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mathematics
calculus

Questions and Answers of Calculus

In Problems a to b, sketch the graph of the given equation and final the area of the region bounded by it. a. r = a, a > 0 b. r = 2a cos θ, a > 0
Sketch the limacon r = 3 - 4 sin θ, and find the area of the region inside its small loop.
Sketch the limacon r = 2 - 4 cos θ, and find the area of the region inside its small loop.
Sketch the limacon r = 2 - 3 cos θ, and find the area of the region inside its large loop.
Sketch the leaf of the four-leaved roser r = 3 cos 2θ, and find the area of the region enclosed by it.
Sketch the three-leaved rose r = 4 cos 3θ, and find the area of the total region enclosed by it.
Sketch the region that is inside the circle r = 3 sin θ and outside the cardioid r = 1 4- sin θ, and find its area.
Sketch the region that is outside the circle r = 2 and in-side the lemniscate r2 = 8 cos 2θ, and find its area.
Sketch the limacon r = 3 - 6 sin θ, and find the area of the region that is inside its large loop, but outside its small loop.
Sketch the region in the first quadrant that is inside the cardioid r = 3 + 3 cos θ and outside the cardioid r = 3 + 3 sin θ, and find its area.
Sketch the region in the second quadrant that is inside the cardioid r = 2 + 2 sin θ and outside the cardioid r = 2 + 2 cos θ, and find its area.
Find the slope of the tangent line to each of the following curves at 0 = rr/3. a. r = 2 cos θ b. r = 1 + sin θ c. r = sin 2θ d. r = 4 - 3 cos θ
Find all points on the cardioid r = a(1 + cos θ) where the tangent line is a. Horizontal, and b. Vertical.
Find all points on the limacon r = 1 - 2 sin θ where the tangent line is horizontal.
Let r = f (Î¸), where f is continuous on the closed interval [Î±, Î²]. Derive the following formula for the length L. of the corresponding polar curve from
Use the formula of Problem 26 to find the perimeter of the cardioid r = a(1 + cos Î¸).
Find the length of the logarithmic spiral r = eθ/2 from θ = 0 to θ = 2π.
Find the total area of the rose r = a cos nθ, where n is a positive integer.
Sketch the graph of the strophoid r = sec θ - 2 cos θ, and find the area of its loop.
Consider the two circles r = 2a sin θ and r = 2b cos θ, with a and b positive. a. Find the area of the region inside both circles. b. Show that the two circles intersect at right angles.
Assume that a planet of mass in is revolving around the sun (located at the pole) with constant angular momentum mr2 dθ/dt. Deduce Kepler's Second Law: The line from the sun to the planet sweeps out
First Old Goat Problem A goat is tethered to the edge of a circular pond of radius a by a rope of length ka (0
Find the lengths of the limacons r = 2 + cos θ and r = 2 + 4 cos θ.
Find the area and length of the three-leaved rose r = 4 sin 3θ
Find the area and length of the leminiscate r2 = 8 cos 2θ
Plot the curve r = 4 sin(3θ / 2), 0 ≤ θ ≤ 4π, and then find its length.
In each of Problems a-c, find the Cartesian equation of the conic with the given properties. a. Vertices (±4; 0) and eccentricity 1 / 2 b. Eccentricity 1, focus (0, -3), and vertex (0, 0) c.
In Problems 1-3, use the process of completing the square to transform the given equation to a standard form. Then name the corresponding curve and sketch its graph. 1. 4x2 + 4y2 - 24x + 36y + 81 =
In each of Problems a to c, name the conic that has the given equation. Find its vertices and foci, and sketch its graph. a. y2 - 6x = 0 b. 9x2 + 4y2 - 36 = 0 c. 25x2 - 36y2 + 900 = 0
A rotation of axes through θ = 45o transforms x2 + 3xy + y2 = 10 into ru2 + sv2 = 10. Determine r and s. name the corresponding conic, and find the distance between its foci.
Determine the rotation angle 6 needed to eliminate the cross-product term in 7x2 + 8xy + y2 = 9. Then obtain the corresponding uv-equation and identify the conic that it represents.
In Problems a to c, a parametric representation of a curve is given. Eliminate the parameter to obtain the corresponding Cartesian equation. Sketch the given curve. a. x = 61 + 2, y = 21; ∞ < t <
In Problems 1 to 2, find the equations of the tangent line at t = 0 1. x = 2t3 - 4t + 7, y = t + ln(t + 1) 2. x = 3e-t, y = 1 / 2et
Find the length of the curve x = 1 + t3/2, y = 2 + t3/2, from t = 0 to t = 9.
Find the length of the curve x = cost t + t sin t, y = sin t - t cos t from 0 to π. Make a sketch.
In Problems a to c, analyze the given polar equation and sketch its graph. a. r = 6 cos θ b. r = 5 / sin θ c. r = cos 2θ
Find a Cartesian equation of the graph of r2 - 6r(cos θ + sin θ) + 9 = 0
Find a Cartesian equation of the graph of r2 cos 2θ = 39 and then sketch the graph.
