| № |
Condition |
free/or 0.5$ |
| m52529 | A graph of the temperature in New York City on September 19, 2009 is shown. Use Simpson s Rule with n = 12 to estimate the average temperature on that day. |
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| m52530 | (a) Graph the curve y = x 3√(4-x), 0 ≤ x ≤ 4.
(b) Compute the lengths of inscribed polygons with n = 1,2 and 4 sides. (Divide the interval into equal subintervals.) Illustrate by sketching these polygons (as in Figure 6).
(c) Set up an integral for the length of the curve.
(d) Use your calculator to find the length of the curve to four decimal places. Compare with the approximations in part (b). |
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| m52531 | (a) Graph the epitrochoid with equations
x = 11 cos t - 4 cos (11t / 2)
y = 11 sin t - 4 sin (11t / 2)
What parameter interval gives the complete curve?
(b) Use your CAS to find the approximate length of this curve. |
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| m52536 | A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation
y = 180 - x2/45
Until it hits the ground, where is its height above the ground and is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter. |
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| m52537 | A hole of radius is bored through the middle of a cylinder of radius R > r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out. |
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| m52539 | A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period. |
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| m52541 | (a) How do you find the slope of a tangent line to a polar curve?
(b) How do you find the area of a region bounded by a polar curve?
(c) How do you find the length of a polar curve? |
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| m52542 | (a) How is the graph of y = f(|x) related to the graph of f?
(b) Sketch the graph of y = sin |x|.
(c) Sketch the graph of y = √|x|. |
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| m52543 | (a) How is the inverse sine function f(x) = sin-1x defined? What are its domain and range?
(b) How is the inverse cosine function f(x) = cos-1x defined? What are its domain and range?
(c) How is the inverse tangent function f(x) = tan-1 x defined? What are its domain and range? |
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| m52544 | (a) How is the logarithmic function defined?
(b) What is the domain of this function?
(c) What is the range of this function? |
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| m52545 | (a) How is the number defined?
(b) Express as a limit.
(c) Why is the natural exponential function y = ex used more often in calculus than the other exponential functions y = ax?
(d) Why is the natural logarithmic function y = In x used more often in calculus than the other logarithmic functions y = loga x? |
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| m52546 | (a) How is the number e defined?
(b) Use a calculator to estimate the values of the limits
Correct to two decimal places. What can you conclude about the value of e? |
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| m52547 | (a) How large do we have to take so that 1/x2 < 0.0001?
(b) Taking r = 2 in Theorem 5, we have the statement
Prove this directly using Definition 7. |
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| m52550 | (a) If $3000 is invested at 5% interest, find the value of the investment at the end of 5 years if the interest is compounded (i) annually, (ii) semiannually, (iii) monthly, (iv) weekly, (v) daily, and (vi) continuously.
(b) If A(t) is the amount of the investment at time for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(t). |
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| m52551 | (a) If a > 0, find the area of the surface generated by rotating the loop of the curve 3ay2 = x(a - x)2 about the -axis.
(b) Find the surface area if the loop is rotated about the -axis. |
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| m52552 | (a) If f is an odd function, show that
C0 = c2 = c4 = ... = 0
(b) If f is an even function, show that
C1 = c3 = c5 = ... = 0 |
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| m52553 | (a) If f(x) = √3 - 5x, use the definition of a derivative to find f (x).
(b) Find the domains of f and f .
(c) Graph f and f on a common screen. Compare the graphs to see whether your answer to part (a) is reasonable. |
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| m52554 | (a) If F(x) = 5x/(1 + x2), find F (2) and use it to find an equation of the tangent line to the curve y = 5x/ (1 + x2) at the point (2, 2).
(b) Illustrate part (a) by graphing the curve and the tangent line on the same screen. |
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| m52555 | (a) If f(x) = sec x - x, find f (x).
(b) Check to see that your answer to part (a) is reasonable by graphing both f and f for |x| < π/2. |
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| m52556 | (a) If f(x) = sin(sin x), use a graph to find an upper bound for |f(4)(x)|.
(b) Use Simpson s Rule with n = 10 to approximate
And use part (a) to estimate the error.
(c) How large should n be to guarantee that the size of the error in using Sn is less than 0.0001? |
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