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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series. f(x) 1 + x'
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.f(-4x) = ln (1 + 4x) f(x) = In (1 - x) = -
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.p(x) = 2x6 ln (1 - x) f(x) = In (1 - x) = -
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.f(x3) = ln (1 - x3) f(x) = In (1 - x) = - >
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.h(x) = x ln (1 - x) f(x) = In (1 - x) = - >
Use the power series representationto find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.g(x) = x3 ln (1 - x) f(x) = In (1 - x) = -
Carry out the procedure described in Exercise 85 with the following functions and Taylor polynomials.f(x) = tan x, p1(x) = x, p3(x) = x + x3/3
Carry out the procedure described in Exercise 85 with the following functions and Taylor polynomials.f(x) = ln (1 + x), p1(x) = x, p2(x) = x - x2/2
Carry out the procedure described in Exercise 85 with the following functions and Taylor polynomials.f(x) = e-x, p1(x) = 1 - x, p2(x) = 1 - x + x2/2
Carry out the procedure described in Exercise 85 with the following functions and Taylor polynomials.f(x) = cos x, p2(x) = 1 - x2/2, p4(x) = 1 - x2/2 + x4/24
Explain why the sequence of partial sums for an alternating series is not an increasing sequence.
Determine whether the following statements are true using a proof or counterexample. Assume that u, v, and w are nonzero vectors in R3.(u - v) × (u + v) = 2u × v
Determine whether the following statements are true using a proof or counterexample. Assume that u, v, and w are nonzero vectors in R3.u × (u × v) = 0
Prove in two ways that for scalars a and b, (au) × (bv) = ab(u × v). Use the definition of the cross product and the determinant formula.
Prove that u × u = 0 in three ways.a. Use the definition of the cross product.b. Use the determinant formulation of the cross product.c. Use the property that u × v = -(v × u).
An electron with a mass of 9.1 × 10-31 kg and a charge of -1.6 × 10-19 C travels in a circular path with no loss of energy in a magnetic field of 0.05 T that is orthogonal to the path of the
A horizontally outstretched arm supports a weight of 20 lb in a hand (see figure). If the distance from the shoulder to the elbow is 1 ft and the distance from the elbow to the hand is 1 ft, find the
A set of caliper brakes exerts a force on the rim of a bicycle wheel that creates a frictional force F of 40 N perpendicular to the radius of the wheel (see figure). Assuming the wheel has a radius
Another operation with vectors is the scalar triple product, defined to be u • (v × w), for vectors u, v, and w in R3.Prove that u • (v × w) = (u × v) • w.
Another operation with vectors is the scalar triple product, defined to be u • (v × w), for vectors u, v, and w in R3.a. Consider the parallelepiped (slanted box) determined by the position
Another operation with vectors is the scalar triple product, defined to be u • (v × w), for vectors u, v, and w in R3.Express u, v, and w in terms of their components and show that u • (v × w)
Find the area of the triangle with vertices on the coordinate axes at the points (a, 0, 0), (0, b, 0), and (0, 0, c), in terms of a, b, and c.
Find all vectors u that satisfy the equation(1, 1, 1) × u = (0, 0, 1).
Find all vectors u that satisfy the equation(1, 1, 1) × u = (-1, -1, 2).
Under what conditions is u × v a unit vector?
Find the area of the following triangles T.The vertices of T are O(0, 0, 0), P(1, 2, 3), and Q(6, 5, 4).
Find the area of the following triangles T.The vertices of T are O(0, 0, 0), P(2, 4, 6), and Q(3, 5, 7).
Find the area of the following triangles T.The sides of T are u = (3, 3, 3), v = (6, 0, 6), and u - v.
Find the area of the following triangles T.The sides of T are u = (0, 6, 0), v = (4, 4, 4), and u - v.
Evaluate (a, b, a) × (b, a, b). For what nonzero values of a and b are the vectors (a, b, a) and (b, a, b) parallel?
Find the value of a such that (a, a, 2) × (1, a, 3) = (2, -4, 2).
Use cross products to determine whether the points A, B, and C are collinear.A(-3, -2, 1), B(1, 4, 7), and C(4, 10, 14)
Use cross products to determine whether the points A, B, and C are collinear.A(3, 2, 1), B(5, 4, 7), and C(9, 8, 19)
Determine whether the following statements are true and give an explanation or counterexample.a. The cross product of two nonzero vectors is a nonzero vector.b. |u × v| is less than both |u| and
Answer the following questions about force on a moving charge.A proton (q = 1.6 × 10-19 C) with velocity 2 × 106 j m/s experiences a force in newtons of F = 5 × 10-12 k as it passes through the
Answer the following questions about force on a moving charge.An electron (q = -1.6 × 10-19 C) enters a constant 2-T magnetic field at an angle of 45° to the field with a speed of 2 × 105 m/s.
Answer the following questions about force on a moving charge.A particle with a unit negative charge (q = -1) enters a constant magnetic field B = 5k with a velocity v = i + 2j. Find the magnitude
Answer the following questions about force on a moving charge.A particle with a positive unit charge (q = 1) enters a constant magnetic field B = i + j with a velocity v = 20k. Find the magnitude and
Answer the following questions about torque.A pump handle has a pivot at (0, 0, 0) and extends to P(5, 0, -5). A force F = (1, 0, -10) is applied at P. Find the magnitude and direction of the torque
Answer the following questions about torque.Let Which is greater (in magnitude): the torque about O when a force F = 5i - 5k is applied at P or the torque about O when a force F = 4i - 3j is applied
Answer the following questions about torque.Let A force F = (10, 10, 0) is applied at P. Find the torque about O that is produced. OP = j + 2k. r =
Answer the following questions about torque.LetA force F = (20, 0, 0) is applied at P. Find the torque about O that is produced. OP = i + j + k. r =
A force of 1.5 lb is applied in a direction perpendicular to the screen of a laptop at a distance of 10 in from the hinge of the screen. Find the magnitude of the torque (in ft-lb) that is applied.
