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mathematics
calculus early transcendentals 9th

Calculus Early Transcendentals 9th Edition James Stewart, Daniel K. Clegg, Saleem Watson, Lothar Redlin - Solutions

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) × w = u × w + v × w
Find a · b.|a| = 80, |b| = 50, the angle between a and b is 3π/4
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 × w) = (u × v) × w
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 × w) = (u × v) × w
Use traces to sketch and identify the surface.x = y2 + 4z2
Determine whether the points lie on a straight line.(a) A(2, 4, 2), B(3, 7, −2), C(1, 3, 3)(b) D(0, −5, 5), E(1, −2, 4), F(3, 4, 2)
State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar.(a) a · (b × c)(b) a × (b · c)(c) a × (b × c)(d) a · (b · c)(e) (a · b) × (c · d)(f) (a × b) · (c × d)
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 = 0.
Use traces to sketch and identify the surface.4x2 + 9y2 + 9z2 = 36
Find the distance from (4, −2, 6) to each of the following.(a) The xy-plane(b) The yz-plane(c) The xz-plane(d) The x-axis(e) The y-axis(f) The z-axis
Find |u × v| and determine whether u × v is directed into the page or out of the page. |u| 60° |v|= 8
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 = u × v.
Use traces to sketch and identify the surface.x2 = 4y2 + z2
Find |u × v| and determine whether u × v is directed into the page or out of the page. Ju|= 4 |v|= 3 150°
Find parametric equations for the line.The line through (4, −1, 2) and (1, 1, 5)
Find the angle between the vectors.u = (5, 1), v = (3, 2)
Use traces to sketch and identify the surface.z2 − 4x2 − y2 = 4
The figure shows a vector a in the xy-plane and a vector b in the direction of k. Their lengths are |a| = 3 and |b| = 2.(a) Find |a × b|.(b) Use the right-hand rule to decide whether the components of a × b are positive, negative, or 0. ZA a y
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 represents a line in space.
Find parametric equations for the line.The line through (1, 0, −1) and parallel to the line 1/3(x − 4) = 1/2 y = z + 2
Find the angle between the vectors.a = i − 3j, b = −3i + 4 j
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 {(x, y, z) | x2 + y2 = 1} is a circle.
Find the angle between the vectors.a = (1, −4, 1), b (0, 2, −2)
Use traces to sketch and identify the surface.3x2 + y + 3z2 = 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.
Find the angle between the vectors.a = (−1, 3, 4), b = (5, 2, 1)
Show that the equation represents a sphere, and find its center and radius.x2 + y2 + z2 + 8x − 2z = 8
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.
Find the angle between the vectors.u = i − 4j + k, v = −3i + j + 5k
Use traces to sketch and identify the surface.3x2 − y2 + 3z2 = 0
Show that the equation represents a sphere, and find its center and radius.x2 + y2 + z2 = 6x − 4y − 10z
Show that the equation represents a sphere, and find its center and radius.2x2 + 2y2 + 2z2 − 2x + 4y + 1 = 0
Show that 0 × a = 0 = a × 0 for any vector a in V3.
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 × v = 0, then u = 0 or v = 0.
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. x - 2 L;: 1 y - 3 -2 - 1 -3 x - 3 L2: 1 y + 4 z - 2 3 -7
Use traces to sketch and identify the surface.x = y2 − z2
Show that the equation represents a sphere, and find its center and radius.4x2 + 4y2 + 4z2 = 16x − 6y − 12
Show that (a × b) · b = 0 for all vectors a and b in V3.
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| ≤ |u| |v|.
Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection. y- 1 z - 2 L: -1 3 х - 2 у -3 L2: 2 -2 7
Determine whether the lines given by the symmetric equationsandare parallel, skew, or intersecting. х — 1 у - 2 -3 y z - 3 4.
Determine whether the given vectors are orthogonal, parallel, or neither.(a) u = (−5, 4, −2), v = (3, 4, −1)(b) u = 9i − 6j + 3k, v = −6i + 4j − 2k(c) u = (c, c, c), v = (c, 0, −c)
Prove the specified property of cross productsProperty 4: (a + b) × c = a × c + b × c
Graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.x2 − y2 − z = 0
Find the volume of the solid that lies inside both of the spheresx2 + y2 + z2 + 4x − 2y + 4z + 5 = 0andx2 + y2 + z2 = 4
Find the distance between the spheresx2 + y2 + z2 = 4andx2 + y2 + z2 = 4x + 4y + 4z – 11.
If c = |a|b + |b|a, where a, b, and c are all nonzero vectors, show that c bisects the angle between a and b.
Which of the following four lines are parallel? Are any of them identical?L1: x = 1 + 6t, y = 1 − 3t, z = 12t + 5L2: x = 1 + 2t, y = t, z = 1 + 4tL3: 2x − 2 = 4 − 4y = z + 1L4: r = (3, 1, 5) + t(4, 2, 8)
The figure shows a curve C given by a vector function r(t).(a) Draw the vectors r(4.5) − r(4) and r(4.2) − r(4).(b) Draw the vectors(c) Write expressions for r'(4) and the unit tangent vector T(4).(d) Draw the vector T(4). r(4.5) - г(4) г(4.2) — г(4) and 0.5 0.2
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 curve with vector equation r(t) = t3 i + 2t3 j + 3t3 k is a line.
(a) Sketch the curve with vector functionr(t) = t i + cos πt j + sin πt k t ≥ 0(b) Find r'(t) and r"(t).
A particle P moves with constant angular speed ω around a circle whose center is at the origin and whose radius is R. The particle is said to be in uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point (R, 0) when t = 0. The position vector at
(a) Use Equation 2 to compute the length of the given line segment.(b) Compute the length using the distance formula and compare to your answer from part (a).r(t) = (3 − t, 2t, 4t + 1), 1 ≤ t ≤ 3
The figure shows the path of a particle that moves with position vector r(t) at time t.(a) Draw a vector that represents the average velocity of the particle over the time interval 2 ≤ t ≤ 2.4.(b) Draw a vector that represents the average velocity over the time interval 1.5 ≤ t ≤ 2.(c)
