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free/or 0.5$ |
| m53149 | Boyle s Law states that when a sample of gas is compressed at a constant temperature, the product of the pressure and the volume remains constant: PV = C.
(a) Find the rate of change of volume with respect to pressure.
(b) A sample of gas is in a container at low pressure and is steadily compressed at constant temperature for 10 minutes. Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes? Explain.
(c) Prove that the isothermal compressibility (see Example 5) is given by β = 1/P. |
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| m53150 | Boyle s Law states that when a sample of gas is compressed at a constant temperature, the pressure P and volume V satisfy the equation PV = C, where C is a constant. Suppose that at a certain instant the volume is 600 cm3, the pressure is 150 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant? |
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| m53158 | Calculate. |
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| m53165 | Calculate the average value of f(x) = x sec2x on the interval [0, π/4] |
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| m53166 | Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
a.
b. |
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| m53169 | Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. |
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| m53170 | Calculate the sum of the series whose partial sums ∑(n=1 an whose partial sums are given.
Sn = 2 - 3(0.8) n |
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| m53171 | Calculate, to four decimal places, the first ten terms of the sequence and use them to plot the graph of the sequence by hand. Does the sequence appear to have a limit? If so, calculate it. If not, explain why.
a. an = 3n / 1 + 6n
b. an = 1 + (- 1/2)n |
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| m53172 | Calculate y .
(a) y = (x2 + x3)4
(b) y = x2 - x + 2/ √x
(c) y = x2 sin π x |
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| m53210 | Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
(a) f(x) = log2x
(b) g(x) = 4√x
(c) h(x) = 2x3 / 1 - x2
(d) u(t) = 1 - 1.1t + 2.54 t2
(e) v(t) = 5t
(f) w(θ) = sin θ cos2θ |
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| m53216 | Compare the curves represented by the parametric equations. How do they differ?
(a) x = t3, y= t2
(b) x = t6, y= t4
(c) x = e-3t, y = e-2t |
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| m53217 | Compare the functions f(x) = x10 and g(x) = ex by graphing both f and in several viewing rectangles. When does the graph of finally surpass the graph of f? |
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| m53235 | Compute Δy and dy for the given values of x and dx = Δx. Then sketch a diagram like Figure 5 showing the line segments with lengths dx, dy, and Δy.
(a) y = 2x - x2, x = 2 Δx = - 0.4
(b) y = 2/x, x = 4, Δx = 1
Figure 5 |
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| m53236 | Computer algebra systems have commands that differentiate functions, but the form of the answer may not be convenient and so further commands may be necessary to simplify the answer.
(a) Use a CAS to find the derivative in Example 5 and compare with the answer in that example. Then use the simplify command and compare again.
(b) Use a CAS to find the derivative in Example 6. What happens if you use the simplify command? What hap pens if you use the factor command? Which form of the answer would be best for locating horizontal tangents? |
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| m53257 | Consider a population P = P(t) with constant relative birth and death rates α and β, respectively, and a constant emigration rate m, where α, β, and m are positive constants. Assume that α > β. The rate of change of the population at time t is modeled by the differential equation
dP / dt = kP - m where k = α - β |
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| m53401 | Consider the series
(a) Find the partial sums s1, s2, s3, and s4. Do you recognize the denominators? Use the pattern to guess a formula for sn.
(b) Use mathematical induction to prove your guess.
(c) Show that the given infinite series is convergent, and find its sum. |
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| m53402 | Consider the series whose terms are the reciprocals of the positive integers that can be written in base 10 notation without using the digit 0. Show that this series is convergent and the sum is less than 90. |
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| m53497 | Derive Equations 1 for the case π/2 < θ < π |
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| m53507 | Describe how we can find the hydrostatic force against a vertical wall submersed in a fluid. |
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| m53508 | Describe several ways in which a limit can fail to exist. Illustrate with sketches. |
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