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mathematics
precalculus

Questions and Answers of Precalculus

Find the magnitude of the resultant force and the angle it makes with the positive x-axis. Уд 20 lb 45° 30° х 16 lb
A quarterback throws a football with angle of elevation 40° and speed 60 ft/s. Find the horizontal and vertical components of the velocity vector.
If a child pulls a sled through the snow on a level path with a force of 50 N exerted at an angle of 38 8 above the horizontal, find the horizontal and vertical components of the force.
If v lies in the first quadrant and makes an angle π/3 with the positive x-axis and |v| = 4, find v in component form.
What is the angle between the given vector and the positive direction of the x-axis?8i + 6j
What is the angle between the given vector and the positive direction of the x-axis?i + √3 j
Find the vector that has the same direction as (6, 2, -3) but has length 4.
Find a unit vector that has the same direction as the given vector.8i - j + 4k
Find a unit vector that has the same direction as the given vector.-5i + 3j - k
Find a unit vector that has the same direction as the given vector. (6, –2)
Find a + b, 4a + 2b, |a |, and |a - b|. (8, 1, —4), Ь — (5, — 2, 1)
Find a + b, 4a + 2b, |a |, and |a - b|. + 2k, b= 2i – 4k a = 4i – 3j
Find a + b, 4a + 2b, |a|, and |a - b|. b = -i - 2j a = 5i + 3j,
Find a + b, 4a + 2b, |a|, and |a - b|. а — (-3, 4), Ь — (9, —1}
Find the sum of the given vectors and illustrate geometrically. (1, 3, –2), (0, 0, 6)
Find the sum of the given vectors and illustrate geometrically. (3, 0, 1), (0, 8, 0)
Find the sum of the given vectors and illustrate geometrically. (3, – 1), (-1,5)
Find the sum of the given vectors and illustrate geometrically. (-1,4) ), (6, –2)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(0, 6, -1), B(3, 4, 4)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(0, 3, 1), B(2, 3, -1)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(3, 2), B(1, 0)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(3, -1), B(2, 3)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(-5, -1), B(-3, 3)
Find a vector a with representation given by the directed line segment AB(vector). Draw AB(vector) and the equivalent representation starting at the origin.A(-2, 1), B(1, 2)
If the vectors in the figure satisfy |u| = |v| = 1 and u + v + w = 0, what is |w|? u W V
In the figure, the tip of c and the tail of d are both the midpoint of QR. Express c and d in terms of a and b. a R
Copy the vectors in the figure and use them to draw the following vectors. (b) а — Ь (d) — ЗЬ (f) 2b – a (а) а + Ь (c) a (e) a + 2b b
Copy the vectors in the figure and use them to draw the following vectors. (a) u + v (b) u + w (d) u - v (f) u - w – v (c) v + w (e) v + u + w
Write each combination of vectors as a single vector. (a) AB + BĆ (b) CD + DB (d) DC (c) DB – AB + CÁ + AB D
Name all the equal vectors in the parallelogram shown. A, D
What is the relationship between the point (4, 7) and the vector k4, 7 l? Illustrate with a sketch.
Are the following quantities vectors or scalars? Explain.(a) The cost of a theater ticket(b) The current in a river(c) The initial flight path from Houston to Dallas(d) The population of the world
Describe and sketch a solid with the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a
Find the distance between the spheresand .2 x² + y? + z? = 4 x² + y? + z? = 4x + 4y + 4z – 11.
Find the volume of the solid that lies inside both of the spheresand x? + y? + z? + 4x – 2y + 4z + 5 = 0 x² + y? + z? = 4 .2 ,2
Find an equation of the set of all points equidistant from the points A(-1, 5, 3) and B(6, 2, -2). Describe the set.
Consider the points P such that the distance from P to A(-1, 5, 3) is twice the distance from P to B(6, 2, -2). Show that the set of all such points is a sphere, and find its center and radius.
Write inequalities to describe the region.The solid upper hemisphere of the sphere of radius 2 centered at the origin
Write inequalities to describe the region.The region consisting of all points between (but not on) the spheres of radius r and R centered at the origin, where r < R
Write inequalities to describe the region.The solid cylinder that lies on or below the plane z = 8 and on or above the disk in the xy-plane with center the origin and radius 2
Write inequalities to describe the region.The region between the yz-plane and the vertical plane x = 5
Describe in words the region of R3 represented by the equation(s) or inequality. x² + y? + z? > 2z
Describe in words the region of R3 represented by the equation(s) or inequality. x? + z? < 9
Describe in words the region of R3 represented by the equation(s) or inequality.x = z
Describe in words the region of R3 represented by the equation(s) or inequality. |1
Describe in words the region of R3 represented by the equation(s) or inequality. x? + y? + z? < 4 4
Describe in words the region of R3 represented by the equation(s) or inequality. x² + y? + z? = 4
Describe in words the region of R3 represented by the equation(s) or inequality. x² + y? = 4|
Describe in words the region of R3 represented by the equation(s) or inequality. z = -1 x? + y? = 4,
