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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

A vector field on R2 is defined by F(x, y) = –yi + xj. Describe F by sketching some of the vectors F(x, y) as in Figure 3.
Find the radius of gyration about the x-axis of the disk in Example 4.
The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line is called a cycloid. If the circle has radius r and rolls along the x-axis and if one
Find the remaining trigonometric ratios.csc θ = −4/3, 3π/2 < θ < 2π
Find the volume of the solid obtained by rotating the region bounded by y = x3, y = 8, and x = 0 about the y-axis.
Figure 8 shows velocity curves for two cars, A and B, that start side by side and move along the same road. What does the area between the curves represent? Use the Midpoint Rule to estimate
A 200-lb cable is 100 ft long and hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building?
Find the volume of the solid obtained by rotating the region bounded by y = x – x2 and y = 0 about the line x = 2.
The region R enclosed by the curves y = x and y = x2 is rotated about the x-axis. Find the volume of the resulting solid.
A tank has the shape of an inverted circular cone with height 10 m and base radius 4 m. It is filled with water to a height of 8 m. Find the work required to empty the tank by pumping all of the
Find the area of the region bounded by the curves y = sin x, y = cos x, x = 0, and x = π/2.
Find the volume of the solid obtained by rotating the region in Example 4 about the line y = 2.Data from Example 4The region R enclosed by the curves y = x and y = x2 is rotated about the x-axis.
Find the volume of the solid obtained by rotating the region in Example 4 about the line x = –1.Data from Example 4The region R enclosed by the curves y = x and y = x2 is rotated about the x-axis.
Figure 12 shows a solid with a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid.Figure 12 У
Find the area enclosed by the line y = x – 1 and the parabola y2 = 2x + 6.
Find the volume of a pyramid whose base is a square with side L and whose height is h.
A wedge is cut out of a circular cylinder of radius 4 by two planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of 30° along a diameter of the
Determine whether the integral ∫1∞(1/x) dx is convergent or divergent.
Find dx. I - X x + st
Find ∫x sin x dx.
Evaluate ∫ cos3x dx.
The region bounded by the curves y = arctan x, y = 0, and x = 1 is rotated about the y-axis. Find the volume of the resulting solid.
Use the Table of Integrals to find ∫x3 sin x dx.
Find the area of the surface generated by rotating the curve y = ex, 0 ≤ x ≤ 1, about the x-axis.
Plot the points whose polar coordinates are given. (a) (1, 5π/4) (b) (2, 3π)(c) (2, –2π/3) (d) (–3, 3π/4)
Sketch the polar curve θ = 1
Sketch the curve r = 1 + sin θ.
Sketch the curve r = cos 2θ.
(a) For the cardioid r = 1 + sin θ of Example 7, find the slope of the tangent line when θ = π/3.(b) Find the points on the cardioid where the tangent line is horizontal or vertical.Data from
Express 1/(1 – x)2 as a power series by differentiating Equation 1. What is the radius of convergence?
Express the number –1 + 3i/2 + 5i in the form a + bi.
Find an equation of the circle with radius 3 and center (2, –5).
(a) Find the radian measure of 60°. (b) Express 5π/4 rad in degrees.
Describe and sketch the regions given by the following sets. (a) {(x, y) |x ≥ 0} (b) {(x,y) | y = 1} (c) {(x, y)||y| < 1}
(a) By comparing areas, show that 1/2 < ln 2 < In 3/4.(b) Use the Midpoint Rule with n = 10 to estimate the value of ln 2.
Solve the inequality 1 + x < 7x + 5.
Write the sum 23 + 33 + · · · + n3 in sigma notation.
Find the roots of the equation x2 + x + 1 = 0.
Sketch the graph of the equation x2 + y2 + 2x – 6y + 7 = 0 by first showing that it represents a circle and then finding its center and radius.
(a) If the radius of a circle is 5 cm, what angle is subtended by an arc of 6 cm? (b) If a circle has radius 3 cm, what is the length of an arc subtended by a central angle of 3π/8 rad?
Solve the inequalities 4 ≤ 3x – 2 < 13.
Find Σ 1. i=1
Find an equation of the line through (1, –7) with slope –1/2.
Write the following numbers in polar form. (a) z = 1 + i (b) w = √3 – i
Draw the graph of the parabola y = x2.
Find the exact trigonometric ratios for θ = 2π/3.
