mae101 mathematics for engineering exercises

At what rate is the areaof the square increasing when the area of the square is 16cm2?29.Find the linear approximationLxtoy = f xnearx = afor the function.af x =x3− x2+ 3,a = −2... a Fin

MAE101: MATHEMATICS FORENGINEERING

Ho Chi Minh City

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c)h(x) = 4|x − 5|

a)f (x) = ln(x + 1) −√x

x − 1 b)g(x) =√

a)f (x − 2) + 3 b) 1

2f (x) − 3 c)2 (f x) + 5

a) Slope= −6, pass through(1 3), b) Slope= 25,x- intercept= 8.

a)f (x) = 3x, g(x) = x + 5 b)f (x) = 3

2x + 1,g(x) =2x

b) lim

(1 +h)2−1h

c) lim

√x + 4 − 1

x + 3

d) lim

x3− 1x2− 1

a) lim

x2+ 2x − 6x − 2

b) lim

x6− 1x10− 1

c) lim

2 − x|x + 2|

d) lim

tan(3 )xtan(5 )x

a)f (x) =

8x2 ifx ≥ 1ax − 5 ifx < 1

b)f (x) =

ax2+ 2x ifx < 2x3− ax ifx ≥ 2

c)f (x) =

2x − x −5 3x2− 9 ifx 6= 3a ifx = 3

d)f (x) =x2− 1√

x − 1 ifx > 1ax + 1 ifx ≤ 1

f (x) =

1 +x2 ifx ≤02 − x if0 < x ≤ 2(x− 2)2 ifx >2

Chapter 2: Derivatives

15.Suppose thatf (2) =−3, g(2) = 4, f0(2) =−2andg0(2) = 7.Findh0(2).a)h(x) = 5f (x) − 4g( )x

b)h(x) =f (x)g(x)

c)h(x) = f (x)g(x)

d)h(x) = g(x)1 + f ( )x

16.Use the given graph to estimate thevalue of each derivativef0.

24.Findy0by implicit differentiation.

a)√x + √y = 1 b)3x3+ 9xy2= 5x3 c)y sin(xy) = y2+ 2

25.Use the information in the following table to findh0(a)at the given value fora

a)h(x) = f (g(x)),a = 0.b)h(x) = g(f (x)), a= 0.

c)h(x) = (1 + g(x)), a = 2.d)h(x) = g(2 + f (x2)), a= 1.

26.Ifz2= x2+ y2,dx/dt = 2anddy/dt = 3, finddz/dtwhenx = 5andy = 12.27.If V is the volume of a cube with edge length x and the cube expands as timepasses, finddV /dtin terms ofdx/dt.

28.Each side of a square is increasing at a rate of 6cms At what rate is the areaof the square increasing when the area of the square is 16cm2?

29.Find the linear approximationL(x)toy = f (x)nearx = afor the function.a)f (x) =x3− x2+ 3,a = −2 b)f (x) =√3

1 + x,a = 0.

30.A particle moves on a vertical line so that its coordinate at time tisy =t3− 12t + 3, t ≥ 0.Find the velocity and acceleration functions.

Chapter 3: APPLICATIONS OFDIFFERENTIATION

31.Use the graph to state the absolute and local maximum and minimum valuesoff, gfunctions.

32.Find the critical numbers of the function.a)f (x) = 5x2+ 4x

b)f (x) = 4√x − x2

c)y = ln(x − 2)

d)f (x) = x

2− 1x2+ 2x − 3

33.Find the absolute maximum and absolute minimum values off on the giveninterval.

a)f (x) = 3x2− 12x + 5, [0, 3]

b)f (x) = (x − x2)2, [− , 1]1

c)f (t) = t√ [ 14 − t2, − , 2]

d)f (x) = ln(x2+ x + 1), [ 1− , 1]

34.Iff (1) = 10andf0(x) ≥ 2for1 ≤ ≤x 4, how small canf (4)possibly be?35.Find all numbers that satisfy the conclusion of Rolle’s Theorem.a)f (x) = 5 − 12x + 3x2,[1 3],

37.Suppose the derivative of a functionf isf0(x) = (x+ 1)2(x − 3) (5x− 6)4 Onwhat interval isf increasing?

