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Name:...........................................Class:...........................................Mathematics for EngineeringExercise BookTrần Thanh Hiệp - 2021CALCULUSChapter 1: Function and Limit Find the domain of each function: a.f x   x  2 b.  21 f x x x  c.  ln 1  1x f x x x   2...

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Name:........................................... Class:...........................................

Mathematics for Engineering Exercise Book

Trần Thanh Hiệp - 2021 1

CALCULUS Chapter 1: Function and Limit

1. Find the domain of each function:

a.

f  x  x  2

b.

x 2  1 (  2; )

f  x 

1 2 x  x

c.

f  x  ln  x  1 

(  ; 0)  (1; )

x x 1

(1; )

2. Find the range of each function: a.

f  x  x  1

b.

f  x   x2  2 x

(0; )

f  x  sin x

c.

(  1; )

[-1;1]

3. Determine whether is even, odd, or neither

x f  x  2 x 1 a.

b.

f  x 

Odd

x2 x 4 1

c.

f  x 

Even

x x 1

Neither

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

f  x  4

b.

Left 4

f  x  3

c.

Up 3

Down 3 Left 2

5. Suppose that the graph of

f  x  x

f  x  2  3

d.

f  x  5  4

Down 4 Right 5

is given. Describe how the graph of the function

y  x  1  2 can be obtained from the graph of f .

Left 1 Up 2 6. Let

f  x  x

and

g x  2 x

. Find each function 2

a. f og 2 x

7. Let

c. g o g

b. g 0 f

f  x 

2

2

x

d. f o f 2 x

4

x

x2  x  1 x . Find

1  f  x  x a. 

b. f  2 x  1

x2 1 x   1 x x1

2x 

1 2x  1

8. Use the table to evaluate each expression x f(x) g(x)

1 3 6

a. f(g(1))

2 1 3

4 2 1

b. g(f(1))

5 e.

3 4 2

1

f.

6 5 3

c. f(f(1))

2

g o f  3

5 2 2

d. g(g(1))

4

3

g o f 6  2

9. Evaluate the following limits x 2  x  12 lim a. x  3 x  3 7 t2  9  3 lim t2 e. t  0

x6  1 lim 10 b. x  1 x  1 3 5

x2  x  12 lim 3 f. x   x  3

tan 3x lim c. x  0 tan 5 x

d.

3 5

1 1 lim    x 0  x x   g.

2 3  h  lim h 0

 9

h

6

h.

x2 1 lim x 1 x  1

3

1 12

0





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



[-3;3];[-2;3]

11. The graph of f is given.

a. Find each limit, or explain why it does not exist. 4

i.

lim f  x 

x 0 

iii.

lim f  x  x 1

,

lim f  x

x 0 

and

and

lim f  x  x 0

lim f  x 

1;4

x 4

b. At what numbers is discontinuous? 12. Determine where the function

a.

f  x 

2x 2  x  1 x 2

(  ; 2)  (2; )

2;0; 

b.

f  x

f  x 

0;2 is continuous x 9 2

4x  4x  1

c.

f  x ln  2 x  5 (

R

5 ; ) 2

13. Find the constant m that makes f continuous on R 2 mx  2 x , x  2 f  x   3 x  mx, x  2 b.

x 2  m 2 , x  4 f  x    mx  20, x  4 a.

m

m=-2

2 3

 x2  1 , x 1  f  x   x  1  mx  1, x 1  d.

e 2 x  1 , x 0  f  x   x m , x 0  c. m=2

m=3

14. Find the numbers at which the function

 x  2, x  0  f  x   2 x 2 , 1  x 0  2  x, x  1 

is discontinuous.

x= 0,x=1

5

Chapter 2: Derivatives

1. Use the given graph to estimate the value of each derivative a.

f '   3

b.

f '  1

9 2

c.

0

f ' 0  

d.

9 4

f '  3 9 2

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

a.

y

x 1 , x 2

 3, 2

b.

y

g= 5-x

2x , x 1 2

 0,0

g=2x 2

c. y 3  2 x  x ,

x 1

3  2x , y x 1 d.

g=2

y  1

g=1-x

3. Find y '

1 y x 2  x x   2 x a. 1

2x 

3 2 1 x  2 2 x

b. y  x  x 2 x 1 2 4 x x x

c.

y

x2 x 1

x 2  2x (x  1)2

6

d. y  x x  2 x2 

e.

y ln  x2  1 

1 x

2x 1  2 2 x 1 x

x 2 x2

y e x sin  2 x 1 

f.

x x e sin(2 x 1)  2 e cos(2x+1)

4. Find y " 3 b. y  2 x  1

3 x 1 a. y  xe

x c. y e cos x

5 8 (2 x 1) 3 9

6e3 x 1  9xe3 x 1

2e  x sinx

5. Find dy / dt for: 3 a. y  x  x  2, dx / dt 2 and x 1

2 b. y ln x, dx / dt 1 and x e

1 e2

8

 y sin      cos  t 3 d.  and

2 t 16 c. y  tan t and 4 

 3

6. Find dy for: a.

y

1 x 1 2

b. y  x  1, x 3

c.

