Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 36 potx

... chances of winning? (e) Sketch a graph of P (n). Use [0, 100] as the range of the graph. Explain the practical significance of any asymptotes. 9.4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTION As an ... 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 cons...

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... 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 an odd function is said to have symmetry ... 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,...

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... 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. As h gets closer and closer to ... rate of change over the tiny interval approaches the instantaneous rate of change at x = a. The instantaneous rate of change of f at x = a ≈ f(a +h) − f(a) (a + h) − a for h...

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... equation of the parabola with the specifications given. 17. x-intercepts of 3 and −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. ... (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...

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... Animals eat plants and so have the same ratio of C 14 to C 12 as do the plants and the atmosphere. Thus, during the lifetime of a living organism the ratio of C 14 to C 12 in the organism is fixed; ... 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 grap...

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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...

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... 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 Simpson’s rule to approximate ... 2y 4 +···+2y 2n−2 +4y 2n−1 +y 2n )x. The answer to part (c) is given at the end of the section. Notice that because Simpson’s rule requires both the midpoint and trapezoid...

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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...

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