Calculus/Multivariable and differential calculus:Exercises

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Multivariable and differential calculus:Exercises

Parametric Equations[edit]

1. Find parametric equations describing the line segment from P(0,0) to Q(7,17).

x=7t and y=17t, where 0 ≤ t ≤ 1

2. Find parametric equations describing the line segment from to .

3. Find parametric equations describing the ellipse centered at the origin with major axis of length 6 along the x-axis and the minor axis of length 3 along the y-axis, generated clockwise.

Polar Coordinates[edit]

20. Convert the equation into Cartesian coordinates:

Making the substitutions and gives

21. Find an equation of the line y=mx+b in polar coordinates.

Making the substitutions and gives

Sketch the following polar curves without using a computer.

22.
2-2sin-t.svg
23.
R-square-eq-4cos-t.svg
24.
2sin-5t.svg

Sketch the following sets of points.

25.
Polar-set-answer-1.svg
26.
Polar-set-answer-2.svg

Calculus in Polar Coordinates[edit]

Find points where the following curves have vertical or horizontal tangents.

40.

Horizontal tangents occur at points where . This condition is equivalent to .

Vertical tangents occur at points where . This condition is equivalent to .

The condition for a horizontal tangent gives:

Horizontal tangents occur at which correspond to the Cartesian points and .

The condition for a vertical tangent gives:

Vertical tangents occur at which correspond to the Cartesian points and .

Horizontal tangents at (2,2) and (2,-2); vertical tangents at (0,0) and (4,0)

41.

Horizontal tangents occur at points where . This condition is equivalent to .

Vertical tangents occur at points where . This condition is equivalent to .

The condition for a horizontal tangent gives:

Horizontal tangents occur at which correspond to the Cartesian points , , , and . Point corresponds to a vertical cusp however and should be excluded leaving , , and .

The condition for a vertical tangent gives:

Vertical tangents occur at which correspond to the Cartesian points , , and .

Horizontal tangents at (r,θ) = (4,π/2), (1,7π/6) and (1,-π/6); vertical tangents at (r,θ) = (3,π/6), (3,5π/6), and (0,3π/2)

Sketch the region and find its area.

42. The region inside the limaçon

Given an infinitesimal wedge with angle and radius , the area is . The total area is therefore .

9π/2
2-plus-cos-t.svg
43. The region inside the petals of the rose and outside the circle

There are 4 petals, as seen in the image below. The area of just one of the petals needs to be computed and the multiplied by .

It is first necessary to compute the angular limits of one of the petals. The petals start and end at points where . The bounds on one of the petals are .

Given an annular wedge with angle , inner radius , and an outer radius of , the area is . The total area of all 4 petals is therefore .

4cos-2t-and-2.svg

Vectors and Dot Product[edit]

60. Find an equation of the sphere with center (1,2,0) passing through the point (3,4,5)

The general equation for a sphere is where is the location of the sphere's center and is the sphere's radius.

It is already known that the sphere's center is . The sphere's radius is the distance between (1,2,0) and (3,4,5) which is .

Therefore the sphere's equation is: .

61. Sketch the plane passing through the points (2,0,0), (0,3,0), and (0,0,4)
Plane-intercepts-2-3-4.svg
62. Find the value of if and

.

Therefore: .

63. Find all unit vectors parallel to

The length of is . Therefore is a unit vector that points in the same direction as , and is a unit vector that points in the opposite direction as .

are the unit vectors that are parallel to .

64. Prove one of the distributive properties for vectors in :

.

65. Find all unit vectors orthogonal to in

Rotating counterclockwise gives . is orthogonal to , and the normalization of and its negative are the only unit vectors that are orthogonal to .

The magnitude of is so the only unit vectors that are orthogonal to are .

66. Find all unit vectors orthogonal to in

All vectors that are orthogonal to must satisfy .

The set of possible values of is . The restriction that becomes .

The set of possible and is an ellipse with radii and . One possible parameterization of and is and where . This parameterization yields where as the complete set of unit vectors that are orthogonal to .

Re-parameterizing by letting gives the set

67. Find all unit vectors that make an angle of with the vector

The angle that makes with the x-axis is counterclockwise.

Making a both a clockwise and a counterclockwise rotation of gives

Cross Product[edit]

Find and

80. and

81. and

Find the area of the parallelogram with sides and .

82. and

The cross product of vectors and is a vector with length where is the angle between and . is the area of the parallelogram with sides and , so this area is found by computing .

83. and

The cross product of vectors and is a vector with length where is the angle between and . is the area of the parallelogram with sides and , so this area is found by computing .


84. Find all vectors that satisfy the equation

The cross product is orthogonal to both multiplicand vectors. should be orthogonal to both and . However, so and are not orthogonal. The equation is never true, and therefore the set of vectors that satisfy the equation is "None".

85. Find the volume of the parallelepiped with edges given by position vectors , , and

The volume of a parallelepiped with edges defined by the vectors , , and is the absolute value of the scalar triple product: .

86. A wrench has a pivot at the origin and extends along the x-axis. Find the magnitude and the direction of the torque at the pivot when the force is applied to the wrench n units away from the origin.

