Exercise Book MAE 101 2022 PDF

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Name:...........................................Class:...........................................Mathematics for EngineeringExercise BookTrần Thanh Hiệp - 2022CALCULUSChapter 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   D...

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

Mathematics for Engineering Exercise Book

Trần Thanh Hiệp - 2022 1

CALCULUS Chapter 1: Function and Limit

1. Find the domain of each function:

a.

f  x  x  2

b.

f  x 

1 2 x  x

c.

f  x  ln  x  1 

x x 1

D = {x R:….}= * A : A 0 A * : B 0 B *ln A : A  0

2. Find the range of each function:

f  x  x 2  2 x x 2  2 x 1  1  x  1  1  1 2

a.

f  x  x  1

c.

f  x sin x

b.  r [  1, )

d.

f  x  cos x D R   ,   r   1,1

3. Determine whether is even, odd, or neither

a.

f  x 

x 2 x 1

b.

f  x 

x2 4 x 1

c.

f  x 

x x 1

2

f   x   f  x   even f   x   f  x   odd

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

f  x  4

b.

f  x  3

5. Suppose that the graph of

c.

f  x  x

f  x  2  3

d.

f  x  5  4

is given. Describe how the graph of the function

y  x  1  2 can be obtained from the graph of f . 6. Let

f  x  x

and

g  x  2  x

. Find each function

f  g  x f  g  x   f  f  x  f  f  x   g  g  x   g  g  x  g  f  x  g  f  x  

a.

f og  x   f  g  x    f

b. g0 f

7. Let





2 x 

c. g o g f  x 

2  x  4 2 x

d. f o f

x2  x  1 1  x 1  x x .

Find 1  f x  x a. 

b.

f  2 x  1

8. Use the table to evaluate each expression

x f(x) g(x) a. f(g(1)) e.

go f  3

1 3 6

2 1 3 b. g(f(1)) f.

3 4 2

4 2 1

5 2 2

c. f(f(1))

6 5 3 d. g(g(1))

go f  6  3

9. Evaluate the following limits

 lim f x L x a    lim f x L   x a   lim f  x   L  x a  c 0  c  0 0 0 c   c 

e   e   0 sin u 1 lim u 0 u tan u 1 lim u 0 u 0  ; : 0  f x f 'x  lim lim xa g x   x a g'  x

 3 h  lim

x2  x  12 lim a. x 3 x  3

x6  1 lim 10 b. x 1 x  1

tan 3x lim c. x 0 tan 5x

d.

t2 9  3 lim t2 e. t  0

x2  x  12 lim 3 f. x   x  3

1 1  lim    x 0  x x   g.

x2  1 lim x 1 x  1 h.

A 7

B 3/5

C 3/5

D 6

E 1/6

F 0

h 0

G -inf

2

9

h

H 2

10. Determine whether each curve is the graph of a function of x. If it is, state the domain and range of the function. a): No; 4

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

11. The graph of f is given.

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

lim f  x

x0

iii.

lim f  x x1

lim f  x lim f  x  , x 0 and x  0 and

lim f  x  x 4

5

b. At what numbers is discontinuous? 12. Determine where the function 2

2 x  x 1 f  x  x 2 a.

b.

f  x

f  x 

is continuous x9 4 x2  4 x  1

c.

f  x ln  2 x  5

13. Find the constant m that makes f continuous on R  x2  m 2 , x  4 f  x   mx  20, x  4 a.

2  mx  2 x, x  2 f  x   3  x  mx, x 2 b.

 e2 x  1 , x 0  f  x   x m , x 0  c.

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

 x  2, x  0  f  x   2x 2 , 1 x 0 2  x , x  1  14. Find the numbers at which the function is discontinuous.

15.

If lim f  x  =L then y  L is the horizontal asymptotes. x 

If lim f  x  =  then x  a is the vertical asymptotes. x a

15.a) find vertical asymptoes

6

2 2x  x  1   x 1:vertiacal asym. 2 x 1 x  x  2 2 2x x  1 * lim 2   x  2 :vertiacal asym. x   2 x  x 2

*lim

x2  1 x1 2 2x x 1 *lim lim 2  x 1is NOT a vertiacal asym t. x 1 x  1 x 1 1 f  x 

7

15.b) 9 x 9 x * lim  lim x   4x 2  3x  2 x    4x 2  3x  2 x 9 9 1 1 1 x x  lim   lim   x   2 3 2 4 x 2  3 x  2 x   4  2 2 x x x 1 y  : horizontal asymptotes 2 9 1 x 9 x * lim  lim x   4x 2  3x  2 x   4x 2  3x  2 x 9 9 1 1 1 x x   lim  lim x  2 4 x 2  3 x  2 x  4 3 2   2 2 x x x 1  y  : horizontal asymptotes 2 1

