study guide for exam 1--new

8/2/2019 Study Guide for Exam 1--New

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StudyGuideforExam1I. Functions:

You must be able to do the following:

1. Find the Domain and Graph the following functions and their families,

and find.

A. The inverse trigonometric functionsB. The hyperbolic functions.

C. The inverse hyperbolic functions.

Examples:Graph each of the following functions:

a. f(x) = 2 arcsin(3x)b. f(x) = arctanh( x - 3 )

c. f(x) = - arctan(-x)

d. f(x) = - sechx

e. f(x) = coshx + 3

2. Find the value, if exists (must explain the reason if it does not) of anyof the six inverse trigonometric functions, the six hyperbolic functions ata point and other values related to it.

Examples:Find the value of each of the following:a. arcsin(-1/2)

b. arcos( 3/2 )

e. arctccot (-2)

f. cos[arctan(-5)]

g. sin[arccos(-1/5)]h. arctan[cos(5/4)]i. arccos[tan(-/4))]

j. cosh(ln5)

k. tanh(-5)

l. arcsec[sec(5 /4)]

m. arcsec[sec(- /4)]

3. Prove all hyperbolic identities

Examples:


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Prove each of the following:

a. cosh2x = sinh

2x + 1

b. coth2x = cschh2x + 1

b. tanh2x + sech

2x = 1

c. cosh2x = cosh2x +sinh

2x

d. sinh2x = ( cosh2x 1 ) / 2

4. Prove the logarithmic expressions for the inverse hyperbolic functionsExamples:

a. sinh ln 1b. cosh ln 1c. tanh ln

5. Prove and use the following

a. sec cos

b. sech cosh c. csch sinh d. tan cot

II. Derivatives


1. Differentiate (Explicitly, implicitly and logarithmically) functions

involving all the studied functions (In the case of inverse hyperbolicfunctions the formula will given)

Examples:Differentiate each of the following functions

a. y = arctan8[ ln (cosx) ]

b.

logarcsin

c. y =arcsectan2d. y = csc(e3x) arccotx7e. y = sinh8[ ln (cosx) ]f. y = e3x cschx7g. y = arcsec7 + x cosh 3

h. sinh


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2. Find the derivative the hyperbolic functions.

3. Use the idea of implicit differentiation to prove the following formulas:

A. The derivatives formulas for the inverse trigonometric functions.

B. The derivatives formulas for the inverse hyperbolic functions.

III. Integrals


1. Integration involving inverse trigonometric and hyperbolic functions.

Examples:

a.

b. 3/1 9c. earctan5x dx / (1 + 25x2 )d. ex dx / [(4x + 1 ) arctan5ex ])

.sech3/19f. sech2x dx / (tanh2x + 1 )g. x2 tanh x3 dxh. x cschx2

i. /5 2j.

2. Evaluate all of the following types of integrals, using the method of

integration by parts.

A. xn

cos(ax) dx, xn

sin(ax) dx, xm

eax

dx ,xn

cosh(ax) dx, xn

sinh(ax) dx

where a is a constant

B. xm

ln(xn) dx, where n and m are real numbers (by parts)

C. cos(ax) eax

dx, cos(ax) eax

dx, ,cos(bx)cosh(ax) dx,sin(bx)sinh(ax) dx (using integration by parts twice)

D. sec

n

x dx,

csc

n

x dx where n is positive integer number greaterthan 3.

E. tan, arcsin, tanh, sinhF. sinlnI. Find the reduction formulas for sin, cos, tan, sec and use them.


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3. Evaluate all of the following types oftrigonometric and hyperbolic

Integrals from scratch. No use of the reduction formulas or any readymade formulas is accepted]. Evaluate these integrals when x is replacedby cx, for a constant c.

A. sinmx cos

nx dx, where either m or n is an odd natural number

B. sinm

x cosnx dx, where both m and n are even natural numbers

C. secm

x tannx dx, where n is an odd natural number

D. secmx tan

nx dx, where m is an even natural number

E. cscm

x cotnx dx, where n is an odd natural number

F. cscmx cot

nx dx, where m is an even natural number

G. sinax cosbx dx ; a bH. sinax sinbx dx ; a bI. cosax cosbx dx ; a b

3. Use the method oftrigonometric substitution to evaluate:

A. Integrals, where the integrand contains an expression of the form ( ax2

+ b)n/2

, where n is an odd number and one or both of the constants a andb being positive.

Examples:

Evaluate each of the following integrals

dxx

xa .

925.

2

3

x

dxxb

94.

2

dxxx

c .925

1.

22

dx

xxd .

2591.

22

2

249.

x

dxxe


5/5

5

dxx

xf .

49.

2

3

dxxx

g .49 2

3

B. Integrals, where the integrand contains a quadratic expression of theform ( cx

2+dx+ k)

n/2, where n is an odd number and the polynomial cx

2

+dx+ k can be transformed to the form ax2

+ b , where n is an oddnumber and one or both of the constants a and b being positive

Examples:

Evaluate each of the following integrals

74

.2

xx

dxa

2/32 )45(.

xx

dxb

xx

dxc

421

.2

32 )156(

.xx

dxd

32 )166(

.xx

dxe

32

)616(

.

xx

dxf

study guide for exam 1--new

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