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Condition |
free/or 0.5$ |
| m52497 | (a) Find the slope of the tangent line to the trochoid
x = rθ - d sin θ, y = r - d cos θ in term of θ.
(b) Show that if d < r, then the trochoid does not have a vertical tangent. |
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| m52498 | (a) Find the slope of the tangent to the curve y = 3 + 4x2 - 2x3 at the point where x = a.
(b) Find equations of the tangent lines at the points (1, 5) and (2, 3).
(c) Graph the curve and both tangents on a common screen. |
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| m52499 | (a) Find the vertical and horizontal asymptotes.
(b) Find the intervals of increase or decrease.
(c) Find the local maximum and minimum values.
(d) Find the intervals of concavity and the inflection points.
(e) Use the information from parts (a)-(d) to sketch the graph of f.
f(x) = 1 + 1/x - 1/x2 |
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| m52500 | (a) Find y by implicit differentiation.
(b) Solve the equation explicitly for and differentiate to get y in terms of x. |
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| m52502 | A fish farmer has 5000 catfish in his pond. The number of catfish increases by 8% per month and the farmer harvests 300 catfish per month.
(a) Show that the catfish population Pn after months is given recursively by
Pn = 1.08Pn-1 - 300 Po = 5000
(b) How many catfish are in the pond after six months? |
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| m52507 | (a) For the limit limx →1 (x3 + x + 1) = 3, use a graph to find a value of δ that corresponds to ε = 0.4.
(b) By using a computer algebra system to solve the cubic equation x3 + x + 1 = 3 + ε, find the largest possible value of that works for any given ε > 0.
(c) Put ε = 0.4 in your answer to part (b) and compare with your answer to part (a). |
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| m52508 | (a) For what values of does the function y = erx satisfy the differential equation y + y - y = 0? |
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| m52509 | (a) For what values of is the function f(x) = |x2 - 9| differentiable? Find a formula for f .
(b) Sketch the graphs of f and f . |
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| m52510 | A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length? |
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| m52511 | A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm? |
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| m52512 | A formula for the derivative of a function is given. How many critical numbers does have?
f (x) = 5e-0.1|x| sin x - 1 |
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| m52514 | (a) From the graph of , state the numbers at which f is discontinuous and explain why.
(b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither. |
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| m52517 | A function f is defined by
Find the minimum value of f. |
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| m52518 | A function f is defined by
f(x) = 1 + 2x + x2 + 2x3 + x4 + ...
that is, its coefficients are c2n = 1 and c2n+1 = 2 for all n ( 0. Find the interval of convergence of the series and find an explicit formula for f(x). |
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| m52519 | A function is a ratio of quadratic functions and has a vertical asymptote x = 4 and just one -intercept, x = 1. It is known that has a removable discontinuity at x = -1 and limx→-1 f(x) = 2. Evaluate
(a) f(0) |
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| m52520 | A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.
(a)
(b)
(c)
(d) f (x) = x2 - 2x |
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| m52523 | A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of the gate. |
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| m52525 | (a) Give a definition of an ellipse in terms of foci.
(b) Write an equation for the ellipse with foci (( c, 0) and vertices (( a, 0) and vertices (( a, 0) |
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| m52526 | A glucose solution is administered intravenously into the bloodstream at a constant rate r. As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration C = C(t) of the glucose solution in the bloodstream is
dC/dt = r - kC
where k is a positive constants.
(a) Suppose that the concentration at time t = 0 is C0.
Determine the concentration at any time t by solving the differential equation.
(b) Assuming that C0 < r/k, find limt→∞ C(t) and interpret your answer. |
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| m52528 | A graph of a population of yeast cells in a new laboratory culture as a function of time is shown.
(a) Describe how the rate of population increase varies.
(b) When is this rate highest?
(c) On what intervals is the population function concave upward or downward?
(d) Estimate the coordinates of the inflection point. |
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