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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest degree.)2i – j + 2k
Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x – y2 + 4z2 = 0
Ropes 3 m and 5 m in length are fastened to a holiday decoration that is suspended over a town square. The decoration has a mass of 5 kg. The ropes, fastened at different heights, makeangles of 52°
Find an equation of the plane.The plane through the origin and the points (2, –4, 6) and (5. 1, 3)
(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) Find the area of triangle PQR.P(–1, 3, 1), Q(0, 5, 2), R(4, 3, –1)
Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 + y2 + 4z2 – 4y – 24z + 36 = 0
Find an equation of the plane.The plane through the points (3, –1,2), (8, 2, 4), and (–1, –2, –3)
A clothesline is tied between two poles, 8 m apart. The line is quite taut and has negligible sag. When a wet shirt with a mass of 0.8 kg is hung at the middle of the line, the midpoint is pulled
Find the volume of the parallelepiped determined by the vectors a, b, and c.a = 〈6, 3, –1〉, b = 〈0, 1, 2〉, c = 〈4, –2, 5〉
Find an equation of the plane.The plane that passes through the point (1, 2, 3) and contains the line x = 3t, y = 1 + t, z = 2 – t
Find the scalar and vector projections of b onto a.a = 〈3. –4〉, b = 〈5, 0〉
Find an equation of the plane.The plane that passes through the point (6, 0, –2) and contains the line x = 4 – 2t, y = 3 + 5t, z = 7 + 4t
Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS.P(2, 0, –1). Q(4, 1, 0). R(3, –1, 1), S(2, –2, 2)
Find the scalar and vector projections of b onto a.a = 〈1, 2〉, b = 〈–4, 1〉
Find an equation of the plane.The plane that passes through the point (1, –1, 1) and contains the line with symmetric equations x = 2y = 3z
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 – y2 + z2 – 2x + 2y + 4z + 2 = 0
(a) Find the unit vectors that are parallel to the tangent line to the curve y = 2 sin x at the point (π/6, 1). (b) Find the unit vectors that are perpendicular to the tangent line. (c) Sketch the
Find the scalar and vector projections of b onto a.a = 〈3, 6, –2〉, b = 〈1, 2, 3〉
Find an equation of the plane.The plane that passes through the point (–1, 2, 1) and contains the line of intersection of the planes x + y – z = 2 and 2x – y + 3z = 1
Find the scalar and vector projections of b onto a.a = 〈–2, 3, –6〉, b = 〈5, –1, 4〉
Let C be the point on the line segment AB that is twice as far from B as it is from A. If a = OA(vector), b = OB(vector), and c = OC(vector), show that c = 2/3a + 1/3b.
Find the scalar and vector projections of b onto a.a = 2i – j + 4k, b = j + 1/2k
Find the scalar and vector projections of b onto a.a = i + j + k, b = i – j + k
Find the point at which the line intersects the given plane.x = 3 – t, y = 2 + t, z = 5t; x – y + 2z = 9
Find an equation for the surface obtained by rotating the parabola y = x2 about the y-axis.
Find the point at which the line intersects the given plane.x = 1 + 2t, y = 4t, z = 2 – 3t; x + 2y – z + 1 = 0
Find an equation for the surface obtained by rotating the line x = 3y about the x-axis.
Find the point at which the line intersects the given plane.x = y –1 = 2z; 4x – y + 3z = 8
First make a substitution and then use integration by parts to evaluate the integral.∫cos √x dx
Find the centroid of the region bounded by the given curves.y = sin x, y = cos x, x = 0, x = π/4
Find the exact length of the polar curve.r = θ, 0 ≤ θ ≤ 2π
Graph the curve and find its length.r = cos2(θ/2)
Test the series for convergence or divergence. 00 Σ k=1 2*k! (k + 2)!
Where does the line through (1, 0, 1) and (4. –2, 2) intersect the plane x + y + z = 6?
