Calculus/Multivariable and differential calculus:Exercises

Parametric Equations

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

2. Find parametric equations describing the line segment from

P(x_{1},y_{1})

Q(x_{2},y_{2})

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

20. Convert the equation into Cartesian coordinates:

r=\sin(\theta )\sec ^{2}(\theta ).

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

Sketch the following polar curves without using a computer.

22.

r=2-2\sin(\theta )

23.

r^{2}=4\cos(\theta )

24.

r=2\sin(5\theta )

Sketch the following sets of points.

25.

\{(r,\theta ):\theta =2\pi /3\}

26.

\{(r,\theta ):|\theta |\leq \pi /3{\mbox{ and }}|r|<3\}

Calculus in Polar Coordinates

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

40.

r=4\cos(\theta )

41.

r=2+2\sin(\theta )

Sketch the region and find its area.

42. The region inside the limaçon

2+\cos(\theta )

43. The region inside the petals of the rose

4\cos(2\theta )

and outside the circle

r=2

Vectors and Dot Product

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

|\mathbf {u} +3\mathbf {v} |

\mathbf {u} =\langle 1,3,0\rangle

and

\mathbf {v} =\langle 3,0,2\rangle

63. Find all unit vectors parallel to

\langle 1,2,3\rangle

64. Prove one of the distributive properties for vectors in

\mathbb {R} ^{3}

c(\mathbf {u} +\mathbf {v} )=c\mathbf {u} +c\mathbf {v}

65. Find all unit vectors orthogonal to

3\mathbf {i} +4\mathbf {j}

\mathbb {R} ^{2}

66. Find all unit vectors orthogonal to

3\mathbf {i} +4\mathbf {j}

\mathbb {R} ^{3}

67. Find all unit vectors that make an angle of

\pi /3

with the vector

\langle 1,2\rangle

Cross Product

Find $\mathbf {u} \times \mathbf {v}$ and $\mathbf {v} \times \mathbf {u}$

80.

\mathbf {u} =\langle -4,1,1\rangle

and

\mathbf {v} =\langle 0,1,-1\rangle

81.

\mathbf {u} =\langle 1,2,-1\rangle

and

\mathbf {v} =\langle 3,-4,6\rangle

Find the area of the parallelogram with sides $\mathbf u$ and ${\mathbf v}$ .

82.

\mathbf {u} =\langle -3,0,2\rangle

and

\mathbf {v} =\langle 1,1,1\rangle

83.

\mathbf {u} =\langle 8,2,-3\rangle

and

\mathbf {v} =\langle 2,4,-4\rangle

84. Find all vectors that satisfy the equation

\langle 1,1,1\rangle \times \mathbf {u} =\langle 0,1,1\rangle

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

\langle 5,0,0\rangle

\langle 1,4,0\rangle

, and

\langle 2,2,7\rangle

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

\mathbf {F} =\langle 1,2,3\rangle

is applied to the wrench n units away from the origin.

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

87.

\mathbf {u} \times (\mathbf {u} \times \mathbf {v} )=\mathbf {0}

88.

\mathbf {u} \cdot (\mathbf {v} \times \mathbf {w} )=\mathbf {w} \cdot (\mathbf {u} \times \mathbf {v} )

89.

(\mathbf {u} -\mathbf {v} )\times (\mathbf {u} +\mathbf {v} )=2(\mathbf {u} \times \mathbf {v} )

Calculus of Vector-Valued Functions

100. Differentiate

\mathbf {r} (t)=\langle te^{-t},t\ln t,t\cos(t)\rangle

101. Find a tangent vector for the curve

\mathbf {r} (t)=\langle 2t^{4},6t^{3/2},10/t\rangle

at the point

t=1

102. Find the unit tangent vector for the curve

\mathbf {r} (t)=\langle t,2,2/t\rangle ,\ t\neq 0

103. Find the unit tangent vector for the curve

\mathbf {r} (t)=\langle \sin(t),\cos(t),e^{-t}\rangle ,\ t\in [0,\pi ]

at the point

t=0

104. Find

\mathbf {r}

\mathbf {r} '(t)=\langle {\sqrt {t}},\cos(\pi t),4/t\rangle

and

\mathbf {r} (1)=\langle 2,3,4\rangle

105. Evaluate

\displaystyle \int _{0}^{\ln 2}(e^{-t}\mathbf {i} +2e^{2t}\mathbf {j} -4e^{t}\mathbf {k} )dt

