mathematics for engineering

Explain how the following graphs are obtained from the graph of fxa.. Determine whether each curve is the graph of a function of... Use the given graph to estimate the value ofeach deriv

Mathematics for Engineering

Exercise Book

Trần Thanh Hiệp - 2023

CALCULUSChapter 1: Function and Limit

1 Find the domain of each function:

a f x  x 2 b   21

f xxx

A AA

A A



2 Find the range of each function:

a f x  x 1 b

  2 2 2 2 1 1  12 1 1[ 1, )

f xxxxxxr

   

c f x sinx

d  

D Rr

      

3 Determine whether is even, odd, or neither

a   21

xf x

f xx

xf x

   

fxf xevenfxf xodd

4 Explain how the following graphs are obtained from the graph of f(x)

a f x 4 b f x 3 c f x 23 d f x  5 4

5 Suppose that the graph of f x  x is given Describe how the graph of the function

1 2

y x  can be obtained from the graph of f

6 Let f x  x and g x  2 x Find each function

        

a 1

f xx

   

f xLf xL

f xL

  

 

   

0; :0

 

a

 

tan 3lim

tan 5

9 3lim

  

10 Determine whether each curve is the graph of a function of If it is, state the domain x

and range of the function.a): No;

b) Yes; D=[-3,3]; r=[-2,3]

11 The graph of is given f

a Find each limit, or explain why it does not exist.i lim0  

xf x

 , lim0  

xf x

 

xf x

  c f x ln 2 x5

13 Find the constant that makes continuous on Rmf

a

xmxf x

15.

  

If lim =L then is the horizontal asymptotes.If lim = then is the vertical asymptotes

 

   

     





1 Use the given graph to estimate the value ofeach derivative

a f' 3  b f' 1 c f' 0  d f' 3 

2 Find an equation of the tangent line to the curve at the given point:

c y 32xx2, x1 d 3 2

 

  

 

' 0

' ;'sin ' coscos ' sin

1tan '

cos1ln '

' ' ' ' ' '.

   

 

  

'tan '

cos'ln '



  

khk h

f xsin 2 1

yex4 Find y"

d sincos

 

 

6 Find dy (differential) for:

 b y x1,x3 c ylnx21 , x1 and dx2

7 The graph of is given State the numbers at which is not differentiable

  



  

' '' '

15 If x2y225 and dy dt/6, find dx dt when y = 4 and x > 0./

16 If z2 x2 y2 z0 , dx dt/ 2,dy dt/ 3, find dz dt when / x5,y1217 Find the linearization L(x) of the function y = f(x) at x = a.

v ts ta tv t

19 A water tank has the shape of an inverted circular cone with base radius 2 m and height4 m If water is being pumped into the tank at a rate of 2 m /min, find the rate at which the3

water level is rising when the water is 3 m deep.

20 The length of a rectangle is increasing at a rate of 7 cm/s and its width is increasing at a rate of 5 cm/s When the length is 22 cm and the width is 12 cm, how fast is the area of the rectangle increasing?

Chapter 3: Applications of Differentiation

1 Find the absolute maximum and absolute minimum values of the function on the given interval

a f x 3x212x5, 0;3  b f x  x3 3x5,  0;3

c f x x 4x2, 1;2 d   ln , 1;22

 2 Find the critical numbers of the function:

  

;x is critical value of the function f(x) if:* ' 0 or

* ' : does not exist.

a f x 5x24x b   21

xf x

  c f x xlnx

3 Find all numbers that satisfy the conclusion of the Rolle's Theoremf'’(x) = 0

a f x x x2, 2;0 b f x   x 2x2,  0;24 Find all numbers that satisfy the conclusion of the Mean Value Theorem

 all numbers (x) that satisfy the conclusion of the Mean Value Theorem on interval [a,b]

If f' xf b  f a b a

 and xa b, 

a f x 3x22x5,1;1 b f x e2x, 0;3 

5 If f 1 10 and f' x   2, x  1;4 , how small can f 4 possibly be?6 Find where the function f x 3x44x312x21 is increasing and where it is decreasing.