38.Letf (x) = x

x2− 1

a) Find the vertical and horizontal asymptotes.b) Find the intervals of increase or decrease.c) Find the local maximum and minimum values.d) Find the intervals of concavity and the inflection points.

39.Determine intervals wherefis concave up or concave down, and determine theinflection points of f

43.Find the numbers whose product is 100 and whose sum is minimum.44.Find the dimensions of a rectangle with area1000 m2whose perimeter is assmall as possible.

45.Use Newton’s method to approximate the given number correct to eight mal places.

20 b) 100√100

46.Use Newton’s method with the specified initial approximationx1to findx3,the third approximation to the root of the given equation (Give your answer tofour decimal places).

a)x3+ 2x − 4 = 0,x1= 1.b)x5− x − 1 = 0,x1= 1.

48.Find the antiderivativeFof f that satisfies the given condition.

a)f (x) = 5x − 2x, F (0) = 4 b)f ( ) = 4 − 3(1 + xx ) , F (1) = 0.49.A particle is moving with the given data Find the position of the particle.a)v(t) = sin(t) − cos(t), s(0) = 0 b)a(t) = t2− 4t + 6,s(0) = 0, s(1) = 20

50.The graph of a functionfis shown Which graph is an antiderivative offandwhy?

Chapter 4-6: INTEGRALS & TECHNIQUESOF INTEGRATION

51.Estimate the area under the graph off (x) =x2+ 1fromx = −1tox = 2usingfour rectangles and

a) left endpoints b) right endpoints c) midpoints

52.Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule toapproximate the given integral with the specified value of (Round your answersn

to six decimal places.)I)R2√4

e)F (x) =Rx2xtdt

f)F (x) =R√x

g)F (x) =R0sin x

√1 − t2dt

h)F (x) =Rcos x1

√1 − t2dt

i)F (x) =R√x1

a)f (x) = 4x − x,[0, 4] b)f ( ) = sin(4 ), [x x −π, π].

60.The velocity function (in meters per second) is given for a particle movingalong a line Find (a) the displacement and (b) the distance traveled by the particleduring the given time interval.

I)v(t) = 3t − 5,0 ≤ t ≤ 3 II) v(t) = t2− t − 8 1 ≤ t ≤ 62 ,

61.Water flows from the bottom of a storage tank at a rate off (t) = 200 − 4tlitersper minute, where0 ≤ t ≤ 50 Find the amount of water that flows from the tankduring the first 10 minutes.

62.First make a substitution and then use integration by parts to evaluate theintegral.

d)R0∞ xdx(x2+2)2

f)R−∞−1e−2tdt

x1+ 2x2−x3+ 2x4+x5= 0x1+ 2x2+ 2x3+x5= 02x1+ 4x2− x2 3+ 3x4+ x5= 0

x1+ 2x2−x3+ x4+x5= 0−x1− 2x2+ 2x3+ x5= 0−x1− 2x2+ 3x3+ x4+ 3x5= 0

x1+x2−x3+ 2x4+x5= 0x1+ 2x2−x3+x4+x5= 02x1+ 3x2−x3+ 2x4+ x5= 04x1+ 5x2− x2 3+ 5x4+ 2x5= 0

x1+x2− 2x3− 2x4+ 2x5= 02x1+ 2x2− x4 3− x4 4+ x5= 0x1−x2+ 2x3+ 4x4+x5= 0−2x1− 4x2+ 8x3+ 10x4+ x5= 0

Name: .Class: .

a bc d

c − 3d −d2a + d a + b

a −b b−cc − d d − a

= 2

1 1−3 1

+ 2

82.Compute the following:

3 2 15 1 0

− 5

3 0 −21 −1 2

− 5

+ 7

−2 13 2

− 4

1 −20 −1

+ 3

2 −3−1 −2

òd) 3 −1 2 − 2 9 3 4 + 3 11 −6

1 −5 4 02 1 0 6

0 −1 21 0 −4−2 4 0

83.FindAif:

a)A+ B = 3 + 2BA b)2A − B = 5(A + 2 )B

85.Compute the following matrix products.

