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| m52474 | (a) Fibonacci posed the following problem: Suppose that rabbits live forever and that every month each pair produces a new pair which becomes productive at age 2 months. If we start with one newborn pair, how many pairs of rabbits will we have in the nth month? Show that the answer is fn, where {fn} is the Fibonacci sequence defined in Example 3(c).
(b) Let an = fn+1/fn and show that an - 1. Assuming that is convergent, find its limit. |
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| m52476 | (a) Find a number δ such that if |x -2| < δ, then |4x - 8 < ε, where ε = 0.01.
(b) Repeat part (a) with ε = 0.01. |
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| m52477 | (a) Find a positive continuous function f such that the area under the graph of f from 0 to t is A(t) = t2 for all t > 0.
(b) A solid is generated by rotating about the -axis the region under the curve y = f(x), where f is a positive function and x ≥0. The volume generated by the part of the curve from x = 0 to x = b is b2 for all b > 0. Find the function f. |
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| m52478 | (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such that f(2) = 1 and sketch several members of the family.
(c) Which function belongs to both families? |
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| m52479 | (a) Find an equation of the tangent line to the curve y = 2x sin x at the point (π/2, π).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. |
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| m52480 | (a) Find an equation of the tangent line to the curve y = 2/(1 + e-x) at the point (0, 1).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. |
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| m52482 | (a) Find the approximations T10, M10 and S10 for
And the corresponding errors ET, EM, and ES
(b) Compare the actual errors in part (a) with the error estimates given by 3 and 4.
(c) How large do we have to choose n so that the approximations Tn, Tn, and Sn to the integral in part (a) are accurate to within 0.00001? |
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| m52483 | (a) Find the approximations TS and MS for the integral
(b) Estimate the errors in the approximations of part (a).
(c) How large do we have to choose n so that the approximations Tn and Mn to the integral in part (a) are accurate to within 0.0001? |
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| m52484 | (a) Find the average rate of change of the area of a circle with respect to its radius as changes from
(i) 2 to 3
(ii) 2 to 2.5
(iii) 2 to 2.1
(b) Find the instantaneous rate of change when r = 2.
(c) Show that the rate of change of the area of a circle with respect to its radius (at any ) is equal to the circumference of the circle. Try to explain geometrically why this is true by drawing a circle whose radius is increased by an amount Δr. How can you approximate the resulting change in area ΔA if Δr is small? |
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| m52485 | (a) Find the average value of f on the given interval.
(b) Find such that fave = f(c).
(c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f.
(1) f(x) = (x - 3)2, [2, 5]
(2) f(x) = 2 sin x - sin 2x, [0, π] |
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| m52486 | (a) Find the differential dy
(b) evaluate dy for the given values of and dx.
1. y = ex/10, x = 0, dx = 0.1
2. y = √3 + x2, x = 1, dx = - 0.1 |
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| m52487 | (a) Find the domain of f(x) = ln (ex - 3).
(b) Find f-1 and its domain. |
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| m52488 | (a) Find the eccentricity and directrix of the conic r = 1/(1 - 2 sin () and graph the conic and its directrix?
(b) If this conic is rotated counterclockwise about the origin through an angle 3(/4, write the resulting equation and graph its curve? |
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| m52489 | (a) Find the eccentricity
(b) Identify me conic
(c) give an equation of the directrix, and
(d) sketch the conic.
R = 4/5-4 sin θ |
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| m52490 | (a) Find the intervals of increase or decrease.
(b) Find the local maximum and minimum values.
(c) Find the intervals of concavity and the inflection points.
(d) Use the information from parts (a)-(c) to sketch the graph. Check your work with a graphing device if you have one.
F(x) = x2 - 12x + 2 |
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| m52491 | (a) Find the intervals on which f is increasing or decreasing.
(b) Find the local maximum and minimum values of f.
(c) Find the intervals of concavity and the inflection points.
F(x) = 2x3 + 3x2 - 36x |
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| m52492 | (a) Find the length of the curve
y = x4/16 + 1/2x2, 1 ≤ x ≤ 2
b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis. |
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| m52493 | (a) Find the linearization of f(x) = 3√1 + 3x at a = 0. State the corresponding linear approximation and use it to give an approximate value for 3√1.03.
(b) Determine the values of for which the linear approximation given in part (a) is accurate to within 0.1. |
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| m52495 | (a) Find the slope of the tangent line to the parabola y = 4x - x2 at the point (1, 3)
(i) Using Definition 1
(ii) Using Equation 2
(b) Find an equation of the tangent line in part (a).
(c) Graph the parabola and the tangent line. As a check on your work, zoom in toward the point (1, 3) until the parabola and the tangent line are indistinguishable. |
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| m52496 | (a) Find the slope of the tangent line to the curve y = 9 - 2x2 at the point (2, 1).
(b) Find an equation of this tangent line. |
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