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mathematics
precalculus

Questions and Answers of Precalculus

Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k(x). Identify each curve and explain your choices. УА a b х
Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k(x). Identify each curve and explain your choices.
Use a graphing calculator or computer to graph both the curve and its curvature function k(x) on the same screen. Is the graph of K what you would expect?y = x-2
Use a graphing calculator or computer to graph both the curve and its curvature function k(x) on the same screen. Is the graph of K what you would expect?y = x4 - 2x2
(a) Is the curvature of the curve C shown in the figure greater at P or at Q? Explain.(b) Estimate the curvature at P and at Q by sketching the osculating circles at those points. УА х
Find an equation of a parabola that has curvature 4 at the origin.
At what point does the curve have maximum curvature?What happens to the curvature as x → ∞?y = ex
At what point does the curve have maximum curvature?What happens to the curvature as x → ∞?y = ln x
Use Formula 11 to find the curvature.y = xex
Use Formula 11 to find the curvature.y = tan x
Use Formula 11 to find the curvature.y = x4
Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0).
Find the curvature of r(t) = (t, t2, t3) at the point (1, 1, 1).
Find the curvature of r(t) = (t2, ln t, t ln t) at the point (1, 0, 0).
Use Theorem 10 to find the curvature.r(t) = √6 t2 i + 2t j + 2t3 k
Use Theorem 10 to find the curvature.r(t) = ti + t2 j + etk
Use Theorem 10 to find the curvature.r(t) = t3 j + t2 k
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.r(t) = (t, 1/2 t2, t2)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.r(t) = (√2 t, et, e-t)
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.r(t) = (t 2, sin t - t cos t, cos t + t sin t), t > 0
(a) Find the unit tangent and unit normal vectors T(t) and N(t).(b) Use Formula 9 to find the curvature.r(t) = (t, 3 cos t, 3 sin t)
Reparametrize the curvewith respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the
Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = 3 sin t, y = 4t, z = 3 cos t in the positive direction. Where are you now?
(a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b)
(a) Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b)
Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 + y2 = 4 and the plane x + y + z = 2.
Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36).
Graph the curve with parametric equations x = sin t, y = sin 2t, z = sin 3t. Find the total length of this curve correct to four decimal places.
Find the length of the curve correct to four decimal places.(Use a calculator to approximate the integral.)r(t) = (cos πt, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0)
Find the length of the curve correct to four decimal places.(Use a calculator to approximate the integral.)r(t) = (t, e-t, te-t l, + < t < 3
Find the length of the curve correct to four decimal places.(Use a calculator to approximate the integral.)r(t) = (t2, t3, t4), 0 < t < 2
Find the length of the curve.r(t) = t - i + 9t j + 4t3/2 k, 1 < t < 4
Find the length of the curve.r(t) = i + t - j + t3 k, 0 < t < 1
Find the length of the curve.r(t) = cos t i + sin t j + ln cos t k, 0 < t < π/4
Find the length of the curve.r(t) = √2 t i + et j + e-t k, 0 < t < 1
Find the length of the curve.r(t) = (2t, t2, 1/3t3), 0 < t < 1
Find the length of the curve. r(t) = (t, 3 cos t, 3 sin t), -5
Prove that c b.c (а X Ь) : (с х d) a ·d b·d
Use Exercise 50 to prove thata x (b x c) + b x (c x a) + c x (a x b) = 0
Prove Property 6 of cross products, that is,a x (b x c) = (a • c)b - (a • b)c
Prove that (a - b) x (a + b) = 2(a x b).
If a + b + c = 0, show that a x b = b x c = c x a
Show that |a x b|2 = |a|2 |b |2 - (a • b)2.
(a) Let P be a point not on the plane that passes through thepoints Q, R, and S. Show that the distance d from P to theplane isd = |a • (b x c)|/|a x b|where(b) Use the formula in part (a) to find
(a) Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L isd = |a x b|/|a|(b) Use the formula in part (a) to find the
(a) Find all vectors v such that(1, 2, 1) x v = (3, 1, -5)(b) Explain why there is no vector v such that(1, 2, 1) x v = (3, 1, 5)
If a • b = √3 and a x b = (1, 2, 2), find the angle between a and b.
Let v = 5j and let u be a vector with length 3 that starts at the origin and rotates in the xy -plane. Find the maximum and minimum values of the length of the vector u x v. In what direction does u
A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction (0, 3, -4) at the end of the wrench. Find the magnitude of the force needed to
