Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 9 potx
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 ChangeDefinitionA is inversely proportional to B ifA =kBfor some constant k. As B increases A decreases, and vice versa.This type of relationship ... return to the four functions we introduced at the beginning of this section:f(x)=x, g(x) = x2, h(x) =|x|, and j(x) =1x(a) Which of functions f , g, h, and j are even?(b) Which of functions...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 20 potx
... 2 and we aresupplied with the following:5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change 173t (in seconds) s (position in feet)1 .9 198 .241 .99 192 .63842 192 2.01 191 .35842.1 ... f(a) )and (b, f(b)).The instantaneous rate of change of f at x = a can be approximated by the averagerate of change of f on [a, a + h] for h very close to zero. As h gets closer and closer to ... time, we can find the average velocityover the intervals, say, [2, 2.0001] and [1 .99 99, 2] to get lower and upper bounds. We couldget better and better approximations by taking smaller and smaller...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 26 potx
... −2; maximum value of 118. x-intercepts of π and 3π; y-intercept of 6 19. x-intercepts of π and 3π; y-intercept of −220. Vertex at (1, 5); y-intercept of 16.4 The Free Fall of an Apple: A Quadratic ... (i) and three arbitrary points we can obtain a system of threelinear equations that we can solve for the constants a, b, and c. We can use the x-interceptsr1 and r2in equation (ii) and use ... Quadratics and Their Graphs 2316.3 QUADRATICS AND THEIR GRAPHSGraphing ParabolasBeginning with the basic parabola y = x2 and using the results of Section 3.4 on shifting,stretching, shrinking, and...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 35 potx
... atoms; therefore plants contain the same ratio of C14 to C12atoms as exists inthe atmosphere. Animals eat plants and so have the same ratio of C14 to C12as do the plants and the atmosphere. ... sketch the graphs of f(x)=2x,g(x) =3x, and h(x) = 10x. Then, on another set of axes, draw a roughsketch of the graphs of f(x), g(x), and h(x). The graphs of f ,g, and h all intersect ... the slope of the tangent at a point by the slope of a secant line. For instance, to approximate the derivative of f(x)=2xat x = 1, use the slope of the secant line between(1, f(1)) and (1 +...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 36 potx
... range of the graph. Explain the practicalsignificance of any asymptotes. 9. 4 THE DERIVATIVE OF AN EXPONENTIAL FUNCTIONAs an exploratory problem you investigated the derivatives of 2x,3x ,and ... average orat instantaneous rates of change, we can state the following.Exponential functions grow at a rate proportional to themselves.Exponential functions have a constant percent change.Question: ... measures.(b) The average cost of tuition in 199 4– 199 5 was $2865 for in-state students. Whatwas it in 198 0– 198 1?(c) Starting with an initial value of one unit for each of the three quantities, averagetuition...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 49 potx
... Logarithmic and Exponential Functions We are looking for the slope of the tangent line to ln x at x = b. Let L1be the tangentline to exat (c, b) and L2be the tangent line to ln x at (b, ... slope at any point on the graph of exis given by the y-coordinate of that point), the slope of the tangent line to exat (c, b)isb.Since the roles of x and y areinterchanged on the graph of ln ... the graph of log3x and build itup from there. log3x and 3xare inverse functions, so their graphs are reflections about theline y = x.13.4 Graphs of Logarithmic Functions: Theme and Variations...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 55 potx
... instantaneousrate of change of y with respect to x at x = 2.Notation: y, f, f(x), y(x),dydx,dfdxall mean the same thing.ddxis an operator that means“take the derivative of ... They can get to the store by acombination of boat and foot. Where should they build their dockso the commute to the store takes the minimum amount of time?home6 milesRIVER1 mileStore ... will save you time and energy. (Of course, there are times when it is necessary to dig in and get your hands all dirty. If itis necessary, do it with relish!) The meaning of this advice is clarified...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 72 potx
... 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 radianswith the ground and at an initial velocity of v0meters ... 3πradiansminute.dθdt= 3π radians/minuteStrategy We knowdθdt and we want to finddxdt. Our strategy is to write an equation relatingθ and x.11We then differentiate with respect to time to get a relationship ... = 0. 099 8334 ≈0.1 and sin 0.02 = 0.0 199 986 ≈0.02,but sin 3 = 0.1411200 ,which is not close to 3.yy = xxsin xFigure 21 .9 Historically the approximation sin x ≈ x for very small values of x...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 85 potx
... M50= 1.6 091 8221 T50= 1.6 099 495 7 Simpson’s rule =2M50+T503= 1.6 094 3 799 (d) n = 100: M100= 1.6 093 7 393 T100= 1.6 095 65 89 Simpson’s rule =2M100+T1003= 1.6 094 3 791 ◆Like the ... estimatingthe area of a plot of land or body of water.Suppose our aim is to approximatebaf(x)dx.Wecan collect values of f(x)at equallyspaced intervals and use Simpson’s rule to approximate ... 2y4+···+2y2n−2+4y2n−1+y2n)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 trapezoidal rules and the relevant data points for each of these...
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Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 112 potx
... between F and L.The axis of symmetry runsthrough F and perpendicular to L.FVLd1d1d2d2d3d3Axis of symmetryFigure 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 orientation of the plane relative to the ... on (x1, x2) and continuous onProof by Induction 1 097 12+ 22+ 32+ 42+···+k2+(k + 1)2=(k + 1)[(k + 1) + 1][2(k + 1) + 1]6.The right-hand side of this equation can be written...
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