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| m97688 | Write the converse and the contrapositive to the following statements.
(a) (Let a, b, and c be the lengths of sides of a triangle.) If a2 b2 = c2, then the triangle is a right triangle.
(b) If angle ABC is acute, then its measure is greater than 0° and less than 90°. |
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| m97690 | Write the equation of the line in Problem 16 in parametric form.
Find the points where the line of intersection of the planes x - 2y + 4z - 14 = 0 and -x + 2y - 5z + 30 = 0 pierces the yz- and xz-planes. |
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| m97691 | Write the equation of the line through (- 2, 1) that
(a) Goes through (7, 3);
(b) Is parallel to 3x - 2y = 5;
(c) Is perpendicular to 3x + 4y = 9;
(d) Is perpendicular to y = 4;
(e) Has y-intercept 3. |
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| m97692 | Write the equation of the plane through the point (-5, 7, -2) that satisfies each condition. a. Parallel to the xz-plane
b. Perpendicular to the x-axis
c. Parallel to both the x- and y-axes
d. Parallel to the plane 3x - 4y + z = 7 |
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| m97693 | Write the equation of the sphere with the given center and radius.
a. (1, 2, 3);5
b. (-2, -3, -6); √5
c. (π, e, √2); √π |
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| m97694 | Write the first two terms in the expansions of the following:
(a) (a + b)3
(b) (a + b)4
(c) (a + b)5 |
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| m97696 | Write the following Cartesian equations in cylindrical co-ordinate form.
a. x2 + y2 = 9
b. x2 + 4y2 = 16
c) x2 + y2 = 9z
d. x2 + y2 + 4z2 = 10 |
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| m97697 | Write the following equations in spherical coordinate form.
a. x2 + y2 + z2 = 4
b. 2x2 + 2y2 - 2z2 = 0
c. x2 - y2 - z2 = 1
d. x2 + y2 = z |
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| m97698 | Write the Maclaurin series for f(x) = sin x + cos x. For what values of x does it represent f? |
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| m97702 | Write the triple iterated integrals for the volume of a sphere of radius a in each case.
(a) Cartesian coordinates
(b) Cylindrical coordinates
(c) Spherical coordinates |
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| m97750 | You suspect that the xy-data that you collect lie on either an exponential curve y = Abx or a power curve y = Cxd. To check, you plot ln y against x on one graph and ln y against ln x on another graph. Explain how these graphs will help you to come to a conclusion. |
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| m97754 | Your computer algebra system (CAS) may allow the use of parameters in defining functions. In each case, draw the graph of y = f(x) for the specified values of the parameter k, using the same axes and - 5 ≤ x ≤ 5.
(a) f(x) = |k, x|0.7 for k = 1, 2, 0.5, and 0.2.
(b) f(x) = |x - k|0.7 for k = 0, 2, - 0.5, and - 3.
(c) f(x) = |x|k for k = 0.4, 0.7, 1, and 1.7. |
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| m97784 | Let I := [0; π/2] and let f : I → R be defined by f(x) := sup{x2, cos x} for x ∈ I. Show there exists an absolute minimum point x0 ∈ I for f on I. Show that x0 is a solution to the equation cos x = x2. |
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| m97785 | Let I := [0; π/2] and let f : I → R be defined by f(x) := sup{x2, cos x} for x ∈ I. Show there exists an absolute minimum point x0 ∈ I for f on I. Show that x0 is a solution to the equation cos x = x2. |
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| m97792 | Let f be continuous, increasing, and concave down on [1, () as in Figure 4. Furthermore, let An be the area of the shaded region. Show that An is increasing with n, that An ( T where T is the area of the outlined triangle, and thus that exits.
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| m97793 | Let f be continuous, increasing, and concave down on [1, () as in Figure 4. Furthermore, let An be the area of the shaded region. Show that An is increasing with n, that An ( T where T is the area of the outlined triangle, and thus that exits.
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| m97794 | Let f be continuous, increasing, and concave down on [1, () as in Figure 4. Furthermore, let An be the area of the shaded region. Show that An is increasing with n, that An ( T where T is the area of the outlined triangle, and thus that exits.
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| m97806 | Show this as indicated. Compare the amount of work.
(a) ey (sin h x dx + cos h x dy) = 0 as an exact ODE and by separation.
(b) (1 + 2x) cos y dx + dy/cos y = 0 by Theorem 2 and by separation.
(c) (x2 + y2) dx – 2xy dy = 0 by Theorem 1 or 2 and by separation with v = y/x.
(d) 3x2 y dx – 4x3 dy = 0 by Theorems 1 and 2 and by separation.
(e) Search the text and the problems for further ODE’s that can be solved by more than one of the methods discussed so far. Make a list these ODEs. Find further cases of your own. |
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| m97920 | Estimate the errors involved in Exercise 63, parts (a) and (b). How large should be in each case to guarantee an error of less than 0.00001? |
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| m97921 | Evaluate
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