Suppose you apply a force of 20 N to a 0.25-meter-long wrench attached to a bolt in a direction perpendicular to the bolt. Determine the magnitude of the torque when the force is applied at an angle
Find a vector orthogonal to the given vectors.(6, -2, 4) and (1, 2, 3)
Find a vector orthogonal to the given vectors.(8, 0, 4) and (-8, 2, 1)
Find a vector orthogonal to the given vectors.(1, 2, 3) and (-2, 4, -1)
Find a vector orthogonal to the given vectors.(0, 1, 2) and (-2, 0, 3)
Find the cross products u × v and v × u for the following vectors u and v.u = 2i - 10j + 15k, v = 0.5i + j - 0.6k
Find the cross products u × v and v × u for the following vectors u and v.u = 3i - j - 2k, v = i + 3j - 2k
Find the cross products u × v and v × u for the following vectors u and v.u = (3, -4, 6), v = (1, 2, -1)
Find the cross products u × v and v × u for the following vectors u and v.u = (2, 3, -9), v = (-1, 1, -1)
Find the cross products u × v and v × u for the following vectors u and v.u = (-4, 1, 1), v = (0, 1, -1)
Find the cross products u × v and v × u for the following vectors u and v.u = (3, 5, 0), v = (0, 3, -6)
For the given points A, B, and C, find the area of the triangle with vertices A, B, and C.A(-1, -5, -3), B(-3, -2, -1), C(0, -5, -1)
For the given points A, B, and C, find the area of the triangle with vertices A, B, and C.A(5, 6, 2), B(7, 16, 4), C(6, 7, 3)
For the given points A, B, and C, find the area of the triangle with vertices A, B, and C.A(1, 2, 3), B(5, 1, 5), C(2, 3, 3)
For the given points A, B, and C, find the area of the triangle with vertices A, B, and C.A(0, 0, 0), B(3, 0, 1), C(1, 1, 0)
Find the area of the parallelogram that has two adjacent sides u and v.u = 8i + 2j - 3k, v = 2i + 4j - 4k
Find the area of the parallelogram that has two adjacent sides u and v.u = 2i - j - 2k, v = 3i + 2j - k
Find the area of the parallelogram that has two adjacent sides u and v.u = -3i + 2k, v = i + j + k
Find the area of the parallelogram that has two adjacent sides u and v.u = 3i - j, v = 3j + 2k
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.2j × (-5)i
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.-2i × 3k
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.3j × i
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.-j × k
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.i × k
Compute the following cross products. Then make a sketch showing the two vectors and their cross product.j × k
Compute |u × v| if |u| = 3 and |v| = 4 and the angle between u and v is 2π/3.
Compute |u × v| if u and v are unit vectors and the angle between u and v is π/4.
Sketch the following vectors u and v. Then compute |u × v| and show the cross product on your sketch.u = (0, -2, -2), v = (0, 2, -2)
Sketch the following vectors u and v. Then compute |u × v| and show the cross product on your sketch.u = (3, 3, 0), v = (3, 3, 3√2)
Sketch the following vectors u and v. Then compute |u × v| and show the cross product on your sketch.u = (0, 4, 0), v = (0, 0, -8)
Sketch the following vectors u and v. Then compute |u × v| and show the cross product on your sketch.u = (0, -2, 0), v = (0, 1, 0)
Find the cross product u × v in each figure. Z. v = (0, 0, 2) u = (-4, 0, 0) || y
Find the cross product u × v in each figure. Z. = (3, 0, 0) v = (0, 5, 0)
Explain how to find the torque produced by a force using cross products.
Explain how to use a determinant to compute u × v.
If u and v are orthogonal, what is the magnitude of u × v?
What is the magnitude of the cross product of two parallel vectors?
Explain how to find the direction of the cross product u × v.
Explain how to find the magnitude of the cross product u × v.
Use projections to find a general formula for the (least) distance between the point P(x0, y0) and the line ax + by = c.
Consider the parallelogram with adjacent sides u and v.a. Show that the diagonals of the parallelogram are u + v and u - v.b. Prove that the diagonals have the same length if and only if u • v =
The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and
The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and
The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and
The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and
The definition u • v = |u| |v| cos θ implies that |u • v| ≤ |u||v| (because |cos θ| ≤ 1). This inequality, known as the Cauchy–Schwarz Inequality, holds in any number of dimensions and
Let v = (a, b, c) and let α, β, and γ be the angles between v and the positive x-axis, the positive y-axis, and the positive z-axis, respectively (see figure).a. Prove that cos2 α + cos2 β +
Recall that two lines y = mx + b and y = nx + c are orthogonal provided mn = -1 (the slopes are negative reciprocals of each other). Prove that the condition mn = -1 is equivalent to the
For fixed values of a, b, c, and d, the value of proj(ka, kb) (c, d) is constant for all nonzero values of k, for (a, b) ≠ (0, 0).
a. Show that (u + v) • (u + v) = |u|2 + 2 u • v + |v|2.b. Show that |u + v| • |u + v| = |u|2 + |v|2 if u is orthogonal to v.c. Show that (u + v) • (u - v) = |u|2 - |v|2.
Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Prove the following vector properties, where c is a scalar.u • (v + w) = u • v + u • w Distributive property

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