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mathematics
calculus 6th edition
Calculus 6th Edition James Stewart - Solutions
Find an equation of the ellipse with foci (0, ±2) and vertices (0, ±3).
Sketch the conic 1 = 12 2 + 4 sin 0
What curve is represented by the polar equation r = 2?
Find the length of the cardioid r = 1 + sin θ.
Find the foci and asymptotes of the hyperbola 9x2 – 16y2 = 144 and sketch its graph.
Find parametric equations for the circle with center (h, k) and radius r.
Sketch the curve with parametric equations x = sin t, y = sin2t.
Find the foci and equation of the hyperbola with vertices (0, ±1) and asymptote y = 2x.
(a) Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about 0.017 and the length of the major axis is about 2.99 x 108 km. (b) Find the distance from the earth to the sun at perihelion and at aphelion.
Show that the surface area of a sphere of radius r is 4πr2.
(a) Sketch the curve with polar equation r = 2cos θ.(b) Find a Cartesian equation for this curve.
Use a graphing device to graph the curve x = y4 – 3y2.
Sketch the conicand find its foci. 9x²4y²72x + 8y + 176 = 0
Find an equation of the ellipse with foci (2, –2), (4, –2) and vertices (1, –2), (5, –2).
Intelligence Quotient (IQ) scores are distributed normally with mean 100 and standard deviation 15. (Figure 6 shows the corresponding probability density function.)(a) What percentage of the population has an IQ score between 85 and 115?(b) What percentage of the population has an IQ above
Find the centroid of the region bounded by the line y = x and the parabola y = x2.
A torus is formed by rotating a circle of radius about a line in the plane of the circle that is a distance R(> r) from the center of the circle. Find the volume of the torus.
Show that every member of the family of functionsis a solution of the differential equation y' = 1/2(y2 – 1). y 1 + ce¹ 1 - ce¹
(a) Sketch the direction field for the differential equation y' = x2 + y2 – 1. (b) Use part (a) to sketch the solution curve that passes through the origin
Use the sum of the first 100 terms to approximate the sum of the series ∑1/(n3 + 1). Estimate the error involved in this approximation.
Use (3) with n = 10 to estimate the sum of the series 00 1 3 n=1 n°
Indicate which tests should be used. k=1 2k 저
Find the radius of convergence and interval of convergence of the series Σ n=0 n(x + 2)" 3"+1
Find lim sin(π/n). n→∞
Show that the series is convergent, and find its sum. 00 Σ n=1 1 n(n + 1)
Find a power series representation for In(1 – x) and its radius of convergence.
Indicate which tests should be used. 00 1 Σ n=1 2 + 3"
Find the Maclaurin series for the function f(x) = x cos x.
Discuss the convergence of the sequence an = n!/nn, where n! = 1 · 2 · 3 · · ·n.
Show that the harmonic seriesis divergent. 511+ ܐܐ 1=n + 1 2 3 + 4
Find a power series representation for f(x) = tan–1 x.
Represent f(x) = sin x as the sum of its Taylor series centered at π/3.
For what values of r is the sequence {rn} convergent?
Show that the series diverges. Σ n=1 την 5n² + 4
(a) Evaluate [1/(1 + x7)]dx as a power series. (b) Use part (a) to approximate ∫0 0.5 [1/(1 + x7)]dx correct to within 10–7.
Why the sequence is decreasing? 3 n+ 5
Find the Maclaurin series for f(x) = (1 + x)k, where k is any real number.
Find the Maclaurin series for the function and its radius of convergence. f(x) = 1 √4-x
Show that the sequence is decreasing. an n n² + 1
(a) Evaluate ∫e–x2 dx as an infinite series. (b) Evaluate ∫01 e–x2 dx correct to within an error of 0.001.
Evaluate lim x0 et - 1 - x करे
Investigate the sequence {an} defined by the recurrence relation
Find the first three nonzero terms in the Maclaurin series for (a) ex sin x (b) tan x.
(a) Find a vector equation and parametric equations for the line that passes through the point (5, 1, 3) and is parallel to the vector i + 4j – 2k. (b) Find two other points on the line.
Draw the sum of the vectors a and b shown in Figure 5.Figure 5 a b
What surfaces in R3 are represented by the following equations?(a) z = 3 (b) y = 5
Sketch the graph of the surface z = x2.
If the vectors a and b have lengths 4 and 6, and the angle between them is π/3, find a · b.
Find a vector perpendicular to the plane that passes through the points P(1, 4, 6), Q(–2, 5, –1), and R(1, –1, 1).
(a) Find parametric equations and symmetric equations of the line that passes through the points A(2, 4, –3) and B(3, –1, 1). (b) At what point does this line intersect the xy-plane?
If a and b are the vectors shown in Figure 9, draw a – 2b.Figure 9 a b
Describe and sketch the surface in R3 represented by the equation y = x.
Identify and sketch the surfaces. (a) x2 + y2 = 1 (b) y2 + z2 = 1
Show that the lines L1 and L2 with parametric equationsare skew lines; that is, they do not intersect and are not parallel (and therefore do not lie in the same plane). x = 1 + t x = 2s y = −2+ 3t y = 3+ s z = 4-t z = -3 + 4s
Find the area of the triangle with vertices P(1, 4, 6), Q(–2, 5, –1), and R(1, –1,1).
Use traces to sketch the quadric surface with equation x2 믕 +
Find the vector represented by the directed line segment with initial point A(2, –3, 4) and terminal point B(–2, 1, 1).
