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mathematics
precalculus

Questions and Answers of Precalculus

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (sin πt, t, cos πt)
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (t, 2 - t, 2t)
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (t2 - 1, t)
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (sin t, t)
Find the limit. t3 + t 1 t sin lim ( te, 213 – 1
Find the limit. 1 + t? lim 2t tan't, - 12 1 - t 00
Find the limit. sin Tt -k In t i + vt + 8 j + lim t - 1
Find the limit. j + cos 2t k sin?t lim (e-3i i + -3' i
Find the domain of the vector function.r(t) = cos t i + ln t j + 1/t - 2k
Find the domain of the vector function. r(t) = ( In(t + 1), · V9 – t2 2'
A solid has 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 square. If the rays are
Find an equation of the largest sphere that passes through the point (-1, 1, 4) and is such that each of the points (x, y, z) inside the sphere satisfies the condition x? + y? + z? < 136 + 2(x + 2y +
A plane is capable of flying at a speed of 180 km/h in still air. The pilot takes off from an airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the
Let L be the line of intersection of the planes cx + y + z = c and x - cy + cz = -1, where c is a real number.(a) Find symmetric equations for L.(b) As the number c varies, the line L sweeps out a
Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L, W, and H.
Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each
A surface consists of all points P such that the distance from P to the plane y = 1 is twice the distance from P to the point (0, -1, 0). Find an equation for this surface and identify it.
An ellipsoid is created by rotating the ellipse 4x2 + y2 = 16 about the x-axis. Find an equation of the ellipsoid.
Identify and sketch the graph of each surface.x = y2 + z2 - 2y - 4z + 5
Identify and sketch the graph of each surface.4x2 + 4y2 - 8y + z2 = 0
Identify and sketch the graph of each surface.y2 + z2 − 1 + x2
Identify and sketch the graph of each surface.-4x2 + y2 - 4z2 = 4
Identify and sketch the graph of each surface.4x - y + 2z = 4
Identify and sketch the graph of each surface.x2 = y2 + 4z2
Identify and sketch the graph of each surface.y = z2
Identify and sketch the graph of each surface.x = z
Identify and sketch the graph of each surface.x = 3
Find the distance between the planes 3x + y - 4z = 2 and 3x + y - 4z = 24.
(a) Find an equation of the plane that passes through the points A(2, 1, 1), B(-1, -1, 10), and C(1, 3, -4).(b) Find symmetric equations for the line through B that is perpendicular to the plane in
Find an equation of the plane through the line of intersection of the planes x - z = 1 and y + 2z = 3 and perpendicular to the plane x + y - 2z = 1.
(a) Show that the planes x + y - z = 1 and 2x - 3y + 4z = 5 are neither parallel nor perpendicular.(b) Find, correct to the nearest degree, the angle between these planes.
Determine whether the lines given by the symmetric equationsx - 1/2 = y - 2/3 = z - 3/4 and x + 1/6 = y - 3/-1 = z + 5/2 are parallel, skew, or intersecting.
Find the distance from the origin to the linex = 1 + t, y = 2 - t, z = -1 + 2t.
Find the point in which the line with parametric equations x = 2 - t, y = 1 + 3t, z = 4t intersects the plane 2x - y + z = 2.
Find an equation of the plane.The plane through (1, 2, -2) that contains the line x = 2t, y = 3 - t, z = 1 + 3t
Find an equation of the plane.The plane through (3, -1, 1), (4, 0, 2), and (6, 3, 1)
Find an equation of the plane.The plane through (2, 1, 0) and parallel to x + 4y - 3z = 1
Find parametric equations for the line.The line through (-2, 2, 4) and perpendicular to the plane 2x - y + 5z = 12
Find parametric equations for the line.The line through (1, 0, -1) and parallel to the line 1/3 (x - 4) = 1/2y = z + 2
Find parametric equations for the line.The line through (4, 21, 2) and (1, 1, 5)
Find the magnitude of the torque about P if a 50-N force is applied as shown. 50 N 30° 40 cm Pl
A boat is pulled onto shore using two ropes, as shown in the diagram. If a force of 255 N is needed, find the magnitude of the force in each rope. 20° 255 N 30°
A constant force F = 3i + 5j + 10k moves an object along the line segment from (1, 0, 2) to (5, 3, 8). Find the work done if the distance is measured in meters and the force in newtons.
(a) Find a vector perpendicular to the plane through the points A(1, 0, 0), B(2, 0, -1), and C(1, 4, 3).(b) Find the area of triangle ABC.
Given the points A(1, 0, 1), B(2, 3, 0), C(-1, 1, 4), and D(0, 3, 2), find the volume of the parallelepiped with adjacent edges AB, AC, and AD.
Find the acute angle between two diagonals of a cube.
Show that if a, b, and c are in V3, then(a x b) • [(b x c) x (c x a)] = [a • (b x c)]2
