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