Calculus/Multivariable and differential calculus:Exercises

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Multivariable and differential calculus:Exercises

Contents

Parametric EquationsEdit

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 CoordinatesEdit

20. Convert the equation into Cartesian coordinates:

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

Sketch the following polar curves without using a computer.

22.
23.
24.

Sketch the following sets of points.

25.
26.

Calculus in Polar CoordinatesEdit

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

40.

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

41.

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/4)

Sketch the region and find its area.

42. The region inside the limaçon
9π/2
43. The region inside the petals of the rose and outside the circle

Vectors and Dot ProductEdit

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

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

63. Find all unit vectors parallel to

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

Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \begin{eqnarray} c(\mathbf u + \mathbf v) &=& c(\langle u_1, u_2, u_3\rangle + \langle v_1, v_2, v_3\rangle)\\ &=& c\langle u_1+v_1, u_2+v_2, u_3+v_3\rangle\\ &=& \langle c(u_1+v_1), c(u_2+v_2), c(u_3+v_3)\rangle\\ &=& \langle cu_1+cv_1, cu_2+cv_2, cu_3+cv_3\rangle\\ &=& \langle cu_1, cu_2, cu_3\rangle + \langle cv_1, cv_2, cv_3\rangle\\ &=& c\mathbf u + c\mathbf v. \end{eqnarray}}

65. Find all unit vectors orthogonal to in

66. Find all unit vectors orthogonal to in

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

Cross ProductEdit

Find and

80. and

81. and

Find the area of the parallelogram with sides and .

82. and

83. and


84. Find all vectors that satisfy the equation

None

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

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.

, so the torque is directed along

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

87.

False:

88.

Once expressed in component form, both sides evaluate to

89.

Failed to parse (unknown function "\begin{eqnarray}"): {\displaystyle \begin{eqnarray}(\mathbf u-\mathbf v)\times(\mathbf u+\mathbf v)&=&(\mathbf u-\mathbf v)\times\mathbf u + (\mathbf u-\mathbf v)\times\mathbf v\\&=&\mathbf u\times\mathbf u - \mathbf v\times\mathbf u + \mathbf u\times\mathbf v - \mathbf v\times\mathbf v\\&=&\mathbf u\times\mathbf v-\mathbf v\times\mathbf u\\&=&\mathbf u\times\mathbf v + \mathbf u \times \mathbf v\\&=&2(\mathbf u\times\mathbf v)\end{eqnarray}}

Calculus of Vector-Valued FunctionsEdit

100. Differentiate .

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

102. Find the unit tangent vector for the curve .

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

104. Find if and .

105. Evaluate

Motion in SpaceEdit

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 CurvesEdit

Find the length of the following curves.

140.

141.

Parametrization and Normal VectorsEdit

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

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

Equations of Lines And PlanesEdit

160. Find an equation of a plane passing through points

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

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 ContinuityEdit

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 DerivativesEdit

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 RuleEdit

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 PlanesEdit

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

240.

241.

242.

243.

Maximum And Minimum ProblemsEdit

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 RegionsEdit

Evaluate the given integral over the region R.

280.

281.

282.

Evaluate the given iterated integrals.

283.

284.

Double Integrals over General RegionsEdit

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 CoordinatesEdit

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 IntegralsEdit

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 CoordinatesEdit

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 CentroidEdit

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 FieldsEdit

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 .

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

403. Find the gradient field for the potential function

Line IntegralsEdit

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 FieldsEdit

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 TheoremEdit

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 CurlEdit

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 IntegralsEdit

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' TheoremEdit

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 TheoremEdit

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