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calculus

Questions and Answers of Calculus

Use Problem 32 and the anti commutative law to prove the right distributive law. u × (v + w) = (u × v) + (u × w)
Use Example 3 to develop a formula for the area of the triangle with vertices P(a, 0, 0), Q(0, b, 0), Q(0, b, 0), and R(0, 0, c) shown in the left half of Figure 6.
Show that the triangle in the plane with vertices (x1, y1), (x2, y2), and (x3, y3) has area equal to one-half the absolute value of the determinant
Let vectors a, b, and c with common initial point determine a tetrahedron, and let m, n, p, and q be vectors perpendicular to the four faces, pointing outward, and having length equal to the area of
Let a, b, and a - b denote the three edges of a triangle with lengths a, b, and c, respectively. Use Lagrange's Identity together with 2a ˆ™ b = ||a||2 + ||b||2 - ||a - b||2 to prove
Find all vectors perpendicular to both of the vector a = -2i + 5j - 2k and b = 3i - 2k + 4k.
Find all vectors perpendicular to the plane determined by the three points (1, 3, 5), (3, - 1, 2), (4, 0, 1).
Find all vectors perpendicular to the plane determined by the three points (-1, 3, 0), (5, 1, 2), and (4, -3, -1).
Find the area of the parallelogram with a = -i + j - 3k and b = 4i + 2j - 4k as the adjacent sides.
Find the area of the parallelogram with a = 2i + 2j - k and b = -i + j - 4k as the adjacent sides.
Find the area of the triangle with (3, 2, 1), (2, 4, 6), and (-1, 2, 5) as vertices.
In Problem a-c, find the required limit or indicate that it does not exist.a.b. c.
State the domain of each of the following vector-valued functions:a. r(t) = ln(t - 1)i + ˆš 20 - t jb. r(t) = ln(t-1)i + tan-1 tj + t kc.
For what values of t is each function in Problem 9 continuous?a.b. (|| ]Denotes the greatest integer function.) c.
For what values of t is each function in Problem 10 continuous?a. r(t) = ln(t - 1)i + ˆš 20 - t jb. r(t) = ln(t-1)i + tan-1 tj + t kc.
Find Dt r(t) and Dt2 r(t) for each of the following: a. r(t) = (3t + 4)3i + et2j + k b. r(t) = sin2t i + cos 3t j + t2k
Find r'(t) and r''(t) for each of the following: a. r(t) = (et + e-t2)i + 2tj + t k b. r(t) = tan 2t I + arctan t j
If r(t) = e-tj - ln(t2)j, find Dt[r(t)] ∙ r''(t)].
If r(t) = sin 3t i - cos 3t j, find Dt[r(t) ∙ r' (t)].
If r(t) = √t - 1 i + ln(2t2)j and h(t) = e-3t, find Dt[h(t)r(t)].
If r(t) = sin 2t i + cosh t j and h(t) = ln (3t - 2), find Dt[h(t)r(t)].
In Problem a-b, find the velocity v, acceleration a, and speed s at the indicated time t = t1. a. r(t) = 4ti + 5(t2 - 1)j + 2tk; t1 = 1 b. r(t) = ti + (t - 1)2j + (t - 3)3k; t1 = 0
Show that if the speed of a moving particle is constant its acceleration vector is always perpendicular to its velocity vector.
Prove that ||r(t)|| is constant if and only if r(t) ∙ r'(t) = 0.
In Problems a to b, find the length of the curve with the given vector equation. a. r(t) = t i + sin t j + cos t k; 0 ≤ t ≤ 2 b. r(t) = t cos t i + t sin t j + √2t k; 0 ≤ t ≤ 2
In Problems a-b, F(t) = f(u(t)), Find F'(t) in terms of t. a. f(u) = cos u i + e3uj and u(t) = 3t2 - 4 b. f(u) = u2i + sin2 uj and u(t) = tan t
Evaluate the integrals in Problems a to b. a. ∫01(eti + e-tj) dt b. ∫-11[(1 + t)3/2i + (1 - t)3/2j] dt
A point moves around the circle x2 + y2 = 25 at constant angular speed of 6 radians per second starting at (5, 0). Find expressions for r(t), v(t), ||v(t)||, and a(t).
Consider the motion of a particle along a helix given by r(t) = sin t i + cos t j + (t2 - 3t + 2)k, where the k component measures the height in meters above the ground and t ≥ 0. a. Does the
Describe in general terms the following "helical'' type motions: a. r(t) = sin t i + cos t j + tk b. r(t) = sin t3 i + cos t3 j + t3 k c. r(t) = sin (t3 + π) i + t3 j + cos(t3 + π)k d. r(t) = t sin
When no domain is given in the definition of a vector-valued function, it is to be understood that the domain is the set of all (real) scalars for which the rule for the function makes sense and
In Problem a-c, find the parametric equations of the line through the given pair of points. a. (1, -2, 3), (4, 5, 6) b. (2, -1, -5), (7, -2, 3) c. (4, 2, 3), (6, 2, -1) d. (5, -3, -3), (5, 4, 2)
Find the parametric equations of the line through (5, -3, 4) that intersects the z-axis at a right angle.
