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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Compute the indefinite integral of the following functions.r(t) = (t4 - 3t, 2t - 1, 10)
Compute r"(t) and r"(t) for the following functions.r(t) = tan t i + (t + 1/t) j - ln (t + 1) k
Compute r"(t) and r"(t) for the following functions. r(t) = Vt + 4i + j - e-† k t + 1
Compute r"(t) and r"(t) for the following functions.r(t) = (e4t, 2e-4t + 1, 2e-t)
Compute r"(t) and r"(t) for the following functions.r(t) = (cos 3t, sin 4t, cos 6t)
Compute r"(t) and r"(t) for the following functions.r(t) = (3t12 - t2, t8 + t3, t-4 - 2)
Compute r"(t) and r"(t) for the following functions.r(t) = (t2 + 1, t + 1, 1)
Compute the following derivatives. ((r' i + 6j – 2Vik) × (3t i – 121°j – 6²k)) 6t k)) dt
Compute the following derivatives. ((3t2 i + Vij – 21- k) • (cos ti + sin 2t j – 3t k)) dt
Compute the following derivatives. ((1'i – 21 j – 2k) × (ti – t² j – t³ k)) dt
Compute the following derivatives. – 21 k) · (e'i + 2e' j – 3e¯† k)) (1²(i + 2j |dt
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.u(t) × v(t)
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.u(t) • v(t)
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.v(√t)
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.u(t4 - 2t)
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.(4t8 - 6t3)v(t)
Let u(t) = 2t3 i + (t2 - 1)j - 8k and v(t) = et i + 2e-t j - e2t k. Compute the derivative of the following functions.(t12 + 3t)u(t)
Find the unit tangent vector at the given value of t for the following parameterized curves.r(t) = (√7et, 3et, 3et), for 0 ≤ t ≤ 1; t = ln 2
Find the unit tangent vector at the given value of t for the following parameterized curves.r(t) = (6t, 6, 3/t), for 0 < t < 2; t = 1
Find the unit tangent vector at the given value of t for the following parameterized curves.r(t) = (sin t, cos t, e-t), for 0 ≤ t ≤ π; t = 0
Find the unit tangent vector at the given value of t for the following parameterized curves.r(t) = (cos 2t, 4, 3 sin 2t), for 0 ≤ t ≤ π; t = π/2
Find the unit tangent vector for the following parameterized curves.r(t) = (e2t, 2e2t, 2e-3t), for t ≥ 0
Find the unit tangent vector for the following parameterized curves.r(t) = (t, 2, 2/t), for t ≥ 1
Find the unit tangent vector for the following parameterized curves.r(t) = (sin t, cos t, cos t), for 0 ≤ t ≤ 2π
Find the unit tangent vector for the following parameterized curves.r(t) = (8, cos 2t, 2 sin 2t), for 0 ≤ t ≤ 2π
Find the unit tangent vector for the following parameterized curves.r(t) = (cos t, sin t, 2) , for 0 ≤ t ≤ 2π
Find the unit tangent vector for the following parameterized curves.r(t) = (2t, 2t, t), for 0 ≤ t ≤ 1
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = 2et i + e-2t j + 4e2t k, t = ln 3
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = 2t4 i + 6t3/2 j + 10/t k, t = 1
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = (2 sin t, 3 cos t, sin (t/2)), t = π
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = (t, cos 2t, 2 sin t), t = π/2
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = (et, e3t, e5t), t = 0
Find a tangent vector at the given value of t for the following parameterized curves.r(t) = (t, 3t2, t3), t = 1
Differentiate the following functions.r(t) = ((t + 1)-1, tan-1 t, ln (t + 1))
Differentiate the following functions.r(t) = (te-t, t ln t, t cos t)
Differentiate the following functions.r(t) = tan t i + sec t j + cos2 t k
Differentiate the following functions.r(t) = et i + 2e-t j - 4e2t k
Differentiate the following functions.r(t) = (4, 3 cos 2t, 2 sin 3t)
Differentiate the following functions.r(t) = (2t3, 6√t, 3/t)
Differentiate the following functions.r(t) = 4eti + 5j + ln t k
Differentiate the following functions.r(t) = (cos t, t2, sin t)
How do you evaluate r(t) dr? dt? a
How do you find the indefinite integral of r(t) = (f(t), g(t), h(t))?