Suppose that u • (v x w) = 2. Find(a) (u x v) • w (b) u • (w x v)(c) v • (u x w) (d) (u x v) • v
Find two unit vectors that are orthogonal to both j + 2k and i - 2j + 3k.
Find the values of x such that the vectors (3, 2, x) and (2x, 4, x) are orthogonal.
If u and v are the vectors shown in the figure, find u • v and |u x v|. Is u x v directed into the page or out of it? |v|= 3 45° |u|=2
Copy the vectors in the figure and use them to draw each of the following vectors.(a) a + b (b) a - b (c) -1/2a (d) 2a + b a
(a) Find an equation of the sphere that passes through the point (6, -2, 3) and has center (-1, 2, 1).(b) Find the curve in which this sphere intersects the yz-plane.(c) Find the center and radius of
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 u and v are in V3, then |u • v | <
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 u • v = 0 and u x v = 0, then u = 0 or v
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 u x v = 0, then u = 0 or v = 0.
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 u • v = 0, then u = 0 or v = 0.
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.In R3 the graph of y = x2 is a paraboloid.
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.The set of points h(x, y, z) |x2 + y2 = 1} is
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.A linear equation Ax + By + Cz + D = 0
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.The vector (3, -1, 2) is parallel to the
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.For any vectors u and v in V3, (u + v) x v =
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.For any vectors u and v in V3, (u x v) • u
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.For any vectors u, v, and w in V3, u x (v x
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.For any vectors u, v, and w in V3, u • (v x
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.For any vectors u, v, and w in V3, (u + v) x
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.For any vectors u and v in V3 and any scalar
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.For any vectors u and v in V3 and any scalar
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.For any vectors u and v in V3, |u x v| = |v x
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.For any vectors u and v in V3, u x v = v x u.
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.For any vectors u and v in V3, u • v = v
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.For any vectors u and v in V3, |u x v| =
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.For any vectors u and v in V3, |u • v | =
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.For any vectors u and v in V3, |u + v | = |u|
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 u = (u1, u2) and v = (v1, v2) , then u •
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.y = x2 - z2 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.x2 + 2z2 = 1 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.y2 = x2 + 2z2 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.y = 2x2 + z2 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.2x2 + y2 - z2 = 1 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.9x2 + 4y2 + z2 = 1 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.x2 - y2 + z2 − 1 п XA Ш IV VI хи х. VII VIII
Match the equation with its graph (labeled I–VIII). Give reasons for your choice.x2 + 4y2 + 9z2 = 1 п XA Ш IV VI хи х. VII VIII
Use traces to sketch and identify the surface.x = y2 - z2
Use traces to sketch and identify the surface.y = z2 - x2
Use traces to sketch and identify the surface.3x2 - y2 + 3z2 = 0
Use traces to sketch and identify the surface.x2/9 + y2/25 + z2/4 = 1
Use traces to sketch and identify the surface.3x2 + y + 3z2 = 0
Use traces to sketch and identify the surface.9y2 + 4z2 = x2 + 36
Use traces to sketch and identify the surface.z2 - 4x2 - y2 = 4
Use traces to sketch and identify the surface.x2 = 4y2 + z2
Use traces to sketch and identify the surface.4x2 + 9y2 + 9z2 = 36
Use traces to sketch and identify the surface.x = y2 + 4z2
(a) Find and identify the traces of the quadric surface 2x2 - y2 + z2 = 1 and explain why the graph looks like the graph of the hyperboloid of two sheets in Table 1.(b) If the equation in part (a) is
(a) Find and identify the traces of the quadric surface x2 + y2 - z2 = 1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. (b) If we change the equation
Describe and sketch the surface.z = sin y
Describe and sketch the surface.xy = 1
Describe and sketch the surface.y = z2
Describe and sketch the surface.z = 1 - y2
Describe and sketch the surface.4x2 + y2 = 4

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