8

Chapter 2: Derivatives

1. Use the given graph to estimate the value of each 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: The equation of the tangent line    to the curve at the given point M  x0 , y0  :

   : y  f '  x0  . x  x0   y0 ; a.

y

x 1 , x 2

 3,2

2 c. y 3  2 x  x ,

y0  f  x0 

b. x 1

d.

y

2x , x 1

 0, 0 

y

3 2 x , x 1

y  1

2

3. Find y '

9

 c  ' 0

 x  ' k .x k

 e  ' e x

 ; 

k 1

'

1 1  x  x 2

x

 sin x  '  cos x  cos x  '  sin x

 c  ' 0

 u  ' k .u  e  ' e .u ' u

u

u' cos2 u u'  ln u  '  u

 tan u  ' 

'

 u  u '.v  v '.u    v2  v

x

.u '

 sin u  ' u '.cos u  cos u '  u '.sin u

 u.v  ' u '.v  v '.u

m n

k 1

k

1  tan x '  2 cos x 1  ln x  '  x   u v   ' u 'v '

n x m

k h k h x .x x ' 1 x  2 x ' u' u  2 u

   

1 y x  x x  2 x a. 2

d. y x x  2

b. y  x  x

e.

y ln  x 2  1 

1 x

x2 y x 1 c.

f.

y e x sin  2x  1

4. Find y " 3 x 1 a. y  xe

3 b. y  2 x  1

x c. y e cos x

5. Find dy/dt: y  f  x  ; x x  t  dy dy dx y '  t   . dt dx dt 10

3 a. y x  x  2, dx / dt 2 and x 1

c. y tan t and

t

2 b. y ln x, dx / dt 1 and x e

2 16

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

b. y  x 1, x 3

c.

6. Find dy for: dy  f '  x  .dx

a.

y

1 x 1 2

y ln  x2  1

, x 1 and dx 2

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

b.

8. A table of values for f , f ', g and g ' is given h  x   f g  x   f  g  x    h ' x  f ' g  x   .g '  x 

a. If

h x f g x 

, find

h ' 1

b. If

H  x  g o f  x 

, find

H '  1 11

c. If

F  x  fof  x

9. If

h  x  4 3 f  x

 x  ' k .x  y  ' k. y k

k 1

k

k 1

y' 

F ' 2

d. If

, where

f  1 7, f '  1 4

, find

G  x   g og  x  , find

h ' 1

, find

G '  3

.

.y'

dy dx

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

13. Find f ' in terms ofg ' f  x  g  3 x 1

a.

f  x  g  sin 2 x 

b.

f  x  g  e1 3 x 

c.

 f ' x g ' 3x  1 . 3x  1 ' 3.g ' 3x  1

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 ?

12

x k ' k .x k 1 .x '

   y  ' k. y k

k 1

.y'

dy dx y'  ; x'  dt dt 2 2 15. If x  y 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 12

17. Find the linearization L(x) of the function at x = a. L x  f ' a . x  a  f  a

a.

f  x 

1 , 2 x

a 2 b.

f  x  3 5  x ,

a  3

s  t   3sin t  4cost  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.

v  t  s ' t  a  t  v '  t 

13

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 x  12 x  5, 0;3

c.

f  x  x 4  x ,

2

 0;3

f  x   x  ln x ,

1   2 ;2

3

b.

  1;2

2

f  x x  3 x  5,

d.

2. Find the critical numbers of the function: x  D; x is critical value of the function f(x) if: * f '  x  0 or * f '  x  : does not exist.

f  x  5 x  4 x 2

a.

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 f'’(x) = 0 a. f  x x x  2,



 2;0

b.

f  x  x  2 x2 ,

 0;2

4. 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 a.

f '  x 

f  b  f  a x   a, b b a and

f  x 3 x2  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

14

7. Find the inflection points for the function

f  x  x 4  4 x  1

a.

8. Find

f  x

for

f '  x   2 x 1

b.

f  x  x 6

and

f  0  1

c.

f  x  xex

.

 x2  A ,x 2 1; 4 9. Find the point  2  on the parabola y 2 x that is closest to the point M   AM 2  xM  xA    yM  yA  2

2

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 x1 to find x3

3 x a.  2 x  4 0, x1 1

c.