Compute r"(t) when r(t) = (t10, 8t, cos t).
Given a tangent vector on an oriented curve, how do you find the unit tangent vector?
Explain the geometric meaning of r'(t).
What is the derivative of r(t) = (f(t), g(t), h(t))?
Use the formula in Exercise 79 to find the (least) distance between the given point Q and line r.Q(6, 6, 72, r(t) = 83t, -3t, 4)Data from Exercise 79Show that the (least) distance d between a point Q
Use the formula in Exercise 79 to find the (least) distance between the given point Q and line r.Q(-5, 2, 9); r(t) = (5t + 7, 2 - t, 12t + 4)Data from Exercise 79Show that the (least) distance d
Use the formula in Exercise 79 to find the (least) distance between the given point Q and line r.Q(5, 6, 1); r(t) = (1 + 3t, 3 - 4t, t + 1)Data from Exercise 79Show that the (least) distance d
Show that the (least) distance d between a point Q and a line r = r0 + t v (both in R3) is where P is a point on the line. |PQ × v| d =
Let r(t) = (f(t), g(t), h(t)).a. Assume thatwhich means that Prove that b. Assume thatand Prove that which means that lim r(t) = L = (L, L2, L3), lim |r(t) – L| = 0.
Find the period of the function in Exercise 76; that is, in terms of m and n, find the smallest positive real number T such that r(t + T) = r(t) for all t.Data from Exercise 76Prove that for integers
Prove that for integers m and n, the curver(t)= (a sin mt cos nt, b sin mt sin nt, c cos mt)lies on the surface of a sphere provided a2 = b2 = c2.
Graph the curve r(t) = (1/2 sin 2t, 1/2(1 - cos 2t), cos t) and prove that it lies on the surface of a sphere centered at the origin.
A golfer launches a tee shot down a horizontal fairway; it follows a path given by r(t) = (at, (75 - 0.1a)t, -5t2 + 80t), where t ≥ 0 measures time in seconds and r has units of feet. The y-axis
Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane.Find a
Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane.Graph the
Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane.Graph the
Consider the curve r(t) = (a cos t + b sin t)i + (c cos t + d sin t)j + (e cos t + f sin t)k, where a, b, c, d, e, and f are real numbers. It can be shown that this curve lies in a plane.Assuming the
Consider the curve described by the vector function r(t) = (50e-t cos t)i + (50e-t sin t)j + (5 - 5e-t)k, for t ≥ 0.a. What is the initial point of the path corresponding to r(0)?b. What is c.
