I. Functions:
The student must be able to do the following:
1. Graph the following functions and their families, and find, from the
graph, the limit, if exists, of any of these function at any point or as x
increases or decreases with no bound (i.e. the limits at infinity and minus
infinity)
A. The inverse sine, the inverse cosine, the inverse tangent and the
inverse secant functions.
B. The hyperbolic sine, the hyperbolic cosine, the hyperbolic tangent and
the hyperbolic secant.
C. The inverse hyperbolic sine, the inverse hyperbolic cosine, the inverse
hyperbolic tangent and the inverse hyperbolic secant.
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 any
of the six inverse trigonometric functions, the six hyperbolic functions at
a 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(5π/4)]
i. arccos[tan(-π/4))]
j. cosh(ln5)
k. tanh(-5)
3. Prove all hyperbolic identities
Examples:
Prove each of the following:
a. cosh2x = sinh2x + 1
b. coth2x = sech2x + 1
c. cosh2x = cosh2x +sinh2x
d. sinh2x = ( cosh2x – 1 ) / 2
II. Derivatives
1. The student must be able to do explicitly, implicitly and
logarithmically differentiate functions involving all the functions studied
in calculus courses so far (power, trigonometric, exponential, logarithmic,
inverse trigonometric and hyperbolic functions.(In the case of inverse
hyperbolic functions the formula will given)
Examples:
Defferentiate each of the following functions
a. y = arctan8[ ln (cosx) ]
b. y = log3(arcsinx5)
c. y = 5arcsec(tan2x)
d. y = (arccos3x)sin5x
e. y = csc(e3x) arccotx7
f. y = (arccos3x)sin5x
g. y = [ 2 arctan5x + 8ln(tanx) ] / [ 2 arcsin5 + 2 arcsin(ex]
h. y = sinh8[ ln (cosx) ]
i. y = log3(coshx5)
j. y = 5tanh2x
k. y = e3x cschx7
l. y = (cosh3x)sin5x
m. y = [ 2 arctan(sihx6 )+ 8ln(coshx) ] / [ 2 coth5 + 2 coth(ex) ]
1. Prove the following formulas:
A. The derivatives formulas for the inverse trigonometric functions.
B. The derivative formulas for the hyperbolic functions.
C. The derivatives formulas for the inverse hyperbolic sine, the inverse
hyperbolic cosine and the inverse hyperbolic tangent.
D. The logarithmic expressions for the inverse hyperbolic sine, the
inverse hyperbolic cosine, the inverse hyperbolic tangent.
III. Integrals
The student must be able to do the following:
1. Integrate using the appropriate simple substitution, when possible.
Examples:
a. ∫ dx / x √ ( 25 – 4ln2 3x )
b. ∫ sec2( arcsin3x) dx / √ ( 1 - 9x2 )
c. ∫ x4 dx / (4 + 5 arcsinx5 ) √ ( 1 – x10 )
d. ∫ ln(arcsin2x) dx / √ ( 1 - 4x2 ) arcsin2x
e. ∫ 2x dx / (4x + 25 )
f. ∫ 2arctan5x dx / (1 + 25x2 )
g. ∫ 2x dx / [(4x + 1 ) arctan2x ]
h. ∫ 2x dx / [(4x + 1 ) arctan52x ])
i. ∫ coshx dx / x√ ( 25 – 4sinh2 3x )
j. ∫ sech2( arcsin3x) dx / √ ( 1 - 9x2 )
k. ∫ sech2xdx / (tanh2x + 25 )
l. ∫sech5x thah5x 2sech5x dx
m. ∫ x2 tanh x3 dx
n. ∫x cschx2
2. Evaluate all of the following types of trigonometric and hyperbolic
Integrals from scratch. No use of the reduction formulas or any ready
made formulas is accepted]. Evaluate these integrals when x is replaced
by cx or a function of x
A.
B.
C.
D.
E.
F.
G.
H..
I.
J.
K.
L..
∫sinmx cosnx dx, where either m or n is an odd natural number
∫sinmx cosnx dx, where both m and n are even natural numbers
∫secmx tannx dx, where n is an odd natural number
∫secmx tannx dx, where m is an even natural number
∫cscmx cotnx dx, where n is an odd natural number
∫cscmx cotnx dx, where m is an even natural number
∫sinhmx coshnx dx, where either m or n is an odd natural number
∫sinhmx coshnx dx, where both m and n are even natural numbers
∫sechmx tanhnx dx, where n is an odd natural number
∫sechmx tanhnx dx, where m is an even natural number
∫cschmx cothnx dx, where n is an odd natural number
∫cschmx cothnx dx, where m is an even natural number
3. Evaluate all of the following types of integrals, using the method of
integration by parts. No use of the reduction formulas or any ready made
formulas is accepted]
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
C. ∫ cos(ax) eax dx, ∫ cos(ax) eax dx, , ∫cos(bx)cosh(ax) dx,
∫sin(bx)sinh(ax) dx
D. ∫ x2n-1 cos(axn) eax dx, ∫ x2n-1 sin(axn),dx , ∫ x2n-1 eaxn
E. ∫arccosx dx, ∫arctanx dx, ∫arsecx dx, ∫ x arctanx dx and
similar integrals
F. ∫ secnx dx, ∫ cscnx dx where n is an odd number greater than 3.
4.
4.1. Use the method of trigonometric substitution the evaluate 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 and b being
positive.
4.2. Use the method of trigonometric substitution to evaluate integrals,
where the integrand contains a quadratic expression of the form ( cx2
+dx+ k)n/2 , where n is an odd number and the polynomial cx2 +dx+ k
can be transformed to the form ax2 + b , where n is an odd number and
one or both of the constants a and b being positive
Examples:
Evaluate each of the following integrals
a.
x3