 2x dx ( x 2 1) 2

y ln  x2  1

, x 1 and dx 2

1

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

l

8. A table of values for f , f ', g and g ' is given

a. If

h  x  f  g  x  

c. If

F  x  fo f  x

9. If

h  x  4  3 f  x

h '  1

b. If

H  x  g o f  x 

F '2

d. If

G  x  go g  x 

, where

f  1 7, f '  1 4

, find

, find

, find

h '  1

, find , find

H ' 1 

G ' 3 

.

2 2 10. For the circle: x  y 25 .

a. Find dy / dx b. Find an equation of the tangent to the circle at the point (3, 4). 11. Let

 L  : x3  y3 6 xy

a. Find dy / dx b. Find an equation of tangent to the curve (L) at the point (3, 3) 12. Find y' by implicit differentiation 4 4 a. x  y 16 x  y b.

x  y 4

3 2 c. x  xy  y

8

13. Find f ' in terms of g ' a.

f  x  g  sin 2 x 

b.

f  x   g  e1 3 x 

14. Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm2 ? 2 2 15. If x  y 25 and dy / dt 6 , find dx / dt when y = 4 and x > 0.

2 2 2 16. If z x  y

 z  0 , dx / dt 2, dy / dt 3 , find dz / dt

when x 5, y 12

17. Find the linearization L(x) of the function at a.

a.

f  x 

1 , 2x

a 2 b.

f  x 3 5  x ,

a  3

s  t  3sin t  4cos t 1 18. The equation of motion is for a particle, where s is in meters 2 and t is in seconds. Find the acceleration (in m/s ) after 3 seconds.

9

Chapter 3: Applications of Differentiation

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

f  x 3 x2  12 x  5,  0;3

c.

f  x x 4  x , 2

b.

  1;2

d.

f  x  x3  3 x  5,

 0;3 

f  x  x  ln x ,

1   ; 2 2 

2. Find the critical numbers of the function

a.

f  x  5 x 2  4 x

b.

f  x 

x1 x  x 1 2

c.

f  x   x ln x

3. Find all numbers that satisfy the conclusion of the Rolle's Theorem a.

f  x  x x  2,

  2;0

b.

 0;2 

f  x   x  2  x 2 ,

4. Find all points that satisfy the conclusion of the Mean Value Theorem a.

f  x 3 x 2  2 x  5,   1;1

5. If

f  1 10

and

f '  x 2, x  1; 4

6. Find where the function decreasing.

f  x  e 2 x ,

 0;3

, how small can

f  4

b.

f  x  3 x4  4 x3  12 x2  1

possibly be?

is increasing and where it is

7. Find the inflection points for the function 4 a. f  x   x  4 x  1

8. Find

f  x

for

f '  x   2 x 1

6 b. f  x x

and

x c. f  x   xe

f  0 1

.

2  1;4 9. Find the point on the parabola y 2 x that is closest to the point

10. Find two numbers whose difference is 100 and whose product is a minimum. 10

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 x1 to find x 3 a.

x 3  2 x  4 0, x1 1

c.

ln  x2 1  2 x  1 0, x1 1

b.

x 5  2 0, x1  1

d.

ln 4  x2  x, x1 1

13. The figure shows the graphs of f , f ' and f " . Identify each curve, and explain your choices a.

b.

14. Find the most general anti-derivative of the function.

a.

c.

f  x  6 x2  2 x  3 f  x 

x2  x  2 x

b.

d.

f  x  6 x 

1 x2

f  x 2 x x2  1

15. Find the anti-derivative of that satisfies the given condition

f  x  5 x  2 x , F  0 4 4

a.

5

b.

f  x  4 

2x , F  0  1 x2  1

16. 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.

v  t  10sin t  3cos t , s    0

c.

v  t  10  3t  3t2 , s  2 10 11

12

Chapter 4 - 6: Integration

1. Estimate the area under the graph of

y  f  x

1 f  x    x x  1, 4   x a. ,

b.

using 6 rectangles and left endpoints

f  x  x 2  2

,

x    1, 2 

c. A table of values for f is given x

1

2

3

4

5

6

7

f(x)

5

6

3

2

7

1

2

3. Repeat part (1) using right endpoints

f x x3 , x   2,2 . Estimate the area under the graph of using four 4. For the function   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. 3

a.



3

n 4

xdx ,

b.

0

sin x dx x 1



, n 6

2

dx I  2 x 1 . Find the approximations L4 , R4 , M 4 , T4 and S 4 for I . 0 6. Let x

7. Find the derivative of the function

g  x   t 2  1 dt 0

8. Find g '

13

x4

x

1 g  x   dt t cos 1 a.

g  x  c.

x2  x2



2x

sin u g  x   du u 1 b. cos x

et dt t

g  x  d.

 1  v 

2 10

dv

sin x

9. Find the average value of the function on the given interval

a.

f  x  x ,

c.

f  x  x x ,  1, 4

2

1 f  x  , x b.