Solve the inequality x2 – 5x + 6 ≤ 0.
Find an equation of the line through the points (–1, 2) and (3, –4).
Prove the formula for the sum of the n first positive integers: M Σ 3 Σ i = 1 + 2 + 3 + ... + n + - i=1 = n(n + 1) 2
Find the distance between the points.(6, –2), (–1, 3)
Find the product of the complex numbers 1 + i and √3 – i in polar form.
Sketch the region bounded by the parabola x = y2 and the line y = x – 2.
Prove the formula for the sum of the squares of the first n positive integers: n Σ i = 1 + 2 + 3 + i=1 + n?. n(n + 1)(2n + 1) 6
If cos θ = 2/5 and 0 < θ < π/2, find the other five trigonometric functions of θ.
Solve x3 + 3x2 > 4x.
Find (1/2 + 1/2i)10.
Use a calculator to approximate the value of x in Figure 12.Figure 12 X 40° 16
Sketch the graph of 9x2 + 16y2 = 144.
Express |3x – 2| without using the absolute-value symbol.
Evaluate M Σ (4i” – 3). 2 i=1
Graph the inequality x + 2y > 5.
Find the six sixth roots of z = –8 and graph these roots in the complex plane.
Sketch the curve 9x2 – 4y2 = 36.
Find all values of x in the interval [0, 2π] such that sin x = sin 2x.
Solve |2x – 5| = 3.
Find the distance between the points.(a, b), (b, a)
Find an equation of the line through the point (5, 2) that is parallel to the line 4x + 6y + 5 = 0.
Find 71 3 lim Σ 11-00 U I=!00
Evaluate: (a) eiπ(b) e–1+iπ/2
Sketch the graph of the equation y = 2x2 – 4x + 1.
Solve |x – 5| < 2.
Show that the lines 2x + 3y = 1 and 6x – 4y – 1 = 0 are perpendicular.
Sketch the curve x = 1 – y2.
Solve |3x + 2| ≥ 4.
If |x – 4| < 0.1 and |y – 7| < 0.2, use the Triangle Inequality to estimate |(x + y) – 11|.
Sketch the graph of the equation.x = 3
Sketch the graph of the equation.y = –2
Sketch the graph of the equation.xy = 0
Prove formula (b) of Theorem 3.Data from Formula b of theorem 3 M (b) Σ c = ne = i=1
Sketch the graph of the equation.|y| = 1
Find the remaining trigonometric ratios.tan α = 2, 0 < α < π/2
Prove formula (e) of Theorem 3 using a method similar to that of Example 5, Solution 1 [start with (1 + i)4 – i4].Data from Formula e of theorem 3Data from Example 5, Solution 1 Τ (e) Σ
Prove the identity.sin(π – x) = sin x
Solve the inequality.|x| < 3
Solve the inequality.|x – 4| < 1
Prove that a b lal 101'
Solve the inequality in terms of intervals and illustrate the solution set on the real number line.–3 < 1/x ≤ 1
Prove formula (e) of Theorem 3 using mathematical induction.Data from Formula e of theorem 3 Τ (e) Σ ;3 i=1 n(n + 1) Ο 2
Prove each equation.(a) Equation 14a (b) Equation 14b 14a 14b tan(x + y) = tan(x - y)= = tan x + tan y 1 - tan x tan y tan x - tan y 1+tan x tan y
Prove each equation.(a) Equation 18a (b) Equation 18b(c) Equation 18c 18a 18b 18c sin x cos y = [sin(x + y) + sin(x - y)] cos x cos y=[cos(x + y) + cos(x - y)] sin x sin y = [cos(x - y) cos(x + y)]
Prove the identity.cos(π/2 – x) = sin x
Prove the identity.sin(π/2 + x) = cos x
Sketch the region in the xy-plane.{(x, y) ||x| ≤ 2}
Solve the inequality.|x| ≥ 3
Evaluate i=1 3 i-1 2¹-
Evaluate Κ Σ (2i + 2'). i=1
Solve the inequality.|x – 6| < 0.1
Show that if a < b then a < (a + b)/2 < b.
Prove that |ab| = |a||b|. [Use Equation 4.]Equation 4 4 √a² = |a|
Evaluate IM Π ΣΙΣ ( «+»]. Σ (i + j) i=1 F-1 Fl

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