The moment arm is , so the torque applied is

The magnitude of the torque is . The torque's direction is .

Prove the following identities or show them false by giving a counterexample.

87.

False:

88.

True: Once expressed in component form, both sides evaluate to

89.

True:

Calculus of Vector-Valued Functions[edit]

100. Differentiate .

101. Find a tangent vector for the curve at the point .

so a possible a tangent vector at is

102. Find the unit tangent vector for the curve .

so the unit tangent vector is

103. Find the unit tangent vector for the curve at the point .

so the unit tangent vector is

At :

104. Find if and .

For an arbitrary the position can be computed by the integral .

105. Evaluate

Motion in Space[edit]

120. Find velocity, speed, and acceleration of an object if the position is given by .

, ,

121. Find the velocity and the position vectors for if the acceleration is given by .

Length of Curves[edit]

Find the length of the following curves.

140.

For an infinitesimal step , the length traversed is approximately .

The total length is therefore:

141.

For an infinitesimal step , the length traversed is approximately .

The total length is therefore:

Parametrization and Normal Vectors[edit]

142. Find a description of the curve that uses arc length as a parameter:

For an infinitesimal step , the length traversed is approximately

Given an upper bound of , the arc length swept out from to is:

The arc length spans a range from to . For an arc length of , the upper bound on that generates an arc length of is , and the point at which this upper bound occurs is:

143. Find the unit tangent vector T and the principal unit normal vector N for the curve Check that TN=0.

A tangent vector is . Normalizing this vector to get the unit tangent vector gives:

A vector that has the direction of the principal unit normal vector is

Normalizing gives the principal unit normal vector:

Equations of Lines And Planes[edit]

160. Find an equation of a plane passing through points

Let denote a plane that contains points , , and . Let denote an arbitrary vector that is orthogonal to , and denote the position vector of an arbitrary point contained by . A point at position vector is contained by if and only if the displacement from is orthogonal to . This yields the equation .

The displacement from to , which is , and the displacement from to , which is , are both contained by so the cross product of these two displacements forms a candidate :

Any of , , and is a candidate . Let

The equation becomes

161. Find an equation of a plane parallel to the plane 2xy+z=1 passing through the point (0,2,-2)

Let denote a plane that is parallel to the plane and contains the point . Let denote an arbitrary vector that is orthogonal to , and denote the position vector of an arbitrary point contained by . A point at position vector is contained by if and only if the displacement from is orthogonal to . This yields the equation .

Any vector that is orthogonal to is also orthogonal to and vice versa. Since , the coefficient vector is orthogonal to , so a candidate is .

Since point is contained by , let .

The equation becomes

162. Find an equation of the line perpendicular to the plane x+y+2z=4 passing through the point (5,5,5).

163. Find an equation of the line where planes x+2yz=1 and x+y+z=1 intersect.

164. Find the angle between the planes x+2yz=1 and x+y+z=1.

165. Find the distance from the point (3,4,5) to the plane x+y+z=1.

Limits And Continuity[edit]

Evaluate the following limits.

180.

−2

181.

1/6

At what points is the function f continuous?

182.

183.

All points (x,y) except for (0,0) and the line y=x+1

Use the two-path test to show that the following limits do not exist. (A path does not have to be a straight line.)

184.

The limit is 1 along the line y=x, and −1 along the line y=−x

185.

The limit is 0 along the line y=0, and along the line x=2y

186.

The limit is 1 along the line y=0, and −1 along the line x=0

187.

The limit is 0 along any line of the form y=mx, and 2 along the parabola

Partial Derivatives[edit]

200. Find if

201. Find all three partial derivatives of the function

Find the four second partial derivatives of the following functions.

202.

203.

Chain Rule[edit]

Find

220.

221.

0

222.

0

Find

223.

Failed to parse (syntax error): {\displaystyle f_s = \cos(s+t)\cos(2s-2t) - 2\sin(s+t)\sin(2s-2t)\\ f_t = \cos(s+t)\cos(2s-2t) + 2\sin(s+t)\sin(2s-2t)}

224.

Failed to parse (syntax error): {\displaystyle \displaystyle f_s = \frac{-2t(t+1)}{(st+s-t)^2}\\ \displaystyle f_t = \frac{2s}{(st+s-t)^2}}


225. The volume of a pyramid with a square base is , where x is the side of the square base and h is the height of the pyramid. Suppose that and for Find

Tangent Planes[edit]

Find an equation of a plane tangent to the given surface at the given point(s).

240.

241.

242.

243.

Maximum And Minimum Problems[edit]

Find critical points of the function f. When possible, determine whether each critical point corresponds to a local maximum, a local minimum, or a saddle point.

260.

Local minima at (1,1) and (−1,−1), saddle at (0,0)

261.

Saddle at (0,0)

262.

Saddle at (0,0), local maxima at local minima at

Find absolute maximum and minimum values of the function f on the set R.

263.