Describe in words the region of R3 represented by the equation(s) or inequality.y2 = 4
Describe in words the region of R3 represented by the equation(s) or inequality.0 < z < 6
Describe in words the region of R3 represented by the equation(s) or inequality.z > -1
Describe in words the region of R3 represented by the equation(s) or inequality.y = -2
Describe in words the region of R3 represented by the equation(s) or inequality.y < 8
Describe in words the region of R3 represented by the equation(s) or inequality.x = 5
Find an equation of the largest sphere with center (5, 4, 9) that is contained in the first octant.
Find equations of the spheres with center (2, -3, 6) that touch(a) The xy-plane, (b) The yz-plane, (c) The xz-plane.
Find an equation of a sphere if one of its diameters has endpoints (5, 4, 3) and (1, 6, -9).
(a) Prove that the midpoint of the line segment from(b) Find the lengths of the medians of the triangle with vertices A(1, 2, 3), B(-2, 0, 5), and C(4, 1, 5). (A median of a triangle is a line
Show that the equation represents a sphere, and find its center and radius. 3x? + 3y? + 3z? = 10 + 6y + 12z
Show that the equation represents a sphere, and find its center and radius. 2x? + 2y? + 2z² = 8x – 24z + 1
Show that the equation represents a sphere, and find its center and radius. x? + y? + z? + 8x – 6y + 2z + 17 = 0
Show that the equation represents a sphere, and find its center and radius. x² + y? + z? – 2x – 4y + 8z = 15
Find an equation of the sphere that passes through the origin and whose center is (1, 2, 3).
Find an equation of the sphere that passes through the point (4, 3, -1) and has center (3, 8, 1).
Find an equation of the sphere with center (2, -6, 4) and radius 5. Describe its intersection with each of the coordinate planes.
Find an equation of the sphere with center (-3, 2, 5) and radius 4. What is the intersection of this sphere with the yz-plane?
Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(2, – 1, 0), Q(4, 1, 1), R(4, –5, 4)
Find the lengths of the sides of the triangle PQR. Is it a right triangle? Is it an isosceles triangle? P(3, –2, –3), Q(7, 0, 1), R(1, 2, 1)
Describe and sketch the surface in R3 represented by the equation x2 + z2 = 9.
Describe and sketch the surface in R3 represented by the equation x + y = 2.
What does the equation y = 3 represent in R3? What does z = 5 represent? What does the pair of equations y = 3, z = 5 represent? In other words, describe the set of points (x, y, z) such that y = 3
What does the equation x = 4 represent in R2? What does it represent in R3? Illustrate with sketches.
What are the projections of the point (2, 3, 5) on the xy-, yz-, and xz-planes? Draw a rectangular box with the origin and (2, 3, 5) as opposite vertices and with its faces parallel to the coordinate
Which of the points A(-4, 0, -1), B(3, 1, -5), and C(2, 4, 6) is closest to the yz-plane? Which point lies in the xz-plane?
Sketch the points (1, 5, 3), (0, 2, -3), (-3, 0, 2), and (2, -2, -1) on a single set of coordinate axes.
Suppose you start at the origin, move along the x-axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?
If f (x) = ex2, show that f(2n)(0) = (2n)!/n!.
Find the Maclaurin series for f and its radius of convergence.You may use either the direct method (definition of a Maclaurin series) or known series such as geometric series, binomial series, or the
Find the radius of convergence and interval of convergence of the series. x" Σ (-1) π n25" η-1
Prove that if the series Σ∞n=1 an is absolutely convergent, then the seriesis also absolutely convergent. п+1 Σ 00 an п п-1
a) Show that the series п" (2n)! п-1 п" lim (2n)! = 0.
(a) Find the partial sum s5 of the series Σ∞n=1 1/n6 and estimate the error in using it as an approximation to the sum of the series.(b) Find the sum of this series correct to five decimal places.
For what values of x does the series Σ∞n=1 (ln x)n converge?
Express the repeating decimal 4.17326326326 . . . as a fraction.
Find the sum of the series.
Find the sum of the series. 1 Σ п(п + 3) п-1 п(n
Determine whether the series is conditionally convergent, absolutely convergent, or divergent.
Determine whether the series is convergent or divergent.
Determine whether the series is convergent or divergent. 00 п In 3n + п31
Determine whether the series is convergent or divergent. 00 Σ n=2 n/In n
Determine whether the series is convergent or divergent.
Determine whether the series is convergent or divergent.
Determine whether the series is convergent or divergent. n2 + 1 n=1 N° + 1
Determine whether the series is convergent or divergent. 00 п Σ 3 + 1 п-1 п' +
Determine whether the sequence is convergent or divergent. If it is convergent, find its limit.
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If an > 0 and ∑an converges, then ∑
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If {an} is decreasing and an > 0 for all
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If {an} and {bn} are divergent, then {anbn}
Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.If {an} and {bn} are divergent, then {an +

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