Show that 2i + 2j – k is perpendicular to 5i – 4j + 2k.
Use the scalar triple product to show that the vectors a = 〈1, 4, –7〉, b = 〈2, –1, 4〉, and c = 〈0, –9, 18〉 are coplanar.
Find an equation of the plane through the point (2, 4, –1) with normal vector n = 〈2, 3, 4〉. Find the intercepts and sketch the plane.
Find an equation of a sphere with radius r and center C(h, k, l).
If a = 〈4, 0, 3) and b = 〈–2, 1, 5〉, find |a| and the vectors a + b, a – b, 3b, and 2a + 5b.
Use traces to sketch the surface z = 4x2 + y2.
A bolt is tightened by applying a 40-N force to a 0.25-m wrench as shown in Figure 5. Find the magnitude of the torque about the center of the bolt.Figure 5 0.25 m 75° 40 N
Find the direction angles of the vector a = 〈1, 2, 3〉.
Find an equation of the plane that passes through the points P(1, 3, 2), Q(3, –1, 6), and R(5, 2, 0).
Show that x2 + y2 + z2 + 4x – 6y + 2z+ 6 = 0 is the equation of a sphere, and find its center and radius.
If a = i + 2j – 3k and b = 4i + 7k, express the vector 2a + 3b in terms of i, j, and k.
Find the scalar projection and vector projection of b = 〈1, 1, 2〉 onto a = 〈–2, 3, 1〉.
Find the point at which the line with parametric equations x = 2 + 3t y = –4t, z = 5 + t intersects the plane 4x + 5y – 2z = 18.
What region in R3 is represented by the following inequalities? 1 < x² + y + z ≤4 z≤0
Sketch the surface 4 +y²-2-1. 4 =
Find the unit vector in the direction of the vector 2i – j – 2k.
A wagon is pulled a distance of 100 m along a horizontal path by a constant force of 70 N. The handle of the wagon is held at an angle of 35° above the horizontal. Find the work done by the force.
A force is given by a vector F = 3i+ 4j + 5k and moves a particle from the point P(2, 1, 0) to the point Q(4, 6, 2). Find the work done.
A 100-lb weight hangs from two wires as shown in Figure 19. Find the tensions (forces) T1 and T2 in both wires and their magnitudes.Figure 19 50° T₁ 100 T₂ 32°
(a) Find the angle between the planes x + y + z = 1 and x – 2y + 3z = 1. (b) Find symmetric equations for the line of intersection L of these two planes.
Identify and sketch the surface 4x2 – y2 + 2z2 + 4 = 0.
Find a formula for the distance D from a point P1(x1, y1, z1) to the plane ax + by + cz + d = 0.
Classify the quadric surface x2 + 2z2 – 6x – y + 10 = 0.
In Example 3 we showed that the linesare skew. Find the distance between them.Data from Example 3Show that the lines L1 and L2 with parametric equationsare skew lines; that is, they do not intersect and are not parallel (and therefore do not lie in the same plane). L₁ x= 1+ t L₂: x=2s y = -2 +
Find the distance between the parallel planes 10x + 2y – 2z = 5 and 5x + y – z = 1.
Find lim r(t), where r(t) = (1 + t³)i + te¯¹j+ 1-0 sin t t k.
(a) Find the derivative of r(t) = (1 + t3)i + te–tj + sin 2t k. (b) Find the unit tangent vector at the point where t = 0.
Find the length of the arc of the circular helix with vector equation r(t) = cos t i+ sin t j + tk from the point (1,0,0) to the point (1, 0, 2π).
The position vector of an object moving in a plane is given by r(t) = t3i + t2j. Find its velocity, speed, and acceleration when t = 1 and illustrate geometrically.
Describe the curve defined by the vector function r(t) = 〈1 + t, 2 + 5t, –1 + 6t〉
For the curve r(t) = √t i + (2 – t) j, find r'(t) and sketch the position vector r(1) and the tangent vector r'(1).
Reparametrize the helix r(t) = cos ti+ sin tj + tk with respect to arc length measured from (1, 0, 0) in the direction of increasing t.
Find the velocity, acceleration, and speed of a particle with position vector r(t) = 〈t2, et, tet).
Find parametric equations for the tangent line to the helix with parametric equationsat the point (0, 1, π/2). x = 2 cost y = sin t z = t
Sketch the curve whose vector equation is r(t) = cos t i + sin t j + t k
A moving particle starts at an initial position r(0) = 〈1, 0, 0〉 with initial velocity v(0) = i – j + k. Its acceleration is a(t) = 4t i + 6t j + k. Find its velocity and position at time t.
Find a vector equation and parametric equations for the line segment that joins the point P(1, 3, –2) to the point Q(2, –1, 3).
Show that if |r(t)| = c (a constant), then r'(t) is orthogonal to r(t) for all t.
Find the curvature of the twisted cubic r(t) = 〈t, t2, t3〉 at a general point and at (0, 0, 0).
An object with mass m that moves in a circular path with constant angular speed ω has position vector r(t) = a cos ωt i + a sin ωt j. Find the force acting on the object and show that it is directed toward the origin.
Find a vector function that represents the curve of intersection of the cylinder x2 + y2 = 1 and the plane y + z = 2.
A projectile is fired with angle of elevation α and initial velocity v0. (See Figure 6.) Assuming that air resistance is negligible and the only external force is due to gravity, find the position function r(t) of the projectile. What value of a maximizes the range (the horizontal distance
Find the curvature of the parabola y = x2 at the points (0, 0), (1, 1), and (2, 4).
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