5 x b.  2 0, x1  1

ln  x2 1  2 x  1 0, x1 1

d.

ln 4  x 2  x, x1 1

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

A Fx

B F’x

C F’’x

1 F’’

2 F’

3 f

15

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

f  x  dx

k .dx kx  C x 1 x dx   C ; k  1;  k 1 1 axb ax b e dx  a .e  C sin xdx  cos x  C k

k

k

k

ax b dx a ln ax  b  C

cos xdx sin x  C  u v dx udx vdx f  x 6 x  2 x  3 2

a.

c.

f  x 

x2  x  2 x

b.

d.

f  x  6 x 

1 x2 x

f  x 2 x x2 1

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

a.

f  x  5 x4  2 x5 , F  0  4

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 s  t  v  t  dt v  t  a  t  dt

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

16

Chapter 4 - 6: Integration Let y  f x  , x   a ,b 

b a xi  1  xi S  f  x  dx  I ;  x  ; xi  n 2 a b

x0 a; x1  x 0   x; x 2  x1  x;....; xn b I Rn x.  f  x1   f  x2   ...  f  xn   {Right point} I Ln x .  f  x0   f  x1   ...  f  x n 1   {Left point I M n x.  f x1  f x 2  ...  f x n  {Mid point}  

 

I Tn 

 

 

x  f (x0 )  2 f (x1 )  2 f (x 2 ) ...  2 f (x n  1 )  f (x n )  2 {Trape

zoidal rule}

x [ f ( x0)  4 f ( x1)  2 f ( x2)  4 f ( x3) 3 ...  2 f ( x n  2 )  4 f ( x n  1)  f ( x n )]  Simpson’s Rule

I S n 

17

1. Estimate the area under the graph of

y  f  x

using 6 rectangles and left endpoints

1 f  x    x x  1, 4   x a. ,

 a=1; b=4; n=6, find L6 4 1 1  6 2 3 5 7 x0 1; x1  ; x2 2; x3  ; x4 3; x5  ; x6 4 2 2 2 1 L6  .  f  x0   f  x1   f  x2   f  x3   f  x4   f  x5   2 1   3  5  7   .  f  1  f    f  2   f    f  3   f    2   2  2  2 

x 

....

f  x  x  2 2

b.

,

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 x , x   2,2 4. For the function   . Estimate the area under the graph of using four approximating rectangles and taking the sample points to be 3

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

3

a.

 0

xdx,

n 4

b.

sin x

x

dx

,n6

1

18

2

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

1 x 1 ; a=0; b=2; n=4. 2

u x

Let g  x  

f (t )dt   

v x

g '  x  u '( x) f ( u( x))  v '( x) f ( v( x)) x

7. Find the derivative of the function

g  x    t 2  1 dt 0

8. Find g ' x4

1 g  x   dt cos t 1 a.

g  x  c.

x2  x2



2x

et dt t

x sin u g  x  du u 1 b.

g  x 

d.

cosx

 1  v  2

10

dv

sin x

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

a.

f  x  x 2,

  1,1 

1 f  x  , x b.

 1,5

19

c.

f  x  x x ,  1, 4

2 f  x  x ln x, 1, e 

d.

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 t2

v  t  dt t1

b. Find the distance traveled during this time period t2

v  t  dt t1

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  a  t  dt

v t t 2  t 12. A particle moves along a line with velocity function   , where is measured in meters per second. Find the displacement and the distance traveled by the particle during the time interval k x dx 

t  0, 2

.

x k1 C k 1

13. Evaluate the integral

x . 2

a.

x 3 1 dx

t  x3  1 t 2 x 3 1 2tdt 3 x 2dx 2 2 t3 2  I  t 2 dt  .  C  . 3 3 3 9



x 3 1



3

C

20

b.

xe

x2

1

dx

c.

y  1  y  2

d.

5

 x 

dy

 x  3 x2  dx  ln x

x e.

dx

 f. t

2

t dt 1

14. Evaluate the integral

u.dv u.v  v.du Let : u ....  du ...dx dv ...dx  v ... Note : u  ln x , x k ,sin x , cos x ,e x ,...

a.

x xe dx

d.

ln xdx

2  x

b.

x e

dx

c.

x sin xdx

1 u  ln x  du  dx x dv 1.dx  v x I x .lnx  1.dx x .lnx  x C

e.

f.

x ln xdx e

x

dx

t x 1 dt  dx  2tdt dx 2 x

21

I e xdx 2te tdt u 2t  du 2dt dv et dt  v et I 2t .e t  2e tdt  2t .et  2et  C  2 x .e x  2e x  C 1

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

f  x  dx 5 0

1

. Find

xf '  x  dx 0

3

16. Suppose f(x) is differentiable, f(1) = 3, f(3) = 1 and average value of f on the interval [1,3]? b

fav 

1 . f  x dx b a  a