| № |
Condition |
free/or 0.5$ |
| m52885 | (a) What is the difference between a sequence and a series?
(b) What is a convergent series? What is a divergent series? |
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| m52886 | (a) What is the eccentricity of a conic section?
(b) What can you say about the eccentricity if the conic section is an ellipse? A hyperbola? A parabola?
(c) Write a polar equation for a conic section with eccentricity e and directrix x = d. what if the directrix is x = - d? y = d? y = - d? |
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| m52888 | (a) What is the volume of a cylindrical shell?
(b) Explain how to use cylindrical shells to find the volume of a solid of revolution.
(c) Why might you want to use the shell method instead of slicing? |
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| m52890 | A window has the shape of a square surmounted by a semi - circle. The base of the window is measured as having width 60 cm with a possible error in measurement of 0.1 cm. Use differentials to estimate the maximum error possible in computing the area of the window. |
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| m52892 | (a) Write a differential equation that expresses the law of natural growth.
(b) Under what circumstances is this an appropriate model for population growth? |
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| m52893 | (a) Write a differential equation that expresses the law of natural growth. What does it say in terms of relative growth rate?
(b) Under what circumstances is this an appropriate model for population growth? |
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| m52894 | (a) Write an equation that defines the exponential function with base a > 0.
(b) What is the domain of this function?
(c) If a ≠ 1, what is the range of this function?
(d) Sketch the general shape of the graph of the exponential function for each of the following cases.
(i) a > 1
(ii) a = 1
(iii) 0 < a < 1 |
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| m52895 | (a) Write an expression for the linearization of f at a.
(b) If y = f(x), write an expression for the differential dy.
(c) If dx = Δx, draw a picture showing the geometric meanings of Δy and dy. |
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| m52896 | (a) Write an expression for the surface area of the surface obtained by rotating the curve y = f(x), a ≤ x ≤ b, about the -axis.
(b) What if x is given as a function of y?
(c) What if the curve is rotated about the y-axis? |
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| m52897 | (a) Write Lotka-Volterra equations to model populations of food fish (F) and sharks (S).
(b) What do these equations say about each population in the absence of the other? |
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| m52898 | (a) Write the logistic equation.
(b) Under what circumstances is this an appropriate model for population growth? |
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| m52899 | (a) Write the solution of the initial-value problem dP/dt = 0.1P(1 - P/2000) P(0) = 100 and use it to find the population when t = 20,
(b) When does the population reach 1200? |
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| m52900 | a. y = 12 - x2, y = x2 - 6
b. y = ex, y = xex, x = 0
c. x = 2y2, x = 4 + y2 |
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| m52901 | a. y = sin x, 0 ≤ x ≤ π
b. x = √y - y , 1 ≤ y ≤ 4 |
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| m52927 | (a)Find the Taylor polynomials up to degree 6 for f(x) = cos x centered at a = 0, Graph f and these polynomials on a common screen.
(b) Evaluate f and these polynomials at x = (/4 (/2, and (.
(c) Comment on how the Taylor polynomials converge to f(x). |
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| m52953 | Allometric growth in biology refers to relationships between sizes of parts of an organism (skull length and body length, for instance). If L1(t) and L2(t) are the sizes of two organs in an organism of age t, then L1 and L2satisfy an allometric law if their specific growth rates are proportional:
1/L1 dL1/dt = k 1/L2 dL2/dt
where k is a constant.
(a) Use the allometric law to write a differential equation relating L1 and L2 solve it to express L1 as a function of L2.
(b) In a study of several species of unicellular algae, the proportionality constant in the allometric law relating B (cell biomass) and V (cell volume) was found to be k = 0.0794. Write B as a function of V. |
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| m52965 | An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find
(a) The hydrostatic pressure on the bottom of the aquarium,
(b) The hydrostatic force on the bottom,
(c) The hydrostatic force on one end of the aquarium. |
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| m52974 | An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are q and - q are and located at a distance d from each other, then the electric field E at the point P in the figure is
By expanding this expression for E as a series in powers of d/D, show that E is approximately proportional to 1/D3 when P is far away from the dipole. |
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| m52975 | An ellipse is cut out of a circle with radius a. The major axis of the ellipse coincides with a diameter of the circle and the minor axis has length 2b. Prove that the area of the remaining part of the circle is the same as the area of an ellipse with semlaxes a and a-b. |
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| m52978 | An equation of motion of the form x = Ae-ct cos(wt + δ) represents damped oscillation of an object. Find the velocity and acceleration of the object. |
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