier in this section to find tangents. In fact, there is a close connection between the tangent problem and the problem of finding velocities. If we draw the graph of the distance function of the ball (as in Figure 5) and we consider the points P a, 4.9a 2 and Q a h, 4.9 a h 2 v on the graph, then the slope of the secant line PQ is mPQ 4 .9 a a h2 h 4.9a 2 a...
58 m s, its height in meters after t seconds is given by h 58t 0.83t 2. (a) Find the average velocity over the given time intervals: (i) [1, 2] (ii) [1, 1.5] (iii) [1, 1.1] (iv) [1, 1.01] (v) [1, 1.001] (b) Find the instantaneous velocity after one second.
7. The displacement (in feet) of a certain particle moving in...
More generally, we have the following law, which is proved as a consequence of Law 10 in Section 2.5.
n 11. lim sf x) n s lim f x)...
The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7. It says that if t x is squeezed between f x and h x near a, and if f and h have the same limit L at a, then t is forced to have the same limit L at a....
expresses the length L of an object as a function of its velocity v with respect to an observer, where L 0 is the length of the object at rest and c is the speed of light. Find lim v l c L and interpret the result. Why is a left-hand limit necessary?
53. If p is a polynomial, show that im xl a p x...
Let’s now apply Theorem 8 in the special case where f x tive integer. Then f tx and
n st x n s lim t x...
Thus, f 1 0 f 2 ; that is, N 0 is a number between f 1 and f 2 . Now f is continuous since it is a polynomial, so the Intermediate Value Theorem says there is a number c between 1 and 2 such that f c 0. In other words, the equation 4 x 3 6x 2 3x 2 0 has at least one root c in the interval 1, 2 . In fact, we can locate a root more precisely by using the Intermediate Value Theorem again. Since f 1.2 0.128 0 and f 1.3 0.548 0...
The meaning of such phrases is given by Definition 1. A more precise definition, similar to the , definition of Section 2.4, is given at the end of this section. L Geometric illustrations of Definition 1 are shown in Figure 2. Notice that there are many ways for the graph of f to approach the line y L (which is called a horizontal asymptote) as we look to the far right of each graph.
y y y...
(c) Was the car slowing down or speeding up at A, B, and C ? (d) What happened between D and E ?
s...
Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t ) as a function of t. What are its units? For times t 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of water remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least?...
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