Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 9 potx

... (−∞,0 )and decreasing on (0, ∞). (d) j is undefined and discontinuous at x = 0. (e) j is 1 -to- 1. Answers to Exercise 2.8 (a) g and h are even functions. (b) f and j are odd functions. (c) The graph of ... Characterizing Functions and Introducing Rates of Change Definition A is inversely proportional to B if A = k B for some constant k. As B increases A decreases, an...

Ngày tải lên: 05/07/2014, 18:20

... 2 and we are supplied with the following: 5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change 173 t (in seconds) s (position in feet) 1 .9 198 .24 1 .99 192 .6384 2 192 2.01 191 .3584 2.1 ... f(a) )and (b, f(b)). The instantaneous rate of change of f at x = a can be approximated by the average rate of change of f on [a, a + h] for h very close to zero. A...

Ngày tải lên: 05/07/2014, 18:20

... −2; maximum value of 1 18. x-intercepts of π and 3π; y-intercept of 6 19. x-intercepts of π and 3π; y-intercept of −2 20. Vertex at (1, 5); y-intercept of 1 6.4 The Free Fall of an Apple: A Quadratic ... (i) and three arbitrary points we can obtain a system of three linear equations that we can solve for the constants a, b, and c. We can use the x-intercepts r 1 an...

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... atoms; therefore plants contain the same ratio of C 14 to C 12 atoms as exists in the atmosphere. Animals eat plants and so have the same ratio of C 14 to C 12 as do the plants and the atmosphere. ... sketch the graphs of f(x)=2 x ,g(x) = 3 x , and h(x) = 10 x . Then, on another set of axes, draw a rough sketch of the graphs of f  (x), g  (x), and h  (x). The graphs...

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... range of the graph. Explain the practical significance of any asymptotes. 9. 4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION As an exploratory problem you investigated the derivatives of 2 x ,3 x ,and ... average or at instantaneous rates of change, we can state the following. Exponential functions grow at a rate proportional to themselves. Exponential functions have a c...

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... Logarithmic and Exponential Functions We are looking for the slope of the tangent line to ln x at x = b. Let L 1 be the tangent line to e x at (c, b) and L 2 be the tangent line to ln x at (b, ... slope at any point on the graph of e x is given by the y-coordinate of that point), the slope of the tangent line to e x at (c, b)isb.Since the roles of x and y are interch...

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... instantaneous rate of change of y with respect to x at x = 2. Notation: y  , f  , f  (x), y  (x), dy dx , df dx all mean the same thing. d dx is an operator that means “take the derivative of ... They can get to the store by a combination of boat and foot. Where should they build their dock so the commute to the store takes the minimum amount of time? home 6 miles R...

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... are the maximum and minimum values of f ? 12. If we ignore air resistance, a baseball thrown from shoulder level at an angle of θ radians with the ground and at an initial velocity of v 0 meters ... 3π radians minute . dθ dt = 3π radians/minute Strategy We know dθ dt and we want to find dx dt . Our strategy is to write an equation relating θ and x. 11 We then differenti...

Ngày tải lên: 05/07/2014, 18:20

... M 50 = 1.6 091 8221 T 50 = 1.6 099 495 7 Simpson’s rule = 2M 50 +T 50 3 = 1.6 094 3 799 (d) n = 100: M 100 = 1.6 093 7 393 T 100 = 1.6 095 65 89 Simpson’s rule = 2M 100 +T 100 3 = 1.6 094 3 791 ◆ Like the ... estimating the area of a plot of land or body of water. Suppose our aim is to approximate  b a f(x)dx.Wecan collect values of f(x)at equally spaced intervals and use...

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... between F and L. The axis of symmetry runs through F and perpendicular to L. F V L d 1 d 1 d 2 d 2 d 3 d 3 Axis of symmetry Figure E.4 An Ellipse An ellipse is the set of points (x, y), the sum of whose ... they can be thought of as the intersection of a plane with a pair of circular cones with vertices joined in an hourglass configuration. Depending upon the orientat...

Ngày tải lên: 05/07/2014, 18:20