Find the slope of the tangent line to the graph of r = 3 + 3 cos θ at the point on the graph where θ = 1/6 π.
Sketch the graph of r = 5 sin θ and r = 2 + sin θ and find their points of intersection.
Find the area of the region bounded by the graph of r = 5 - 5 cos θ.
Find the area of the region that is outside the limacon r = 2 + sin θ and inside the circle r = 5 sin θ.
A racing car driving on the elliptical race track x2/400 + y2/100 = 1 went out of control at the point (16, 6) and thereafter continued on the tangent line until it hit a tree at (14, k). Determine k.
In Problems a to c, sketch a graph of the cylinder or quadric surface. a. x2 + y2 + z2 = 64 b. x2 + z2 = 4 c. z = x2 + 4y2
In Problems a to b, find the maximum and minimum value of the function on the given interval. Use the second derivative test to determine whether each stationary point is a maximum or a minimum. a.
A storage can is to be made in the shape of a right circular cylinder of height h and radius r. Find the surface area of the container (including the circular top and bottom) as a function of r only,
A three-dimensional box without a lid is to be made of a material that costs $1 per square foot for the sides and $3 per square foot for the bottom. The box is to contain 27 cubic feet. Let I, w, and
In Problems 1-2, find the indicated derivative. 1. a. d/dx 2x3 b. d/dx 5x3 c. d/dx kx3 d. d/dx ax3 2. a. d/dx sin 2x b. d/dx sin 17t c. d/dt sin at d. d/dt sin bt
In Problems a-c, say whether the function is continuous and whether the given point. a. f(x) = 1/x2 - 1 at x = 2 b. f(x) = tan x at x = π/2 c. f(x) = |x - 4| at x = 4
Find the equation of the sphere that has (-2, 3, 3) and (4. 1, 5) as end points of a diameter.
Find the unit vectors that are perpendicular to the plane determined by the three points (3, -6, 4), (2, 1, 1), and (5, 0, -2).
Write the equation of the plane through the point (-5, 7, -2) that satisfies each condition. a. Parallel to the xz-plane b. Perpendicular to the x-axis c. Parallel to both the x- and y-axes d.
A plane through the point (2, -4, -5) is perpendicular to the line joining the points (-1, 5, -7) and (4, 1, 1). a. Write a vector equation of the plane. b. Find a Cartesian equation of the plane. c.
Find a Cartesian equation of the plane through the three points (2, 3, -1), (-1, 5, 2), and (-4, -2, 2).
Find the points where the line of intersection of the planes x - 2y + 4z - 14 = 0 and -x + 2y - 5z + 30 = 0 pierces the yz- and xz-planes.
Write the equation of the line in Problem 16 in parametric form. Find the points where the line of intersection of the planes x - 2y + 4z - 14 = 0 and -x + 2y - 5z + 30 = 0 pierces the yz- and
Find symmetric equations of the line through (4, 5, 8) and perpendicular to the plane 3x + 5y + 2z = 30. Sketch the plane and the line.
Write a vector equation of the line through (2, -2, 1) and (-3, 2, 4).
Find the center and radius of the sphere with equation .r2 + y2 + z2 - 6x + 2y - 8z = 0.
Sketch the curve whose vector equation is r(t) = t i + 1/2t2j + 1/3t3k, -2 ≤ t ≤ 3.
Find the symmetric equations for the tangent line to the curve of Problem 20 at the point where t = 2. Also find the equation of the normal plane at this point. r(t) = t i + 1/2t2j + 1/3t3k, -2 ≤ t
Find r'(π/2), T(π/2), and r"(π/2) if r(t) = (t cos t, t sin t, 2t)
Find the length of the curve r(t) = et sin t i + et cos t j + et k, 1 ≤ t ≤ 5
What heading and airspeed are required for an airplane to fly 450 miles per hour due north if a wind of 100 miles per hour is blowing in the direction N 60° E?
If r(t) = (e2t, e-t find each of the following:a.b. c. ˆ«0ln2 r(t) dt d. Dt[tr(t)] e. Dt[r(3t + 10)] f. Dt[r(t) ˆ™ r'(t)]
Find ri(t) and r"(r) for each of the following: a. r(t) = (In t)i - 3t2j b. r(t) = sin t i + cos 2t j c. r(t) = tan ti - t4 j
Suppose that an object is moving so that its position vector at time t is r(t) et i + e-t j + 2t k Find v(t),a(t), and k(t) at t = ln 2.
If r(t) = t i + t2 j + t3 k is the position vector for a moving particle at time t, find the tangential and normal components, aT and aN, of the acceleration vector at t = 1.