Motion in Space

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

\mathbf {r} (t)=\langle 3\sin(t),5\cos(t),4\sin(t)\rangle

121. Find the velocity and the position vectors for

t\geq 0

if the acceleration is given by

\mathbf {a} (t)=\langle e^{-t},1\rangle ,\ \mathbf {v} (0)=\langle 1,0\rangle ,\ \mathbf {r} (0)=\langle 0,0\rangle

Length of Curves

Find the length of the following curves.

140.

\mathbf {r} (t)=\langle 4\cos(3t),4\sin(3t)\rangle ,\ t\in [0,2\pi /3].

141.

\mathbf {r} (t)=\langle 2+3t,1-4t,3t-4\rangle ,\ t\in [1,6].

Parametrization and Normal Vectors

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

\mathbf {r} (t)=\langle t^{2},2t^{2},4t^{2}\rangle \ t\in [1,4].

143. Find the unit tangent vector T and the principal unit normal vector N for the curve

\mathbf {r} (t)=\langle t^{2},t\rangle .

Check that T⋅N=0.

Equations of Lines And Planes

160. Find an equation of a plane passing through points

(1,1,2),\ (1,2,2),\ (-1,0,1).

161. Find an equation of a plane parallel to the plane 2x−y+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+2y−z=1 and x+y+z=1 intersect.

164. Find the angle between the planes x+2y−z=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

Evaluate the following limits.

180.

\displaystyle \lim _{(x,y)\rightarrow (1,-2)}{\frac {y^{2}+2xy}{y+2x}}

181.

\displaystyle \lim _{(x,y)\rightarrow (4,5)}{\frac {{\sqrt {x+y}}-3}{x+y-9}}

At what points is the function f continuous?

182.

f(x,y)=\ln |x-y|

183.

f(x,y)=\displaystyle {\frac {\ln(x^{2}+y^{2})}{x-y+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.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {4xy}{3x^{2}+y^{2}}}

185.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {y}{\sqrt {x^{2}-y^{2}}}}

186.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{3}-y^{2}}{x^{3}+y^{2}}}

187.

\displaystyle \lim _{(x,y)\rightarrow (0,0)}{\frac {x^{2}y^{2}+y^{6}}{x^{3}}}

Partial Derivatives

200. Find

\partial z/\partial x

\displaystyle z(x,y)={\frac {1}{\ln(xy)}}

201. Find all three partial derivatives of the function

\displaystyle f(x,y,z)=xe^{y^{2}+z}

Find the four second partial derivatives of the following functions.

202.

f(x,y)=\cos(xy)

203.

f(x,y)=xe^{y}

Chain Rule

Find $df/dt.$

220.

f(x,y)=x^{2}y-xy^{3},\ x(t)=t^{2},\ y(t)=t^{-2}

221.

f(x,y)={\sqrt {x^{2}+y^{2}}},\ x(t)=\cos(2t),\ y(t)=\sin(2t)

222.

\displaystyle f(x,y,z)={\frac {x-y}{y+z}},\ x(t)=t,\ \displaystyle y(t)=2t,\ z(t)=3t

Find $f_{s},\ f_{t}.$

223.

f(x,y)=\sin(x)\cos(2y),\ x=s+t,\ y=s-t

224.

\displaystyle f(x,y,z)={\frac {x-z}{y+z}},\ x(t)=s+t,\ y(t)=st,\ z(t)=s-t

225. The volume of a pyramid with a square base is

V={\frac {1}{3}}x^{2}h

, where x is the side of the square base and h is the height of the pyramid. Suppose that

\displaystyle x(t)={\frac {t}{t+1}}

and

\displaystyle h(t)={\frac {1}{t+1}}

for

t\geq 0.

