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| m97922 | Explain in your own words what is meant by the equation
Is it possible for this statement to be true and yet f (2) = 3? Explain. |
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| m97923 | Explain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain.
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| m97925 | Fanciful shapes can be created by using the implicit plotting capabilities of computer algebra systems.
(a) Graph the curve with equation y(y2 – 1) (y – 2) = x(x – 1) (x – 2)
At how many points does this curve have horizontal tangents? Estimate the -coordinates of these points.
(b) Find equations of the tangent lines at the points (0, 1) and (0, 2).
(c) Find the exact -coordinates of the points in part (a).
(d) Create even more fanciful curves by modifying the equation in part (a). |
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| m97929 | Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y = r. |
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| m97930 | Find the average value of the function f(x) = sin2x cos3x on the interval [– π, π]. |
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| m97931 | Find the centroid of the region bounded by the curves y = 2x and y = x2, 0 < x < 2, to three decimal places. Sketch the region and plot the centroid to see if your answer is reasonable. |
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| m97932 | Find the centroid of the region shown, not by integration, but by locating the centroid s of the rectangles and triangles (from Exercise 37) and using additively of moments.
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| m97933 | Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.
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| m97934 | Find the vertex, focus, and directrix of the parabola and sketch its graph.
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| m97935 | Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere. |
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| m97951 | If f and g are the functions whose graphs are shown, let u(x) = f (g(x)), v(x) = g(f(x)), and w(x) = g(g(x)). Find each derivative, if it exists. If it does not exist, explain why.
(a) u’ (1) (b) v’ (1) (c) w’ (1)
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| m97976 | Investigate the family of curves defined by the parametric equations x = t2, y = t3 – ct. How does the shape change as c increases? Illustrate by graphing several members of the family. |
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| m97983 | Let T and N be the tangent and normal lines to the ellipse x2/9 + y2/4 = 1 at any point P on the ellipse in the first quadrant. Let xT and yT be the - and -intercepts of T and xN and yN be the intercepts of N. As P moves along the ellipse in the first quadrant (but not on the axes), what values can xT, yT, xN, and yN take on? First try to guess the answers just by looking at the figure. Then use calculus to solve the problem and see how good your intuition is. |
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| m97991 | Match the graph of each function in (a)–(d) with the graph of its derivative in I–IV. Give reasons for your choices.
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| m98012 | Produce graphs of f that reveal all the important aspects of the curve. In particular, you should use graphs of f and f’’ to estimate the intervals of increase and decrease, extreme values, intervals of concavity, and inflection points.
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| m98014 | Prove the formulas given in Table 6 for the derivatives of the following functions.
(a) Csch–1
(b) Tanh–1
(c) Csch–1
(d) Sech–1
(e) Coth–1 |
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| m98079 | The figure shows two circles C and D of radius 1 that touch at P.T is a common tangent line; C1 is the circle that touches C, D, and T; C2 is the circle that touches C, D, and C1; C3 is the circle that touches C, D, and C2. This procedure can be continued indefinitely and produces an infinite sequence of circles {C2}. Find an expression for the diameter of Cn and thus provide another geometric demonstration of Example 6.
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| m98120 | The Pacific halibut fishery has been modeled by the differential equation dy/dt = ky (1 – y/K) where y (t) is the biomass (the total mass of the members of the population) in kilograms at time (measured in years), the carrying capacity is estimated to be K = 8 X 107 kg, and k = 0.71 per year.
(a) If y (0) = 2 X 107 kg, find the biomass a year later.
(b) How long will it take for the biomass to reach 4 X 107kg? |
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| m98132 | The rate of change of atmospheric pressure P with respect to altitude is proportional to P, provided that the temperature is constant. At 15oC the pressure is 101.3 kPa at sea level and 87.14 kPa at m.
(a) What is the pressure at an altitude of 3000 m?
(b) What is the pressure at the top of Mount McKinley, at an altitude of 6187 m? |
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| m98134 | The table gives the population of the United States, in millions, for the years 1900–2000. Use a graphing calculator with exponential regression capability to model the U.S. population since 1900. Use the model to estimate the population in 1925 and to predict the population in the years 2010 and 2020.
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