7 Find the inflection points for the function

 .A gần M nhất khi AM nhỏ nhất2

 2 22

10 Find two numbers whose difference is 100 and whose product is a minimum.11 Find two positive numbers whose product is 100 and whose sum is a minimum.12 Use Newton’s method with the specified initial approximation x1to find x3of equation f(x) =0

f xxx

c   x2 x 2f x

 

cos sin

s tv t dtv ta t dt

a v t sintcos ,t s 0 0b v t 10sint3cos ,t s  0c v t   10 3t 3 ,t2 s 2 10

xxb a

Sf x dxIxxn

xa xxx xxxxbIRxf xf xf xILxf xf xf x

[ ( ) 4 ( ) 2 ( ) 4 ( )3

a f x  1 xx

 

, x 1, 4 a=1; b=4; n=6, find L6

3 Repeat part (1) using right endpoints

4 For the function f x x x3,   2,2 Estimate the area under the graph of using four

approximating rectangles and taking the sample points to be a Right endpoints

b Left endpoints c Midpoints

5 Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to

approximate the given integral with the specified value of n.

a

,n 6

6 Let

220 1

Find the approximations L4, R4,M4,T4 and S4 for I.

  21

f xx

 

g xdtt

a Find the displacement of the particle during the time period 1 ≤ t ≤ 4

v t dt

11 Suppose the acceleration function and initial velocity are a(t)= t + 3 (m/s ), v(0)=52

(m/s) Find the velocity at time t and the distance traveled when 0 ≤ t ≤ 5.

  

v t a t dt

12 A particle moves along a line with velocity function v t  t2 t , where is measured in

meters per second Find the displacement and the distance traveled by the particle duringthe time interval t0,2

cos sin

13 Evaluate the integral

b

xxe dx

 f 2 1

tdtt 

:ln , , sin , cos , , kx

u dvu vv duLet

What is the average value of on the interval f[1,3]?

 

= 132

 

13 3 1 1



f x dxF xF bF aa

; F’(x)=f(x)a

 

  

* If 1, I is convergent* If 1, is divergent

ax bp

  

d  2 20 2

ye dy

j

43 3

k

l



20 Use the Comparison Theorem to determine whether the integral is convergent ordivergent

* If I is divergent, J is divergent* If J is convergent, I is covergent

* If 1, I is convergent* If 1, is divergent

ax bppI



c 1 2

d 1 1 6

f



LINEAR ALGEBRAChapter 1: Systems of Linear Equations

1 Write the augmented matrix for each of the following systems of linear equations and then solve them.

  

 

 

  

  

   

  

 

   

4 Determine the values of m such that the system of linear equations has exactly one

a 2

c

5 Determine the values of m such that the system of linear equations is inconsistent.

a 2

6 Find a, b and c so that the system

 has the solution 3, 1, 2 

7 Consider the matrix

  

12 Solve the system

Chapter 2: Matrix Algebra

f A A  A I

4 Find the inverse of each of the following matrices.

13

6 Find A when

a  1 1 23

8 Find 1

10 Compute 1 3

T RR be a linear transformation, and assume that T  1, 2  1,1

  

c d

         

a Compute T11, 5  b Compute T1,11

c Find the matrix of T d Compute T12,3

* Linear transformation:

  

12 Let T R: 2R2 be a linear transformation such that the matrix of Tis

1 3

  Find T3, 2 

13 The (2;1)-entry of the product

Chapter 3: Determinants and Diagonalization

1 Evaluate the determinant

a

2 Find the minors and the cofactors of the matrix

AadjA LetAkA

A AadjAk AadjA

k AadjAkAadjA

f A12adjA

1 31 3

5 42 1

9 Find the determinant of the matrix

Chapter 4: Vector Geometry

1 Find the equations of the line through the points P (2, 0, 1) and P (4, − 1, 1).01

2 Find the equations of the line through P (3, − 1, 2) 0 parallel to the line with equations: 1 2

1 3 4

3 Determine whether the following lines intersect and, if so, find the point of intersection.

4 Compute ||v|| if v equals:

a (2,-1,2) b 2(1,1,-1) c -3(1,1,2) d (1,2,3) - (4,1,2)5 Find a unit vector in the direction from A(3,-1,4) to B(1,3,5)

 6 Find ||v − 3w|| when ||v|| = 2, ||w|| = 1, and v · w = 2

7 Compute the angle between u = (-1,1,2) and v = (-1,2,1).

8 Show that the points P(3, − 1, 1), Q(4, 1, 4), and R(6, 0, 4) are the vertices of a right

9 Suppose a ten-kilogram block is placed on a flat surface inclined 30◦ to the horizontal as in the diagram Neglecting friction, how much force is required to keep the block from sliding down the surface?