Find

V'(t).

Tangent Planes

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

240.

xy\sin(z)=1,\ (1,2,\pi /6),\ (-1,-2,5\pi /6).

241.

z=x^{2}e^{x-y},\ (2,2,4),\ (-1,-1,1).

242.

z=\tan ^{-1}(x+y),\ (0,0,0).

243.

\sin(xyz)=1/2,\ (\pi ,1,1/6).

Maximum And Minimum Problems

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.

f(x,y)=x^{4}+2y^{2}-4xy

261.

f(x,y)=\tan ^{-1}(xy)

262.

f(x,y)=2xye^{-x^{2}-y^{2}}

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

263.

f(x,y)=x^{2}+y^{2}-2y+1,\ R=\{(x,y)\mid x^{2}+y^{2}\leq 4\}

264.

f(x,y)=x^{2}+y^{2}-2x-2y,

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

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

266. Find the point on the surface

z=x^{2}+y^{2}+10

closest to the plane

x+2y-z=0.

Double Integrals over Rectangular Regions

Evaluate the given integral over the region R.

280.

\displaystyle \iint _{R}(x^{2}+xy)dA,\ R=\{(x,y)\mid x\in [1,2],\ y\in [-1,1]\}

281.

\displaystyle \iint _{R}(xy\sin(x^{2}))dA,\ R=\{(x,y)\mid x\in [0,{\sqrt {\pi /2}}],\ y\in [0,1]\}

282.

\displaystyle \iint _{R}{\frac {x}{(1+xy)^{2}}}dA,\ R=\{(x,y)\mid x\in [0,4],\ y\in [1,2]\}

Evaluate the given iterated integrals.

283.

\displaystyle \int _{0}^{2}\int _{0}^{1}x^{5}y^{2}e^{x^{3}y^{3}}dydx

284.

\displaystyle \int _{1}^{4}\int _{0}^{2}e^{y{\sqrt {x}}}dydx

Double Integrals over General Regions

Evaluate the following integrals.

300.

\displaystyle \iint _{R}xydA,

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

301.

\displaystyle \iint _{R}(x+y)dA,

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

y=x^{2},

and

y=8-x^{2}.

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

z=8-x^{2}-2y^{2}.

303. The solid beneath the cylinder

z=y^{2}

and above the region

R=\{(x,y)\mid y\in [0,1],\ x\in [y,1]\}.

304. The solid bounded by the paraboloids

z=x^{2}+y^{2}

and

z=50-x^{2}-y^{2}.

Double Integrals in Polar Coordinates

320. Evaluate

\displaystyle \iint _{R}2xydA

for

R=\{(r,\theta )\mid r\in [1,3],\ \theta \in [0,\pi /2]\}

321. Find the average value of the function

f(r,\theta )=1/r^{2}

over the region

\{(r,\theta )\mid r\in [2,4]\}.

322. Evaluate

\displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}{\sqrt {x^{2}+y^{2}}}dydx.

323. Evaluate

\displaystyle \iint _{R}{\frac {x-y}{x^{2}+y^{2}+1}}dA

if R is the unit disk centered at the origin.

Triple Integrals

340. Evaluate

\displaystyle \int _{1}^{\ln 8}\int _{0}^{\ln 4}\int _{0}^{\ln 2}e^{-x-y-2z}dxdydz.

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

z=\sin(y)

for

y\in [0,\pi ]

, and the planes y=x and x=0.

343. Evaluate

\displaystyle \int _{0}^{1}\int _{y}^{2-y}\int _{0}^{2-x-y}xydzdxdy.

344. Rewrite the integral

\displaystyle \int _{0}^{1}\int _{-2}^{2}\int _{0}^{\sqrt {4-y^{2}}}dzdydx

in the order dydzdx.