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.2z = 4y – x, 3x – 12y + 6z = 1
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.x + y + z = 1, x – y + z = 1
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.2x – 3y + 4z = 5, x + 6y + 4z = 3
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.x = 4y – 2z. 8y = 1 + 2x + 4z
Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them.x + 2y + 2z = 1, 2x – y + 2z = 1
Find the domain of the vector function.kr(t) = 〈√4 – t2, e–3t, In(t + 1)〉
Find the domain of the vector function.r(t) = t – 2/t + 2 i + sin t j + ln(9 – t2)k
Find the limit. lim 1-0 et - 1 √1+t - 1 t t 3 1+t,
Find the limit. lim e-³¹₁+ 1² sin²t -j + cos 2tk
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
Find a vector equation and parametric equations for the line segment that joins P to Q.P(1, –1, 2), Q(4, 1, 7)
Find the length of the curve.r(t) = 12t i + 813/2j + 3t2k, 0 ≤ t ≤1
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases.r(t) = 〈sin t, t〉
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases.r(t) = 〈1 + t, 3t, –t〉
Find the derivative of the vector function.r(t) = i – j + e4tk
For the curve given by , find(a) The unit tangent vector(b) The unit normal vector(c) The curvature r(t) = ( 31³, 31², t),
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases.r(t) = 〈1, cos t, 2 sin t〉
Find the derivative of the vector function.r(t) = sin–1 ti + √1 – t2 j + k
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases.r(t) = t2i + tj + 2k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = t2i + ln t j + t k
Find the derivative of the vector function.r(t) = et2i – j + ln(1 + 3t) k
Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t.r(t) = 2t i + (1 – 3t)j + (5 + 4t) k
Find the derivative of the vector function.r(t) at cos 3ti + b sin3t j + c cos3t k
Reparametrize the curve with respect to arc length measured from the point where t = 0 in the direction of increasing t.r(t) = e2t cos 2t i + 2 j + e2t sin 2t k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = t sin t i + t cos t j + t2 k
Find a vector equation and parametric equations for the line segment that joins P to Q.kP(0, 0, 0). Q(1, 2, 3)
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.a(t) = i + 2 j, v(0) = k, r(0) = i
Find a vector equation and parametric equations for the line segment that joins P to Q.P(1, 0, 1). Q(2, 3, 1)
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.a(t) = 2i + 6tj + 12t2k, v(0) = i, r(0) = j – k
Find the unit tangent vector T(t) at the point with the given value of the parameter t.r(t) = 〈te–t, 2 arctan t, 2et〉, t = 0
Find the unit tangent vector T(t) at the point with the given value of the parameter t.r(t) = 4√ti + t2j + tk, t = 1
Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.x = cos 4t, y = t, z = sin 4t I III V of ZA A II x4 IV VI N Z ZA
Find a vector equation and parametric equations for the line segment that joins P to Q.P(–2, 4, 0), Q(6, –1, 2)
Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.x = t, y = t2, z = e–t I X- 8 V 00 ZA M II X4 IV VI N Z ZA
Find the unit tangent vector T(t) at the point with the given value of the parameter t.r(t) = 2 sin t i + 2 cos t j + tan t k, t = π/4
Use Theorem 10 to find the curvature.r(t) = t2 i + tkData from Theorem 10The curvature of the curve given by the vector function is k(t) r'(t) x r"(t) | |r' (t) ³
If r(t) = (t, t2, t3), find r'(t), T(1), r"(t), and r'(t) r"(t).
Use Theorem 10 to find the curvature.r(t) = t i + t j + (1 + t2) kData from Theorem 10The curvature of the curve given by the vector function is k(t) r'(t) x r"(t) | |r' (t) ³
Use Theorem 10 to find the curvature.r(t) = 3t i + 4 sin t j + 4 cos tkData from Theorem 10The curvature of the curve given by the vector function is k(t) r'(t) x r"(t) | |r' (t) ³
If r(t) = (e2t, e–2t, te2t), find T(0), r"(0), and r'(t) · r"(t). .
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.x = 1 + 2√t, y = t3 – t, z = t3 + t; (3, 0, 2)
Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.x = cos t, y = sin t, z – sin 5t I III V of ZA A II x4 IV VI N Z ZA
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.x = et, y = tet, z = tet2; (1, 0, 0)
Find the curvature of r(t) = 〈et cos t, et sin t, t〉 at the point (1, 0, 0).
Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices.x = cos t, y = sin t, z = ln t I III V of ZA A II x4 IV VI N Z ZA
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.x = ln t, y = 2√t, z = t2; (0, 2, 1)
Graph the curve with parametric equations x = t, y = 4t3/2 z = –t2 and find the curvature at the point (1, 4, –1).
Use Formula 11 to find the curvature.y = 2x – x2Formula 11 K(x) = [ƒ"(x) | [1 + (f'(x))²]³/²
A gun is fired with angle of elevation 30°. What is the muzzle speed if the maximum height of the shell is 500 m?
Use Formula 11 to find the curvature.y = cos xFormula 11 K(x) = [ƒ"(x) | [1 + (f'(x))²]³/²
A gun has muzzle speed 150m/s. Find two angles of elevation that can be used to hit a target 800 m away.
Use Formula 11 to find the curvature.ky = 4x5/2Formula 11 K(x) = [ƒ"(x) | [1 + (f'(x))²]³/²
Evaluate the integral. f² (16²³i – 9r²j + 25t¹k) dr
Evaluate the integral. 2t £ ( ₁ 4 ,‚ ³ + — ²7² pk) æ j+ dt 0 + 1² 1 +
Find the tangential and normal components of the acceleration vector.r(t) = (3t - t3)i + 3t2 j
Evaluate the integral. √² (1² i + t√/1 − 1 j + t sin 77 k) dt
Evaluate the integral. TT/2 (3 sin²t cos ti + 3 sint cos'tj + 2 sin t cos t k) dt
Find the tangential and normal components of the acceleration vector.r(t) = (1 + t)i + (t2 - 2t) j
Find the tangential and normal components of the acceleration vector.r(t) = t i + t2j + 3t k
Evaluate the integral. f (e'i + 2t j + In 1 k) dt t
Evaluate the integral. | (cos wri+ sin #tj+rk) di
Find the tangential and normal components of the acceleration vector.r(t) = et i + √2 t j + e–tk
Find the tangential and normal components of the acceleration vector.r(t) = t i + cos2t j + sin2t k
Use the Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with respect to t.) Data from Frenet-Serret formulas (a) r" s"T + K(s')²N (b) r' X r" = K(s')³B (c) r" =
Find equations of the normal plane and osculating plane of the curve at the given point.x = 2 sin 3t, y = t, z = 2 cos 3t; (0, π, –2)
Find equations of the normal plane and osculating plane of the curve at the given point.x = 1, y = t2, z = t3; (1, 1, 1)

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