Find the symmetric equations of the line through (2, -4, 5) that is parallel to the plane 3x + y - 2z = 5 and perpendicular to the line
Find the equation of the plane that contains the parallel linesAnd
Show that the linesAnd Interest, and find the equation of the plane that they determine.
Find the equation of the plane containing the line x = 1 + 2t, y = -1 + 3t, z = 4 + t and the point (1, -1, 5).
Find the equation of the plane containing the line x = 3t, y = 1 + t, z = 2t and parallel to the intersection of the planes 2x - y + z = 0 and y + z + 1 = 0.
Find the distance between the skew lines x = 1 + 2t, y = -3 + 4t, z = -1 - t and x = 4 -2t, y = 1 + 3t, z = 2t
Find the symmetric equations of the tangent line to the curve with equation r(t) = 2 cos t i + 6 sub t j + t k at t = π/3.
Find the parametric equations of the tangent line to the curve x = 2t2, y = 4t, z = t3 at t = 1.
Find the equations of the plane perpendicular to the curve x = 3t, y = 2t2, z = t5 at t = -1.
Find the equation of the plane perpendicular to the curve r(t) = t sin t i + 3t j + 2t cos t k at t = π / 2.
Consider the curvea. Show that the curve lies on a sphere centered at the origin. b. Where does the tangent line at t = 1/4 intersect the xz-plane?
Consider the curve r(t) = 2r i + t2j + (1 - t2)k a. Show that this curve lies on a plane and find the equation of this plane. b. Where does the tangent live at t = 2 intersect the xy-plane?
Let P and Q be points on nonintersecting skew lines with directions n1 and n2, and let n = n1 Ã— n2 Figure 8. Show that the distance d between these lines is given byAnd use this result
In Problems a-c, write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. a. (4, 5, 6), (3, 2, 1) b. (-1, 3, -6), (-2, 0,
In Problems a-b, find the symmetric equations of the line of intersection of the given pair of planers. a. 4x + 3y - 7z = 1, 10x + 6y - 5z = 10 b. x + y - z = 2, 3x - 2y + z = 3
Sketch the curve over the indicated domain for t. Find v, a, T, and k at the point where t = t1. r(t) = t i + t2j; 0 ≤ t ≤ 2; t1 = 1.
Find the curvature k, the unit tangent vector T, the unit normal vector N, and the binormal vector B at t = t1. r(t) = 1/2ti + t j + 1/3 t3k; t1 = 2
Find the point of the curve at which the curvature is a maximum. y = ln x
In Problems a-b, find the tangential and normal components (aT and aN) of the acceleration vector at t. Then evaluate at t = t1. a. r(t) = 3t i + 3t2j; t1 = 1/3 b. r(t) = t2 + i + tj; t1 = 1
Sketch the path for a particle if its position vector is r = sin t i + sin 2t j, 0 ‰¤ t ‰¤ 2Ï€ (you should get a figure eight). Where is the acceleration zero? Where
The position vector of a particle at time t ≥ 0 is r(t) = (cos t + t sin t)i + (sin t - t cos t)j a. Show that the speed ds/dt = t. b. Show that aT = 1 and aN = 1.
If, for a particle, aT = 0 for all t, what can you conclude about its speed? If aN = 0 for all t, what can you conclude about its curvature?
Find N for the ellipse r(t) = a cos (t i + b sin (t j
An object moves along the curve y = sin 2x. Without doing any calculating, decide where aN = 0
An object moves along the parabola y = x2 with constant speed of 4. Express a at the point (x, x2) in terms of T and N.
A car traveling at constant speed v rounds a level curve, which we take to be a circle of radius R, If the car is to avoid sliding outward, the horizontal frictional force F exerted by the road on
Consider again the car of Problem 61. Suppose that the curve is icy at its worst spot (Î¼ = 0), but is banked at angle Î¸ from the horizontal Figure 14. Let F be the force
Demonstrate that the second formula in Theorem a Can also be written as k = | y'' cos3 ϕ, where ϕ is the angle of inclination of the tangent line to the graph of y = F(x).
Show that for a plane curve N points to the concave side of the curve. One method is to show thatThen consider the cases dÏ•/ds > 0 (curve bends to the left) and dÏ•/ds
Prove that N = B × T. Derive a similar result for T in terms of N and B.
Show that the curveHas continuous first derivatives and curvature at all points.
Find a curve given by a polynomial P5(x) that provides a smooth transition between y = 0 for x ≤ 0 and y = x for x ≥ 1.
Derive the polar coordinate curvature formulaWhere is the derivatives are with respect to Î¸.