Cylindrical And Spherical Coordinates

360. Evaluate the integral in cylindrical coordinates:

\displaystyle \int _{0}^{3}\int _{0}^{\sqrt {9-x^{2}}}\int _{0}^{\sqrt {x^{2}+y^{2}}}{\frac {1}{\sqrt {x^{2}+y^{2}}}}dzdydx

361. Find the mass of the solid cylinder

D=\{(r,\theta ,z)\mid r\in [0,3],\ z\in [0,2]\}

given the density function

\delta (r,\theta ,z)=5e^{-r^{2}}

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

z={\sqrt {17}}-{\sqrt {1+x^{2}+y^{2}}}

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

\iiint _{D}(x^{2}+y^{2}+z^{2})^{5/2}dV

364. Find the mass of a solid cone

\{(\rho ,\phi ,\theta )\mid \phi \leq \pi /3,\ z\in [0,4]\}

if the density function is

\delta (\rho ,\phi ,\theta )=5-z

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

x^{2}+z^{2}=1,\ y^{2}+z^{2}=1

Center of Mass and Centroid

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

\rho (x)=1+\sin(x)

for

x\in [0,\pi ].

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

\rho (x)=2-x^{2}/16

for

x\in [0,4].

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

x^{2}+y^{2}=16.

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

y=\ln(x)

y=0

, and

x=e

385. Find the center of mass for the region

\{(x,y)\mid x\in [0,4],y\in [0,2]\}

, with the density

\rho (x,y)=1+x/2.

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

\rho (x,y)=1+x+y.

Vector Fields

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

401. Find and sketch the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y)={\sqrt {x^{2}+y^{2}}}

402. Find and sketch the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y)=\sin(x)\sin(y)

for

|x|\leq \pi

and

|y|\leq \pi

403. Find the gradient field

\mathbf {F} =\nabla \phi

for the potential function

\phi (x,y,z)=e^{-z}\sin(x+y)

Line Integrals

420. Evaluate

\int _{C}(x^{2}+y^{2})ds

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

421. Evaluate

\int _{C}(x^{2}+y^{2})ds

if C is the circle of radius 4 centered at the origin

422. Evaluate

\int _{C}(y-z)ds

if C is the helix

\mathbf {r} (t)=\langle 3\cos(t),3\sin(t),t\rangle ,\ t\in [0,2\pi ]

423. Evaluate

\int _{C}\mathbf {F} \cdot d\mathbf {r}

\mathbf {F} =\langle x,y\rangle

and C is the arc of the parabola

\mathbf {r} (t)=\langle 4t,t^{2}\rangle ,\ t\in [0,1]

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

\displaystyle \mathbf {F} ={\frac {\langle x,y,z\rangle }{x^{2}+y^{2}+z^{2}}}

Conservative Vector Fields

Determine if the following vector fields are conservative on $\mathbb {R} ^{2}.$

440.

\langle -y,x+y\rangle

441.

\langle 2x^{3}+xy^{2},2y^{3}+x^{2}y\rangle

Determine if the following vector fields are conservative on their respective domains in $\mathbb {R} ^{3}.$ When possible, find the potential function.

442.

\langle y,x,1\rangle

443.

\langle x^{3},2y,-z^{3}\rangle

Green's Theorem

460. Evaluate the circulation of the field

\mathbf {F} =\langle 2xy,x^{2}-y^{2}\rangle

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

\mathbf {F} =\langle 0,x^{2}+y^{2}\rangle

over the unit circle centered at the origin in two different ways, and compare the answers.

