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).

Let denote an arbitrary plane. 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 . Therefore the equation that defines is .

The equation is equivalent to . This implies that the coefficient vector is orthogonal to the plane defined by . A line that passes through point and is parallel to is parameterized by:

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

To describe the line that forms the intersection between and , both and will be expressed as functions of (it can also be the case that and are functions of , etc.). Let denote equation and denote equation .

To find as a function of subtract 2 times from to get

To find as a function of subtract from to get

Parameterizing with gives the parameterization

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

Let denote an arbitrary plane. 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 . Therefore the equation that defines is .

Let be the plane described by and be the plane described by

Since , the coefficient vector is orthogonal to .

Since , the coefficient vector is orthogonal to .

The angle between and is equivalent to the angle between and :

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

Given a unit length vector , consider an axis oriented in the direction of . The " coordinate" is determined by orthogonally projecting points onto the axis. Given a position vector , the expression computes the coordinate.

The equation is equivalent to

Letting , the plane consists of all points whose coordinate is . The coordinate of is .

The distance between the plane and the point along the axis is

The distance is the distance between the point and plane along a direction that is orthogonal to the plane, and is hence the shortest distance.

Limits And Continuity[edit]

Evaluate the following limits.

180.

181.

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

By repeatedly applying the chain rule:

201. Find all three partial derivatives of the function

Find the four second partial derivatives of the following functions.

202.

The first derivatives are: and

The second derivatives are:

203.

The first derivatives are: and

The second derivatives are:

Chain Rule[edit]

Find

220.

The single derivatives are: ; ; ; and

The chain rule gives:

221.

The single derivatives are: ; ; ; and

The chain rule gives:

222.

The single derivatives are: ; ; ; ; ; and

The chain rule gives:

Find

223.

The single derivatives are: ; ; ; ; ; and

The chain rule gives:

and

Therefore: and

224.

The single derivatives are: ; ; ; ; ; ; ; ; and

The chain rule gives:

and

Therefore: and


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

The single derivatives are: ; ; ; and

The chain rule gives:

Tangent Planes[edit]

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

240.

Start with a point that is on the surface. Perturbing the , , and coordinates by infinitesimal amounts , , and respectively, changes the value of by the infinitesimal amount , and the value of by . To remain in the surface it must be the case that .

To linearly extrapolate the condition to a tangent plane at , replace the infinitesimal perturbations , , and with large perturbations , , and to get . Any point in the tangent plane at can be reached by an appropriate choice of , , and where . Any point in the tangent plane at must satisfy .

The point lies in the surface, and the tangent plane is .

The point lies in the surface, and the tangent plane is .

The tangent planes are therefore:

241.

Start with a point that is on the surface. Perturbing the , , and coordinates by infinitesimal amounts , , and respectively, changes the value of by the infinitesimal amount , and the value of by . To remain in the surface it must be the case that .

To linearly extrapolate the condition