Consider the linesr(t) = (2 + 2t, 8 + t, 10 + 3t) andR(s) = (6 + s, 10 - 2s, 16 - s).a. Determine whether the lines intersect (have a common point)and if so, find the coordinates of that point.b. If
Match functions a–f with the appropriate graphs A–F.a. r(t) = (t, -t, t) b. r(t) = (t2, t, t)c. r(t) = (4 cos t, 4 sin t, 2) d. r(t) = (2t, sin t, cos t)e. r(t) = (sin t, cos t, sin
Find the points (if they exist) at which the following planes and curves intersect. y + x = 0; r(t) = (cos t, sin t, t), for 0 ≤ t ≤ 4π
Find the points (if they exist) at which the following planes and curves intersect. z = 16; r(t) = (t, 2t, 4 + 3t), for -∞ < t < ∞
Find the points (if they exist) at which the following planes and curves intersect. y = 1; r(t) = (10 cos t, 2 sin t, 1), for 0 ≤ t ≤ 2π
Find the point (if it exists) at which the following planes and lines intersect.z = -8; r(t) = (3t - 2, t - 6, -2t + 4)
Find the point (if it exists) at which the following planes and lines intersect.y = -2; r(t) = (2t + 1, -t + 4, t - 6)
Find the point (if it exists) at which the following planes and lines intersect.z = 4; r(t) = (2t + 1, -t + 4, t - 6)
Find the point (if it exists) at which the following planes and lines intersect.x = 3; r(t) = (t, t, t)
Find the domain of the following vector-valued functions. r(t) = V4 – t² i + Vij - k V1 + t
Find the domain of the following vector-valued functions. 12 r(t) = cos 2t i + eV'j +
Find the domain of the following vector-valued functions. r(t) = Vt + 2i + V2 – tj
Find the domain of the following vector-valued functions. 3 j i + r(t) t + 2
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
A pair of lines in R3 are said to be skew if they are neither parallel nor intersecting. Determine whether the following pairs of lines are parallel, intersecting, or skew. If the lines intersect,
Determine an equation of the line that is perpendicular to the lines r(t) = (4t, 1 + 2t, 3t) and R(s) = (-1 + s, -7 + 2s, -12 + 3s) and passes through the point of intersection of the lines r and R.
Determine an equation of the line that is perpendicular to the lines r(t) = (-2 + 3t, 2t, 3t) and R(s) = (-6 + s, -8 + 2s, -12 + 3s) and passes through the point of intersection of the lines r and R.
Determine whether the following statements are true and give an explanation or counterexample.a. The line r(t) = (3, -1, 4) + t(6, -2, 8) passes through the origin.b. Any two nonparallel lines in R3
Evaluate the following limits. 3t -j + Vt + 1 k tan t lim i sin t t
Evaluate the following limits. .2 cos t + t/2 – 1 j+ e' – t – 1 sin t lim -k .2
Evaluate the following limits. 4e sin #t j + lim t→2\t + 1 i - V4t + 1
Evaluate the following limits. 2t j + tan¬ltk t + 1 lim ( ei –
Evaluate the following limits.
Evaluate the following limits. 2t cos 2t i – 4 sin tj + lim →T/2 k TT
Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility.r(t) = cos t sin 3t i + sin t sin 3t j + √t k, for 0 ≤ t
Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility.r(t) = sin t i + sin2 t j + t/(5π) k, for 0 ≤ t ≤ 10π
Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility.r(t) = 2 cos t i + 4 sin t j + cos 10t k, for 0 ≤ t ≤ 2π
Graph the curves described by the following functions. Use analysis to anticipate the shape of the curve before using a graphing utility.r(t) = 0.5 cos 15t i + (8 + sin 15t) cos t j + (8 + sin 15t)
Graph the curves described by the following functions, indicating the positive orientation.r(t) = e-t/10 i + 3 cos t j + 3 sin t k, for 0 ≤ t < ∞
Graph the curves described by the following functions, indicating the positive orientation.r(t) = e-t/20 sin t i + e-t/20 cos t j + t k, for 0 ≤ t < ∞
Graph the curves described by the following functions, indicating the positive orientation.r(t) = 4 sin t i + 4 cos t j + e-t/10 k, for 0 ≤ t < ∞
Graph the curves described by the following functions, indicating the positive orientation.r(t) = t cos t i + t sin t j + t k, for 0 ≤ t ≤ 6π
Graph the curves described by the following functions, indicating the positive orientation.r(t) = 2 cos t i + 2 sin t j + 2 k, for 0 ≤ t ≤ 2π
Graph the curves described by the following functions, indicating the positive orientation.r(t) = cos t i + j + sin t k, for 0 ≤ t ≤ 2π
Graph the curves described by the following functions, indicating the positive orientation.r(t) = (0, 4 cos t, 16 sin t) for 0 ≤ t ≤ 2π

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