1 30 −2

ò ï

2 −10 1

1 −1 22 0 4

2 3 11 9 7−1 0 2

5 0 −71 5 9

1 0 00 1 00 0 1

3 −25 −79 7

e) 1 −1 321−8

−7 1 −1 3

3 15 2

ò ï

2 −1−5 3

2 3 1

a 0 00 b 00 0 c

a 0 00 b 00 0 c

a0 0 00 b0 00 0 c0

86.Finda, b, a1, andb1if:

a ba1 b1

ò ï

3 −5−1 2

1 −12 0

2 1−1 2

ò ï

a ba1 b1

7 2−1 4

ò87.In each case, show that the matrices are inverses of each other.

3 51 2

2 −5−1 3

3 01 −4

4 01 −3

1 2 00 2 31 3 1 ,

7 2 −6−3 −1 32 1 −2

3 00 5

1 −1−1 3

4 13 2

1 0 −13 2 0−1 −1 0

1 −1 2−5 7 −11−2 3 −5

89.In each case, solve the systems of equations by finding the inverse of thecoefficient matrix.

a) 3x − y = 52x + 2y = 1

b)2x − y = 03x − 4y = 1

x + y + 2z = 5x + y + z = 0x + 2y + 4z = −2

x + 4y + 2z = 12x + 3 + 3y z = −14x + y + 4z = 0

a) (3 )A−1=

1 −10 1

b) (2 )AT=

1 −12 3

c)(I + 3A)−1=

2 01 −1

d) I − 2AT −1=

2 11 1

1 −10 1

2 31 1

1 02 1

g) AT− 2I −1= 2

1 12 3

h) A−1− 2I T= −2

1 11 0

ò91.LetT :R3→R2be a linear transformation.

a) FindT837 ifT

b) FindT56−13 ifT

ò92.(Extra) LetT :R4→R3be a linear transformation.

a) FindT13−2−3

ifT110−1 =

23−1 andT

0−111 =

b) FindT5−12−4

ifT1111 =

51−3 andT

−1102 =

201

93.In each case determine allsandtsuch that the given matrix is symmetric:

s 2s stt −1 s

t s2 s d)

2 s t2s 0 s + t

3 3 t

94.In each case find the matrixA.

A + 3

1 −1 01 2 4

òãT =2 10 53 8

3AT+ 2

1 00 2

8 03 1

c) 2A − 3 1 2 0 T= 3AT+ 2 1 −1 T

2AT− 5

1 0−1 2

= 4A− 9

1 1−1 0

ò

Name: .Class: .

Chapter 3: Determinants and Diagonalization

95.Compute the determinants of the following matrices.

−3 2 14 5 62 −3 1

2 −1 10 2 10 0 4

x y 1−1 −2 11 5 1

m −1 01 2 12 m −3

96.Find the minors and the cofactors of the matrix.

a)A =

2 3−2 4

b)B =

−3 4 26 3 1

2 1 0

(Note: The adjugate of matrixA, denotedadj A( ), is the transpose of this cofactormatrix)

a) Find the adjugate and the inverse of the matrix.b) CalculateA(adjA)and(adjA)A.

98.LetA =

1 ∗ ∗ ∗0 −1 ∗ ∗0 0 2 ∗

0 0 0 2

ê Finda) 2A−1

100.LetA, BandCdenoten × nmatrices and assume thatdetA = −1,detB = 2,anddetC = 3.Evaluate:

b)B =

m 1 31 3 2

2 3 1

é103.Find the characteristic polynomial of the matrix

a)A =

3 51 2

b)B =

1 0 0−2 2 1

−1 1 1

é104.Find the eigenvalues and corresponding eigenvectors of the matrix

a)A =

−3 510 2

b)B =

5 42 3

c)C =

1 0 0−2 3 0

1 0 0 −4

106.Find the(1 2), -cofactor and(3, 1)- cofactor of the matrix

−1 3 −24 5 −77 8 1

107.LetA =

1 3 10 1 0

2 −1 x

.For which values ofxisAinvertible?