In Problems a to c, use the formula in Problem 69 to find the curvature k of the following:a. Circle: r = 4 cos Î¸ b. Cardioid: r = 1 + cos Î¸ at Î¸ = 0 c. r =
Show that the curvature of the polar curve r = e6θ is proportional to 1/r.
Show that the curvature of the polar curve r2 = cos 2θ is directly proportional to r for r > 0.
Derive the first curvature formula in Theorem A by working directly with k = ||T'(t)|| / ||r'(t)||.
Draw the graph of x = 4 cos t, y = 3 sin(t + 0.5), 0 ≤ t ≤ 2 2π. Estimate its maximum and minimum curvature by looking at the graph (curvature) is the reciprocal of the radius of curvature).
Show that the unit binormal vector B = T × N has the property that dB/ds is perpendicular to B.
Show that unit binormal vector B = T × N has the property that dB/ds is perpendicular to T.
Shot that for a plane curve the torsion is τ(s) = 0.
Show that for a straight line r(t) = r0 + a0ti + b0tj + c0t k both k and τ are zero.
A fly is crawling along a wire helix so that its position vector is r(t) = 6 cos πt i + 6 sin πt j + 2t k, t ≥ 0. At what point will the fly hit the sphere x2 + y2 + z2 = 100. And how far did it
The DNA molecule in humans is a double helix, each with about 2.9 × 108 complete turns. Each helix has radius about 10 angstroms and rises about 34 angstroms on each complete turn (an angstrom is
In Problems a-c, name and sketch the graph of each of the following equations in three-space. a. 4x2 + 36y = 144 b. y2 + z2 = 15 c. 3x + 2z = 10
The graph of an equation x, y, and z is symmetric with respect to the xy-plane if replacing z by - z results in an equivalent equation. What condition leads to a graph that symmetric with respect to
What condition leads to a graph that is symmetric with respect to the following? a. xz-plane b. y-axis c. x-axis
If the curve z = x2 in the xz-plane is revolved about the z-axis the resulting surface has equation z = x2 + y2, obtained as a result of replacing x by √x2 + y2. If y = 2x2 in the xy-plane is
Find the equation of the surface that results when the curve z = 2y in the yz-plane is revolved about the z-axis.
Find the equation of the surface that result when the curve 4x2 + 3y2 = 12 in the xy-plane is revolved about the y-axis.
Find the equation of the surface that result when curve 4x2 - 3y2 = 12 in the xy-plane is revolved about the x-axis.
Find the coordinates of the foci of the ellipse that is the intersection of z = x2 / 4 + y2 / 9 with the plane z = 4.
Find the coordinates of the focus of the parabola that is the intersection of z = x2 / 4 + y2 / 9 with x = 4.
Find the area of the elliptical cross section cut from the surface x2/a2 + y2/b2 + z2/c2 = 1 by the plane z = h -c < h < c, Recall: The area of the ellipse x2/A2 + y2/B2 = 1 is πAB.
Show that the volume of the solid bounded by the elliptic paraboloid x2/a2 + y2/b2 = h - z, h > 0, and the xy-plane is πabh2/2, that is, the volume is one-half the area of the base times the height.
Show that the projection in the xz-plane of the curve that is the intersection of the surfaces y = 4 - x2 and y = x2 + z2 is an ellipse, and find its major and minor diameters.
Sketch the triangle in the plane y = x that is above the plane z = y/2, below the plane z = 2y, and inside the cylinder x2 + y2 = 8. Then find the area of this triangle.
Show that the spiral r = i cos t i + t sin t j + t k lies on the circular cone x2 + y2 - z2 = 0. On what surface does spiral r = 3t cos t i + t sin t j + t k lie?
Show that the curve determined by r = t i + t j + t2 k is a parabola, and find the coordinates of its focus.
Make a table, like the one just before Example 4, that gives the relationships between cylindrical and spherical coordinates.
In Problems a to d, make the required change in the given equation. a. x2 + y2 = 9 to cylindrical coordinates b. x2 - y2 = 25 to cylindrical coordinates c. x2 + y2 + 4z2 = 10 cylindrical
Change the following from cylindrical to spherical coordinates, a. (1, π/2, 1) b. (-2, π/4, 2)
Change the following from cylindrical to Cartesian (rectangular) coordinates, a. (6, π/6, -2) b. (4, 4π/3, -8)
Find the great-circle distance from St. Paul (longitude 9.31o W, latitude 45o N) to Oslo (longitude 10.5o E, latitude 59.6o N)
Find the great-circle distance from New York (longitude 74o W, latitude 40.4o N) to Greenwich (longitude 0o, latitude 51.3o N)
Find the great-circle distance from St. Paul (longitude 93.1o W, latitude 45o N) to Turin, Italy (longitude 7.4o, latitude 45o N)
What is the distance along the 45o parallel between St. Paul and Turin?

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