25  9 x
b.

c.
x
d.
x
2
.dx
4 x 2  9  dx
x
1
2
25  9 x
2
1
2
9 x 2  25
.dx
.dx
9  4 x 2  dx
x2

e.
f.

g


h.

i.
j.

x3
9  4x
x3
9  4x
2
2
.dx
.dx
dx
( x 2  6 x  15) 3
dx
( x 2  6 x  16)3
dx
(16  x 2  6 x)3
5. Integrate using the method of partial fractions
Examples
a.
b.
dx
 x2  x
8  4x
 ( x2  1)( x2  2 x  1) dx
x2  2x  1
c. 
 dx
2
2
( x  1)
x2  2
d. 
 dx
x 1
x 3dx
e.  2
x 1
f.

15 x  120
dx
x( x  1)( x  2)( x  3)( x  4)( x  5)
IV. Improper Integrals
The student must be able to investigate the convergence of
improper integrals and evaluate a convergent improper integral.
1. improper integrals of the form:

c

f ( x) dx
and


c
f ( x) dx
Examples

(1)


dx
9  25 x 2
(2)
3
5
 23

dx
2
x




(4)
3
1
(5)
x2
1

(3)

dx
3
dx
4 9 x 2
0
dx
x5
(6)
x
xe
 dx

1

(7 )


dx
23 x 2
2. Improper integrals of the form:
1.

b
a
f ( x) dx ;
where f is contoninuous on (a, b]
and x  a is a vertical asymptote for f
2.

b
a
f ( x) dx ;
where f is contoninuous on [a, b)
and x  b is a vertical asymptote for f
3.

b
a
f ( x) dx ;
where f is contoninuous on [a, c) and (c, b]
and x  c is a vertical asymptote for f
Examples
1
(1)

0
10
dx
2
( x 1) 3
( 2)

0
1
(5)

dx
2
( x 1) 3
 ln x dx
0
dx
x 1
1
2
(3)

( 4)

0
dx
x  x
3