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free/or 0.5$ |
| m52438 | A curve has equation y = f(x).
(a) Write an expression for the slope of the secant line through the points P(3, f(3)) and Q(x, f(x)).
(b) Write an expression for the slope of the tangent line at P. |
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| m52442 | A cylindrical tank with radius 5 m is being filled with water at a rate of 3 m3/ min. How fast is the height of the water increasing? |
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| m52444 | A demand curve is given by ( = 450/(x+8). Find the consumer surplus when the selling price is $10 |
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| m52451 | A direction field for the differential equation is y = x cos π y is shown. |
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| m52452 | A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system (as shown in the figure), assume:
(i) The rabbit is at the origin and the dog is at the point (L, 0) at the instant the dog first sees the rabbit.
(ii) The rabbit runs up the -axis and the dog always runs straight for the rabbit.
(iii) The dog runs at the same speed as the rabbit.
(a) Show that the dog s path is the graph of the function y = f(x), where satisfies the differential equation |
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| m52453 | A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 mi/h; she drives the second half at 60 mi/h. What is her average speed on this trip? |
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| m52457 | A drug response curve describes the level of medication in the bloodstream after a drug is administered. A surge function S(t) = AtPe-kt is often used to model the response curve, reflecting an initial surge in the drug level and then a more gradual decline. If, for a particular drug, A = 0.01, p = 4, k = 0.07, and t is measured in minutes, estimate the times corresponding to the inflection points and explain their significance. If you have a graphing device, use it to graph the drug response curve. |
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| m52458 | (a) dy / dx = xy2
(b) xy2 y = x + 1
(c) (y + sin y)y = x + x3 |
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| m52459 | (a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
x = sin 1/2 θ, y = cos 1/2 θ, -π ≤ θ ≤ π. |
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| m52460 | (a) Estimate the area under the graph of f(x) = cos x from x = 0 to x = π/2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate?
(b) Repeat part (a) using left endpoints. |
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| m52461 | (a) Estimate the area under the graph of f(x) = 1 + x2 from x = -1 to x = 2 using three rectangles and right end points. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.
(b) Repeat part (a) using left endpoints.
(c) Repeat part (a) using midpoints.
(d) From your sketches in parts (a) - (c), which appears to be the best estimate? |
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| m52462 | (a) Estimate the value of
by graphing the function f(x) = x/(√1 + 3x - 1).
(b) Make a table of f(x) values of for x close to 0 and guess the value of the limit.
(c) Use the Limit Laws to prove that your guess is correct. |
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| m52463 | (a) Estimate the value of
by graphing the function f(x) = √x2 + x + 1 + x.
(b) Use a table of values of f(x) to guess the value of the limit.
(c) Prove that your guess is correct. |
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| m52464 | (a) Estimate the value of the limit limit limx→0 to five decimal places. Does this number look familiar?
(b) Illustrate part (a) by graphing the function y = (1 + x)1/x. |
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| m52466 | (a) Evaluate the function f(x) = x2 - (2*/1000) for x = 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of
(b) Evaluate f(x) for x = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again. |
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| m52467 | a) Evaluate
Where is a positive integer.
(b) Evaluate
Where and are real numbers with 0 ≤ a < b. |
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| m52468 | (a) Explain how implicit differentiation works.
(b) Explain how logarithmic differentiation works. |
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| m52469 | (a) Explain the meaning of the indefinite integral ∫ f(x) dx..
(b) What is the connection between the definite integral
and the indefinite integral ∫ f(x) dx? |
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| m52470 | (a) Explain why the function
is a probability density function.
(b) Find P(X < 4).
(c) Calculate the mean. Is the value what you would expect? |
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| m52471 | (a) Express the area under the curve y = x5 from 0 to 2 as a limit.
(b) Use a computer algebra system to find the sum in your expression from part (a).
(c) Evaluate the limit in part (a). |
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