(a) Make a large sketch of the curve described by the vector function r(t) = (t2, t), 0 ≤ t ≤ 2, and draw the vectors r(1), r(1.1), and r(1.1) − r(1).(b) Draw the vector r'(1) starting at (1, 1), and compare it with the vectorExplain why these vectors are so close to each other in length and
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 curve r(t) = (0, t2, 4t) is a parabola.
Let r(t) = (√2 − t , (et − 1)/t, ln(t + 1)).(a) Find the domain of r.(b) Find limt→0 r(t).(c) Find r'(t).
A circular curve of radius R on a highway is banked at an angle θ so that a car can safely traverse the curve without skidding when there is no friction between the road and the tires. The loss of friction could occur, for example, if the road is covered with a film of water or ice. The rated
(a) Use Equation 2 to compute the length of the given line segment.(b) Compute the length using the distance formula and compare to your answer from part (a).r(t) = (t + 2) i − t j + (3t − 5) k, −1 ≤ t ≤ 2
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 curve r(t) = (2t, 3 − t, 0) is a line that passes through the origin.
A projectile is fired from the origin with angle of elevation α and initial speed v0. Assuming that air resistance is negligible and that the only force acting on the projectile is gravity, t, that the position vector of the projectile isWe also showed that the maximum horizontal distance of the
Find the length of the curve.r(t) = k t, 3 cos t, 3 sin t), −5 ≤ t ≤ 5
Find the limit. e-31 i + -j+ cos 2tk sin?t lim
If u and v are differentiable vector functions, c is a scalar, and f is a real valued function, write the rules for differentiating the following vector functions.(a) u(t) + v(t)(b) cu(t)(c) f(t) u(t)(d) u(t) · v(t)(e) u(t) × v(t)(f) u(f(t))
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 derivative of a vector function is obtained by differentiating each component function.
(a) A projectile is fired from the origin down an inclined plane that makes an angle θ with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are α and v0, respectively. Find the position vector of the projectile and the parametric equations of the path of
Find the limit. t2 lim i+ vt + 8 j + t - 1 sin mt -k In t
How do you find the length of a space curve given by a vector function r(t)?
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(t) and v(t) are differentiable vector functions, then d [u(t) X v(t)]= u'(f) X v'(t) dt
If r(t) = t2 i + t cos πt j + sin πt k, evaluate ∫10 r(t) dt .
Find the length of the curve.r(t) = √2 t i + et j + e−t k, 0 ≤ t ≤ 1
Find the limit. (1 + t? lim 1- e tan-t, t 1 - 12
(a) What is the definition of curvature?(b) Write a formula for curvature in terms of r'(t) and T'(t).(c) Write a formula for curvature in terms of r'(t) and r"(t).(d) Write a formula for the curvature of a plane curve with equation y = f(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.If r(t) is a differentiable vector function, then d |r(1) | – dt |r'() |
Find the curvature of the curve with parametric equations - [ sin(70) de y - f cos(70*) do X =
Find the length of the curve.r(t) = cos t i + sin t j + ln cos t k, 0 ≤ t ≤ π/4
(a) Write formulas for the unit normal and binormal vectors of a smooth space curve r(t).(b) What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating circle?
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 T(t) is the unit tangent vector of a smooth curve, then the curvature is κ = |dT/dt|.
If a projectile is fired with angle of elevation α and initial speed v, then parametric equations for its trajectory areWe know that the range (horizontal distance traveled) is maximized when α = 45°. What value of α maximizes the total distance traveled by the projectile? (v cos a)t y = (v sin
(a) How do you find the velocity, speed, and acceleration of a particle that moves along a space curve?(b) Write the acceleration in terms of its tangential and normal components.
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 binormal vector is B(t) = N(t) × T(t).
Find the length of the curve.r(t) = t2i + 9tj + 4t3/2k, 1 ≤ t ≤ 4
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (t2 − 1, t)
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = (t2 + t, t2 – t, t3)
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.Suppose f is twice continuously differentiable. At an inflection point of the curve y = f(x), the curvature is 0.
Show that the curve with vector equationlies in a plane and find an equation of the plane. r(t) = (a,t? + b,t + c1, azt? + bzt + c2, azt? + bzt + c3) %3!
Find the length of the curve correct to four decimal places. (Use a calculator or computer to approximate the integral.)r(t) = (t2, t3, t4), 0 ≤ t ≤ 2
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (3 sin t, 2 cos t)
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 κ(t) = 0 for all t, the curve is a straight line.
Find the length of the curve correct to four decimal places. (Use a calculator or computer to approximate the integral.)r(t) = k t, e−t, te−t), 1 ≤ t ≤ 3
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = et i + e−t j
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = √2 t i + et j + e–t k
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 |r(t)| = 1 for all t, then |r'(t)| is a constant.
Find the length of the curve correct to four decimal places. (Use a calculator or computer to approximate the integral.)r(t) = (cos πt, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0)
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = t2 i + 2t j + ln t k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = et(cos t i + sin t j + t k)
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = (3, t, 2 − t2)
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = (t2, sin t – t cos t, cos t + t sin t), t ≥ 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.Different parametrizations of the same curve result in identical tangent vectors at a given point on the curve.
Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases.r(t) = 2 cos ti + 2 sin tj + k

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