Let a = (2, -5), b = (1, 1), and c = (-6, 0). Find each of the following: a. 3a - 2b b. a ∙ b c. a ∙ (b + c) d. (4a + 5b) ∙ 3c e. ||e||e ∙ b f. e ∙ e - ||c||
For each equation in Problems a-c, name and sketch the graph in three-space. a. x2 + y2 = 81 b. x2 + y2 + z2 = 81 c. z2 = 4y
Write the following Cartesian equations in cylindrical co-ordinate form. a. x2 + y2 = 9 b. x2 + 4y2 = 16 c) x2 + y2 = 9z d. x2 + y2 + 4z2 = 10
Find the cosine of the angle between a and b and make a sketch. a. a = 3i + 2j, b = -i + 4j b. a = -5i - 3j, b = 2i - j c. a = (7, 0), b = (5, 1)
Find the Cartesian equation corresponding to each of the following cylindrical coordinate equations. a. r2 + z2 = 9 b. r2 cos2θ + z2 = 4 c. r2 cos2θ + z2 = 1
Write the following equations in spherical coordinate form. a. x2 + y2 + z2 = 4 b. 2x2 + 2y2 - 2z2 = 0 c. x2 - y2 - z2 = 1 d. x2 + y2 = z
Find the (straight-line) distance between the points whose spherical coordinates are (8, π/4, π/6) and (4, π/3, 3π/4).
Find the acute angle between the planes 2x - 4y + z = 7 and 3x + 2y - 5z = 9.
Show that if the speed of a moving particle is constant then its velocity and acceleration vectors are orthogonal.
Let a = -i + J + 2k. b = j - 2k, and e = 3i - J + 4k. Find each of the following if they are defined. a. a + b + c b. b ∙ c c. a ∙ (b × c) d. a × (b ∙ c) e. ||a - b|| f. ||b × c||
Find the angle between each pair of vectors. a. a = (1, 5, -1). b = (0, 1, 3) b. a = -i + 2k, b = i - j + 3k
Sketch the two position vectors a = 2i - J + 2k and b = 5i + j - 3k.Then find each of the following: a. Their lengths b. Their direction cosines c. The unit vector with the same direction as a d. The
Let a = 2i - j + k, b = + 3j + 2k, and c = i + 2j - k. Find each of the following: a. a × b b. a × (b + c) c. a ∙ (b × c) d. a × (b × c)
Plot the points whose coordinates are (1, 2, 3), (2, 0, 1), (-2, 4, 5), (0, 3, 0), and (-1, -2, -3). If appropriate, show the "box" as in Figures 4 and 5.
P(x, 5, z) is on al line through Q(2, -4, 3) that is parallel to one of the coordinate axes. Which axis must it be and what are x and z?
Write the equation of the sphere with the given center and radius. a. (1, 2, 3);5 b. (-2, -3, -6); √5 c. (π, e, √2); √π
Find the equation of the sphere whose center is (1 4, 5) and that is tangent to the xy-plane.
In Problems a-b complete the squares to find the center and ra-dius of the sphere whose equation is given. a. x2 + y2 + z2 - 12x + 14y - 8z + 1 = 0 b. x2 - y2 + z2 + 2x - 6y - 10z + 34 = 0
In Problems a-c, sketch the graphs of the given equations. Begin by sketching the traces in the coordinate planes. a. 2x + 6y + 3z = 12 b. 3x - 4y + 2z = 24 b. x + 3y - z = 6
Follow the directions of Problem 1 for (√3, -3, 3), (0, π, -3), (-2, 1/3, 2), and (0, 0, e).
In Problems 1-3, find the arc length of the given curve. 1. x = t. y = 1, z = 2t; 0 ≤ t ≤ 2 2. x = t/4, y = t/3, z = t/2; 1 ≤ t ≤ 3 3. x = t3/2, y = 3t, z = 4t:1 ≤ t ≤ 4
In Problems a-b, set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with n = 10 or a CAS w approximate the integral. a. x = √t, y = t, = t; 1 ≤ t ≤ 6 b. x
Find the equation of the sphere that has the line segment joining (-2, 3, 6) and (4, -1, 5) as a diameter.
Find the equations of the tangent spheres of equal radii whose centers are (-3, 1, 2) and (5, -3, 6).
Find the equation of the sphere that is tangent to the three coordinate planes if its radius is 6 and its center is in the first octant.
Find the equation of the sphere with center (l, 1, 4) that is tangent to the plane x + y = 12.
Describe the graph in three-space of each equation. a. z = 2 b. x = y c. xy = 0 d. xyz = 0 e. x2 + y2 = 4 f. z = √9 - x2 - y2
The sphere (x - 1)2 + (y + 2)2 + (z + 1)2 = 10 intersects the plane z = 2 in a circle. Find the circle's center and radius.
An object's position P changes so that its distance from (1, 2, -3) is always twice its distance from (1, 2, 3). Show that P is on a sphere and find its center and radius.
An object's position P changes so that its distance from (1.2. -3) always equals its distance from (2. 3. 2). Find the equation of the plane on which P lies
The solid spheres (x - 1)2 + (y - 2)2 + (z - 1)2 ≤ 4 and (x - 2)2 (y - 4)2 + (z - 3)2 ≤ 4 intersect in a solid. Find its volume.
Do Problem 45 assuming that the second solid sphere is (x - 2)2 + (y - 4)2 + (z - 3)2 ≤ 9.

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