(a) A horizontal force of 20 lb is applied to the handle of a gearshift lever as shown. Find the magnitude of the torque about the pivot point P.(b) Find the magnitude of the torque about P if the
A bicycle pedal is pushed by a foot with a 60-N force as shown. The shaft of the pedal is 18 cm long. Find the magnitude of the torque about P. 60 N 70° ) 10° P
Use the scalar triple product to determine whether the points A(1, 3, 2), B(3, -1, 6), C(5, 2, 0), and D(3, 6, -4) lie in the same plane.
Use the scalar triple product to verify that the vectors u = i + 5 j - 2 k, v = 3i - j, and w = 5i + 9 j - 4 k are coplanar.
Graph the surfaces z = x2 + y2 and z = 1 - y2 on a common screen using the domain |x| < 1.2, |y| < 1.2 and observe the curve of intersection of these surfaces. Show that the projection of this
Show that the curve of intersection of the surfaces x2 + 2y2 - z2 + 3x = 1 and 2x2 + 4y2 - 2z2 - 5y = 0 lies in a plane.
Show that if the point (a, b, c) lies on the hyperbolic paraboloid z = y2 - x2, then the lines with parametric equations x = a + t, y = b + t, z = c + 2(b - a)t and x = a + t, y = b - t, z = c - 2(b
A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet (see the photo on page 839). The diameter at the base is 280 m and the minimum diameter, 500 m
Traditionally, the earth’s surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth at the
Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.
Find an equation for the surface consisting of all points that are equidistant from the point (-1, 0, 0) and the plane x = 1. Identify the surface.
Find an equation for the surface obtained by rotating the line z = 2y about the z-axis.
Find an equation for the surface obtained by rotating the curve y = √x about the x-axis.
Sketch the region bounded by the paraboloids z = x2 + y2 and z = 2 - x2 - y2.
Sketch the region bounded by the surfaces z = √x2 + y2 and x2 + y2 = 1 for 1 < z < 2.
Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.x2 - 6x + 4y2 - z
Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.-4x2 - y2 + z2 = 0
Use a computer with three-dimensional graphing software to 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
Use a computer with three-dimensional graphing software to graph the surface. Experiment with viewpoints and with domains for the variables until you get a good view of the surface.-4x2 - y2 + z2 = 1
Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 + y2 + z22 -4x - 8y + 4z + 55 = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 - y2 + z2 - 4x - 2z = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 - y2 - z2 - 4x - 2z + 3 = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 + y2 - 2x - 6y - z + 10 = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.y2 = x2 + 4z2 + 4
Reduce the equation to one of the standard forms, classify the surface, and sketch it.x2 + 2y - 2z2 = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.4x2 - y + 2z2 = 0
Reduce the equation to one of the standard forms, classify the surface, and sketch it.y2 = x2 + 1/9z2
Sketch and identify a quadric surface that could have the traces shown.Traces in x = kTraces in z = k ZA k= ±2 k= ±1 -k = 0 УА k= 0 k= 2 k=2 k=1 х k=0
Sketch and identify a quadric surface that could have the traces shown.Traces in x = kTraces in y = k k= ±1 k=0 k= ±2
A uniformly charged disk has radius R and surface charge density as in the figure. The electric potential V at a point P at a distance d along the perpendicular central axis of the disk iswhere ke
Use the information from Exercise 16 to estimate sin 38° correct to five decimal places.
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
14(a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation f(x) ≈ Tn (x) when x line in the given
Use a computer algebra system to find the Taylor polynomials Tn centered at a for n = 2, 3, 4, 5. Then graph these polynomials and f on the same screen. |f(x) = 1 + x², a = 0 %3D
Use a computer algebra system to find the Taylor polynomials Tn centered at a for n = 2, 3, 4, 5. Then graph these polynomials and f on the same screen. а %3D п/4 cot x, f(x) а %3 п,
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = tan-1x, a = 1
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = xe-2x, a = 0
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = x cos x, a = 0
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = e-x sin x, a = 0
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = sin x, a = π/6
Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.f (x) = ex, a = 1
Find the sum of the series. 3· 23 '5· 25 7: 27 1: 2
Find the sum of the series. 9. 27 81 3 + 2! 3! 4!
Find the sum of the series. (In 2)? 1 - In 2 + 2! (In 2)³ + . 3!
Find the sum of the series. 3" Σ 5" n! п-0
Find the sum of the series.

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