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free/or 0.5$ |
| m52723 | (a) Show that if satisfies the logistic equation 4, then
d2P / dt2 = k2P(1 - P / M) (1 - 2P / M)
(b) Deduce that a population grows fastest when it reaches half its carrying capacity. |
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| m52724 | a. Show that
is divergent.
b Show that
This shows that we can t define |
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| m52725 | (a) Show that J0 (the Bessel function) of order 0 given in Example 4) satisfies the differential equation |
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| m52727 | (a) Show that tan ½ x = cot ½ x - 2 cot x.
(b) Find the sum of the series |
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| m52729 | (a) Show that the function |
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| m52733 | (a) Show that the parametric equations
x = x1 + (x2- x1)t, y = Yl + (y2- yt)t
Where 0 ≤ t ≤ 1 describe the line segment that joins the points p1(x1, yl) and P2(x2, Y2).
(b) Find parametric equations to represent the line segment from (-2,7) to (3,-1). |
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| m52739 | (a) Show that the substitution z = 1 / P transforms the logistic differential equation P = kP(1 - P / M) into the linear differential equation
z + kz = k / M
(b) Solve the linear differential equation in part (a) and thus obtain an expression for P(t). Compare with Equation 9.4.7. |
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| m52740 | (a) Show that the volume of a segment of height h of a sphere of radius r is
V = 1/3 πh2 (3r - h) |
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| m52746 | (a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
(b) Eliminate the parameter to find a Cartesian equation of the curve.
X = 3 - 4t, y = 2 - 3t |
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| m52747 | (a) Sketch the graph of a function on [- 1, 2] that has an absolute maximum but no absolute minimum.
(b) Sketch the graph of a function on [- 1, 2] that is discontinuous but has both an absolute maximum and an absolute minimum. |
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| m52748 | (a) Sketch the graph of a function that has a local maximum at 2 and is differentiable at 2.
(b) Sketch the graph of a function that has a local maximum at 2 and is continuous but not differentiable at 2.
(c) Sketch the graph of a function that has a local maximum at 2 and is not continuous at 2. |
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| m52749 | (a) Sketch the graph of the function f(x) = x |x|.
(b) For what values of is f differentiable?
(c) Find a formula for f. |
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| m52760 | (a) Solve the differential equation y = 2x√1 - y2. |
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| m52763 | A spherical balloon is being inflated. Find the rate of increase of the surface area (S = 4πr2) with respect to the radius r when is
(a) 1 ft,
(b) 2 ft,
(c) 3 ft. What conclusion can you make? |
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| m52764 | A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length.
(a) Explain why the function
is a probability density function for the spinner s values.
(b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral. |
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| m52765 | A spring has natural length 20 cm. Compare the work W1 done in stretching the spring from 20 cm to 30 cm with the work W2 done in stretching it from 30 cm to 40 cm. How are W2 and W1 related? |
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| m52768 | (a) State the Net Change Theorem.
(b) If r(t) is the rate at which water flows into a reservoir, what does
represent? |
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| m52771 | A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm /s.
(a) Express the radius of this circle as a function of the time (in seconds).
(b) If A is the area of this circle as a function of the radius, find A o r and interpret it. |
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| m52773 | A string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius and center 0 and the initial position of P is (r, 0) and if the parameter is θ chosen as in the figure, show that parametric equations of the involute are
x = r (cos θ + θ sin θ) y = r (sin θ - θ cos θ) |
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| m52789 | (a) Suppose f is a one-to-one differentiable function and its inverse function f-1 is also differentiable. Use implicit differentiation to show that
provided that the denominator is not 0.
(b) If f(4) = 5 and f(4) = 2/3 , find (f-1) (5). |
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