10 Find the projection of u =(2,-3,1) on d = (-1,1,3) and express u = u + u where u is 121

parallel to d and u is 2 orthogonal to d.

11 Find an equation of the plane through P (1, − 1, 3) with n = (-3,-1,2) as normal.0

12 Find an equation of the plane through P (3, − 1, 2) that is 0 parallel to the plane with equation 2x − 3y − z = 6.

13 Find the shortest distance from the point M(2, -1, − 3) to the plane with equation (P): 3x − y + 4z = 1 Also find the point H on this plane closest to M

14 Find the equation of the plane through P(1, 3, − 2), Q(1, 1, 5), and R(2, − 2, 3).n= PQxPR

15 Find the shortest distance between the nonparallel lines

ax by cz dM x y zaxbyczdd M



16 Compute u · v where:

a u = (2,-1,3), v = (-1,1,1) b u = (-2,1,4), v = (-1,5,1)

17 Find all real numbers x such that:a (3,-1,2) and (3,-2,x) are orthogonal.b (2,-1,1) and (1,x,2) are at an angle of π/3

18 Find the three internal angles of the triangle with vertices:a A(3, 1, − 2), B(3, 0, − 1), and C(5, 2, − 1)

21 Find the volume of the parallelepiped determined by the vectors u = (1,2,-1),

v = (3,4,5) and w = (-1,2,4).ũxv=(14,-8,-2)

22 In each case show that that T is either projection on a line, reflection in a line, orrotation through an angle, and find the line or angle

23 Determine the effect of the following transformations.

a Rotation through π/2 , followed by projection on the y axis, followed by reflection inthe line y = x.

b Projection on the line y = x followed by projection on the line y = −x c Projection on the x axis followed by reflection in the line y = x.24 Find the reflection of the point P in the line y = 1 + 2x in R if:2

a P = P(1, 1)b P = P(1, 4)

25 Find the angle between the following pairs of vectors.a u = (1,-1,4), v = (5,2,-1)

b u = (2,1,5), v = (0,3,1)

26 In each case, compute the projection of u on v

27 Find the shortest distance between the following pairs of nonparallel lines and find thepoints on the lines that are closest together.

Chapter 5: The Vector Space R n

1 Let x    1, 2, 2 ,u0,1, 4 , v  1,1, 2 and w3,1,2 in R3 Find scalars a, b andc such that xaubvcw

a bb

  

Let Su uuin RS

If r=m: independent;If r<m: dependent

5 Find all values of m such that the set S is a basis of R

6 Find a basis for and the dimension of the subspace Ua U2st s s, , t| ,s tR

   

    

8 Find all values of m for which x lies in the subspace spanned by S

a x 3,2,m and S   1, 1,1 , 2, 3, 4    

3 2det 0

mA

 

rk k k

mmmhas so

 

12 Let u  1, 3, 2 ,v 1,1,0 and w1,2, 3 Compute u v w 

13 Let u v, 3 such that u3,v4 and u v 2 Find

a u v b 2u3v c ||2u - v||

1) Express the limit as a definite integral over [a, b].

2) Let u=(.,.,.), v=(.,.,.) Which of the following vectors belong to span{u,v}.

3) Let T be a linear transformation in the plane, T[x,y] =  ,  1 3 , 32

T x y x yxy Choose the corect statement.

(i) T is projection on the line y = sqrt(3) x

(ii) T is projection on the line y = -sqrt(3) x

(iii) T is rotation through pi/3.

4) Let A be a 3x7 matrix of of rank 2 Which of the following statements are true?(i) A has 2 independent rows

(ii) A has 6 independent columns(iii) The null space of A has dimension 2.

5) Let T R: 2R2 be rotation through pi/2 followed by reflection in the line y = x Then T is:

(i) Rotation through 180 degrees.(ii) Reflection in the y-axis.(iii) Reflection about y = -x(iv) Reflection in the x-axis.

6) Two car start moving from the same point, One travels south at 28 mi/h and the other travels west at 70 mi/h At what rate is the distance betwin the cars increasing 5 hours later? (75.39 mi/h)

7) Let u = (-2,5,1), v=(1,3,7) Let w be such that u – 2v = ||w||.wFind the length of w.