Maximum of 9 at (0,−2) and minimum of 0 at (0,1)

264. R is a closed triangle with vertices (0,0), (2,0), and (0,2).

Maximum of 0 at (2,0), (0,2), and (0,0); minimum of −2 at (1,1)


265. Find the point on the plane xy+z=2 closest to the point (1,1,1).

266. Find the point on the surface closest to the plane

Double Integrals over Rectangular Regions[edit]

Evaluate the given integral over the region R.

280.

281.

282.

Evaluate the given iterated integrals.

283.

284.

Double Integrals over General Regions[edit]

Evaluate the following integrals.

300. R is bounded by x=0, y=2x+1, and y=5−2x.

301. R is in the first quadrant and bounded by x=0, and

Use double integrals to compute the volume of the given region.

302. The solid in the first octant bound by the coordinate planes and the surface

303. The solid beneath the cylinder and above the region

304. The solid bounded by the paraboloids and

Double Integrals in Polar Coordinates[edit]

320. Evaluate for

321. Find the average value of the function over the region

322. Evaluate

323. Evaluate if R is the unit disk centered at the origin.

Triple Integrals[edit]

340. Evaluate

In the following exercises, sketching the region of integration may be helpful.

341. Find the volume of the solid in the first octant bounded by the plane 2x+3y+6z=12 and the coordinate planes.

342. Find the volume of the solid in the first octant bounded by the cylinder for , and the planes y=x and x=0.

343. Evaluate

344. Rewrite the integral in the order dydzdx.

Cylindrical And Spherical Coordinates[edit]

360. Evaluate the integral in cylindrical coordinates:

361. Find the mass of the solid cylinder given the density function

362. Use a triple integral to find the volume of the region bounded by the plane z=0 and the hyperboloid

363. If D is a unit ball, use a triple integral in spherical coordinates to evaluate

364. Find the mass of a solid cone if the density function is

365. Find the volume of the region common to two cylinders:

Center of Mass and Centroid[edit]

380. Find the center of mass for three particles located in space at (1,2,3), (0,0,1), and (1,1,0), with masses 2, 1, and 1 respectively.

381. Find the center of mass for a piece of wire with the density for

382. Find the center of mass for a piece of wire with the density for

383. Find the centroid of the region in the first quadrant bounded by the coordinate axes and

384. Find the centroid of the region in the first quadrant bounded by , , and .

385. Find the center of mass for the region , with the density

386. Find the center of mass for the triangular plate with vertices (0,0), (0,4), and (4,0), with density

Vector Fields[edit]

One can sketch two-dimensional vector fields by plotting vector values, flow curves, and/or equipotential curves.

401. Find and sketch the gradient field for the potential function .

Radial vector field.svg
402. Find and sketch the gradient field for the potential function for and .

403. Find the gradient field for the potential function

Line Integrals[edit]

420. Evaluate if C is the line segment from (0,0) to (5,5)

421. Evaluate if C is the circle of radius 4 centered at the origin

422. Evaluate if C is the helix

423. Evaluate if and C is the arc of the parabola

424. Find the work required to move an object from (1,1,1) to (8,4,2) along a straight line in the force field

Conservative Vector Fields[edit]

Determine if the following vector fields are conservative on

440.

No

441.

Yes

Determine if the following vector fields are conservative on their respective domains in When possible, find the potential function.

442.

443.

Green's Theorem[edit]

460. Evaluate the circulation of the field over the boundary of the region above y=0 and below y=x(2-x) in two different ways, and compare the answers.

461. Evaluate the circulation of the field over the unit circle centered at the origin in two different ways, and compare the answers.

462. Evaluate the flux of the field over the square with vertices (0,0), (1,0), (1,1), and (0,1) in two different ways, and compare the answers.

Divergence And Curl[edit]

480. Find the divergence of

481. Find the divergence of

482. Find the curl of

483. Find the curl of

484. Prove that the general rotation field , where is a non-zero constant vector and , has zero divergence, and the curl of is .

If , then

, and then

Surface Integrals[edit]

500. Give a parametric description of the plane

501. Give a parametric description of the hyperboloid

502. Integrate over the portion of the plane z=2−xy in the first octant.

503. Integrate over the paraboloid

504. Find the flux of the field across the surface of the cone
with normal vectors pointing in the positive z direction.

505. Find the flux of the field across the surface
with normal vectors pointing in the positive y direction.

Stokes' Theorem[edit]

520. Use a surface integral to evaluate the circulation of the field on the boundary of the plane in the first octant.

521. Use a surface integral to evaluate the circulation of the field on the circle

522. Use a line integral to find
where , is the upper half of the ellipsoid , and points in the direction of the z-axis.

523. Use a line integral to find
where , is the part of the sphere for , and points in the direction of the z-axis.

Divergence Theorem[edit]

Compute the net outward flux of the given field across the given surface.

540. , is a sphere of radius centered at the origin.

541. , is the boundary of the tetrahedron in the first octant bounded by

542. , is the boundary of the cube

543. , is the surface of the region bounded by the paraboloid and the xy-plane.

544. , is the boundary of the region between the concentric spheres of radii 2 and 4, centered at the origin.

545. , is the boundary of the region between the cylinders and and cut off by planes and