  1,1

d.

 1,5

f  x  x ln x, 1,e 2 

10. A particle moves along a line so that its velocity at time t is v(t) = t2 – t – 6 (m/s) a. Find the displacement of the particle during the time period 1 ≤ t ≤ 4 b. Find the distance traveled during this time period 11. Suppose the acceleration function and initial velocity are a(t)= t + 3 (m/s 2), v(0)=5 (m/s). Find the velocity at time t and the distance traveled when 0 ≤ t ≤ 5.

v t t2  t , where is measured 12. A particle moves along a line with velocity function   in meters per second. Find the displacement and the distance traveled by the particle during the time interval

t   0, 2 

.

13. Evaluate the integral 2

x . 2

a.

x3  1 dx

0

1

d.

y 1  y 

2 5

b.

xe

e.

x

2

ln x

dy

0

x

1

 x 

dx

c.

dx

 f. t

2

 x  3x 2  dx 

t dt 1

14. Evaluate the integral 1

a.

x xe dx

x e

2  x

b.

0

dx c.

x sin xdx 14

e

ln xdx d. 

e.

x ln xdx

f.

1

e

x

dx

1

f  x  dx 5

15. Suppose f(x) is differentiable, f(1) = 4 and

0

1

xf '  x  dx

. Find

0

/

3

16. Suppose f(x) is differentiable, f(1) = 3, f(3) = 1 and average value of f on the interval [1,3]?    x  1, f  x   2    1 x , 17. Let

18. Find

g ' 0

 3 x 0 0  x 1

a.

1

. What is the

1

. Evaluate

f  x dx

3

for

x2

g  x  e

xf '  x  dx 13

x3 2 t 1

u

dt

b.

x

u  1du

2 x 1

19. Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 

a.

 3x  1 

e.

2

1



dx  b.   2 x  5

e dy 4

f.

i.

e dt

j.

g.

 4

 3





dx 2 x

d.

  2t

1

dx  0 4x  1

c.

1

y 2



1

0

dx

dx x 3

k.

h.

2



1

3

0

2

dx

x

l.

xdx

2

xe

2   x2

2

dx

 1

1





sin d 1

x

dx x

 0

20. Use the Comparison Theorem to determine whether the integral is convergent or divergent 

a.

cos 2 xdx  1 x 2 1



2  e x dx  x 1 b.



c.

dx

x  e

2x

1

15



d.

xdx

 1 x 1

 2

6

cos xdx  e. 0 sin x

1

f.

2dx

 0

x3

16

LINEAR ALGEBRA Chapter 1: Systems of Linear Equations

1. Write the augmented matrix for each of the following systems of linear equations and then solve them.   x  y  2 z 1   2 x  3 y  z  2  a.  5x  4y  2z  4

 2 x  3 y  z 10   2 x  3 y  3z 22  b.  4 x  2 y  3z  2

 x  y  z 0   2 x  y  2 z 0 x z c.   0

 x1  2 x2  x3  x4 0   2 x1  3x2  2x3  3x4 0    3  0 d.  x1 x2 x3 x4

2. Compute the rank of each of the following matrices.  1 1 2   A  3  1 1    1 3 4   a.

  2 3 3   B  3  4 1    5 7 2   b.

 1 1  1 4 C  2 1 3 0  0 1 5 8   c.

 1 1 1 3  D    1 4 5  2  1 6 3 4    d.

3. Find all values of k for which the system has nontrivial solutions and determine all solutions in each case.

a.

 x  y  2z 0    x  y  z 0  x  ky  z 0 

 x  y  z 0   x  y  z 0  0 c.  x  y  kz 

b.

 x  2 y  z 0   x  ky  3z 0  x  6 y  5 z 0 

 x  y  z 0   ky  z 0  0 d.  x  y  kz  17

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

a.

 x  y  2z m    x  y  z 0   x  my  z 1  m 

b.

mx  y  z 1   x  my  z m  2  x  y  mz m

 x  my  mz m  2x  y  z  2  d.  x  y  z 0

 x  y  z 1   x  my  2 z m  c.  x  2 y  z 2

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

a.

 x  y  2z m   x  y  z  0  x  y  3z 1  m 

b.

 x  2 y  2 z m   x  my  z 0  2 x  y  mz  2  m 

 x  ay  cz 0  bx  cy  3 z 1  3,  1, 2  6. Find a, b and c so that the system ax  2 y  bz 5 has the solution   2  1 3   A   4 2 k   4  2 6   7. Consider the matrix

a. If A is the augmented matrix of a system of linear equations, determine the number of equations and the number of variables. b. If A is the augmented matrix of a system of linear equations, find the value(s) of k such that the system is consistent. 8. Find all values of k so that the system of equations has no solution.

a.

 x  y  z 2   2 y  z 3 4 y  2 z k 

x  y  z 1  2 x   k  5  y  2 z 4  x   k  3 y   k  1 z k  3 b. 

18

 x  y  3 z 2  ...