Name: .Class: .

Chapter 4: VECTOR GEOMETRY

108.Find the distance between the following pairs of points.

0 and

2 and

−352 and

40−2 and

109.Find a unit vector in the direction fromA(3, −1, 4)toB(1, ,3 5).110.Find the equations of

a) the line through the pointsP (3,−1 4), andQ(1, 0, −1).

b) the line passing through P (1, ,0 −3) and parallel to the line with parametricequationsx = −1 + 2t, y = 2 − t,andz = 3 + 3t.

111.Find the point of intersection (if any) of the following pairs of lines.

x = 3 + ty = 1 − 2tz = 3 + 3t and

x = 4 + 2sy = 6 + 3sz = 1 + s b)

x = 1 − ty = 2 + 2tz = −1 + 3t and

x = 2sy = 1 + sz = 3

112.Computeu · vwhere:a)u = (2, −1,3), v = (− 1 1).1, ,

117.Calculate the distance from the pointP (3, ,2 −1)to the line

x = 2 + 3ty = 1 − tz = 3 − 2t

and find the pointQon the line closest toP.118.Find an equation of each of the following planes.a) Passing throughA(2, 1, 3), B(3, −1, 5), andC(1, ,2 −3).b) Passing throughA(1, −1, 6), B(0, 0, 1), andC(4, ,7 −11).

c) Passing throughP (2,−3 5), and parallel to the plane with equation3x− y−z = 02 119.Find the area of the triangle with the following vertices.

a)A(3, −1, 2), B(1, 1, 0), andC(1, 2, −1) b)A(3, 0, 1), B(5, 1, 0), andC(7, ,2 −1)

120.Find two orthogonal vectors that are both orthogonal tov = (1, 2, 0).121.Find the volume of the parallelepiped determined byw, u, andvwhen:a)w = (2, ,1 1), v = (1, 0 2), u = (2 1, −1), ,

x = 2 + ty = −1 − tz = 3 + 4t and

x = 1 + ty = −t

z = 1 + 4t b)

x = 3 + 3ty = t

z = 2 and

x = −1 + 3ty = 2 + tz = 2

126.In each case solve the problem by finding the matrix of the operator.a) Find the projection ofv = (1,−2 3), on the plane with equation3x − y + 2z = 05 b) Find the reflection ofv = (1,−2 3), in the plane with equationx − y + 3z = 0.

Name: .Class: .

Chapter 5: The Vector SpaceRn

127.Letu = (2, −1 5 0), , , v = (4, 3 1, , −1), andw = (−6 2 0, , , 3)be vectors inR4.

Findxusing each equation.

3x1+ 3x2+ 15x3+ 11x4= 0x1− 3x2+x3+x4= 02x1+ 3x2+ 11x3+ 8x4= 0

135.Find bases for the row and column spaces ofAand determine the rank of A

a)A =

2 −4 6 82 −1 3 24 −5 9 100 −1 1 2

b)A =

2 −1 1−2 1 14 −2 3−6 3 0

c)A =

1 −1 5 −2 22 −2 −2 5 10 0 −12 9 −3−1 1 7 −7 1

d)A =

1 2 −1 3−3 −6 3 −2

ò136.Which of the following are subspaces ofR3?

a){(2 + a, b − a, b) | a, b ∈ R}

b){(a + b, a, b) | a, b ∈ R}

c){(2a + b, , ab) | a, b ∈ R}0

137.Letu = (1,−3, −2), v = (− 1, 0)1, andw = (1, ,2 −3) Computeku − v + wk

138.Letu, v ∈R3such thatkuk = 3, vk = 4k andu · v = −2 Finda)ku + vk

b)k2u + 3vk

c)k2u − vk