462. Evaluate the flux of the field

\mathbf {F} =\langle y,-x\rangle

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

480. Find the divergence of

\langle 2x,4y,-3z\rangle

481. Find the divergence of

\displaystyle {\frac {\langle x,y,z\rangle }{1+x^{2}+y^{2}}}

482. Find the curl of

\langle x^{2}-y^{2},xy,z\rangle

483. Find the curl of

\langle z^{2}\sin(y),xz^{2}\cos(y),2xz\sin(y)\rangle

484. Prove that the general rotation field

\mathbf {F} =\mathbf {a} \times \mathbf {r}

, where

{\mathbf a}

is a non-zero constant vector and

\mathbf {r} =\langle x,y,z\rangle

, has zero divergence, and the curl of

\mathbf {F}

2\mathbf {a}

$\mathbf {F} =\mathbf {a} \times \mathbf {r} =\langle a_{2}z-a_{3}y,a_{3}x-a_{1}z,a_{1}y-a_{2}x\rangle =\langle f,g,h\rangle$ , and then

$\nabla \cdot \mathbf {F} =\mathbf {f} _{x}+\mathbf {g} _{y}+\mathbf {h} _{z}=0+0+0=0,$

$\nabla \times \mathbf {F} =\langle h_{y}-g_{z},f_{z}-h_{x},g_{x}-f_{y}\rangle =\langle a_{1}+a_{1},a_{2}+a_{2},a_{3}+a_{3}\rangle =2\mathbf {a} .$

Surface Integrals

500. Give a parametric description of the plane

2x-4y+3z=16.

501. Give a parametric description of the hyperboloid

z^{2}=1+x^{2}+y^{2}.

502. Integrate

f(x,y,z)=xy

over the portion of the plane z=2−x−y in the first octant.

503. Integrate

f(x,y,z)=x^{2}+y^{2}

over the paraboloid

z=x^{2}+y^{2},\ z\in [0,4].

504. Find the flux of the field

\mathbf {F} =\langle x,y,z\rangle

across the surface of the cone

z^{2}=x^{2}+y^{2},\ z\in [0,1],

with normal vectors pointing in the positive z direction.

505. Find the flux of the field

\mathbf {F} =\langle -y,z,1\rangle

across the surface

y=x^{2},\ z\in [0,4],\ x\in [0,1],

with normal vectors pointing in the positive y direction.

Stokes' Theorem

520. Use a surface integral to evaluate the circulation of the field

\mathbf {F} =\langle x^{2}-z^{2},y,2xz\rangle

on the boundary of the plane

z=4-x-y

in the first octant.

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

\mathbf {F} =\langle y^{2},-z^{2},x\rangle

on the circle

\mathbf {r} (t)=\langle 3\cos(t),4\cos(t),5\sin(t)\rangle .

522. Use a line integral to find

\iint _{S}(\nabla \times F)\cdot \mathbf {n} dS

where

\mathbf {F} =\langle x,y,z\rangle

S

is the upper half of the ellipsoid

{\frac {x^{2}}{4}}+{\frac {y^{2}}{9}}+z^{2}=1

, and

\mathbf {n}

points in the direction of the z-axis.

523. Use a line integral to find

\iint _{S}(\nabla \times F)\cdot \mathbf {n} dS

where

\mathbf {F} =\langle 2y,-z,x-y-z\rangle

S

is the part of the sphere

x^{2}+y^{2}+z^{2}=25

for

3\leq z\leq 5

, and

\mathbf {n}

points in the direction of the z-axis.

Divergence Theorem

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

540.

\mathbf {F} =\langle x,-2y,3z\rangle

S

is a sphere of radius

{\sqrt {6}}

centered at the origin.

541.

\mathbf {F} =\langle x,2y,z\rangle

S

is the boundary of the tetrahedron in the first octant bounded by

x+y+z=1

542.

\mathbf {F} =\langle y+z,x+z,x+y\rangle

S

is the boundary of the cube

\{(x,y,z)\mid |x|\leq 1,|y|\leq 1,|z|\leq 1\}

543.

\mathbf {F} =\langle x,y,z\rangle

S

is the surface of the region bounded by the paraboloid

z=4-x^{2}-y^{2}

and the xy-plane.

544.

\mathbf {F} =\langle z-x,x-y,2y-z\rangle

S

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

545.

\mathbf {F} =\langle x,2y,3z\rangle

S

is the boundary of the region between the cylinders

x^{2}+y^{2}=1

and

x^{2}+y^{2}=4

and cut off by planes

z=0

and

z=8

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