Math 1431
Section 14819
TTh 10-11:30am 100 SEC
Bekki George
[email protected]
639 PGH
Office Hours:
11:30am – 12:30pm TTh and by appointment
Class webpage:
http://www.math.uh.edu/~bekki/Math1431.html
Find the derivative of a log function with base a.
lnx
log a x =
Use change of base formula:
lna
y = log a x
y = log a u
Find the derivative of each.
y=2
x
y=5
3x
2
ln ( cos x )
y=
2
y = log ( cos x )
y = log 5 ( tan x )
y=
(
( x + 3)
x 4 + 12
)
4
x2 + 5
y = (3x 4 + 2x)(cosx )
Inverse Trigonometric Functions
f (x) = sin x on the interval [–10, 10]









Is this an invertible function?




f (x) = tan x on the interval [–6, 6]










Is this an invertible function?














Restricted versions of these functions.









   
f (x) = sin x on 
, 
 2 2
These ARE invertible functions.
   
f (x) = tan x on 
, 
 2 2

   
Let f (x) = sin (x) for x  
,  . This function is invertible and
 2 2
we denote its inverse by sin 1  x 
or arcsin  x  .






   
sin (x) on 
, 
 2 2




arcsin  x  on [–1, 1]
   
Let f (x) =tan (x) for x  
,  . This function is invertible and
 2 2
we denote its inverse by tan 1  x 
or arctan  x  .






   
tan (x) on 
, 
 2 2
arctan (x) on (–, )
Function
Domain
Range
From PreCalculus:
sin
sin
sin

3

6

4



2
sin

3
4
sin

3
sin
sin
sin
sin
sin
1
1
1
1
1
3

2
1

2
1

2
3

2
 3

2
If y = arcsin x a) find sin y
b) find cos y
c) find y’
Popper15
A
1.
In the given right
triangle, BC =
x
B
A.
x 2 -1
B.
1- x 2
C.
x 2 +1
D.
x +1
E. none of these
1
C
2. sin
1


sin π 
6
A. 1/2
B.
3/2
C. p / 6
D. p / 3
E. none of these
3. A
y = arcsin x find cos y
y  arc sec 5 find tan y
2
si n  arc sec x  




cos 2 arcsin 3 
5
sin 2 arccos 4 
5
f (x) = y = arctan x
f (x) = y = arcsec x
find y’
find y’
Formulas (u is a function of x):
d é arcsinu ù =
û
dx ë
u'
2
1-u
d é arcta nu ù = u'
û
2
dx ë
1+u
d é arc sec u ù =
û
dx ë
u'
2
u u -1
Give the domain of f  x   arctan  ln  x  
and compute its derivative.
x
Give the domain of g  x   arcsin e and find the equation for the
2
tangent line to the graph of this function at x = 0.
Differentiate: y  tan
1
x
Differentiate: f  x   e
tan 1 x
Differentiate: y = sin
-1
2
x +2
Popper15:
4. The graph of y = f ' (x) is shown below. Give the smallest value of x
where the graph has a point of inflection.
a) 3.5
b) -2
c) -3.5
d) 2
e) 0
5-11. Given the graph of f ‘(x) below, determine where f is increasing,
decreasing, intervals of concave up and concave down. List all local maximum
and minimums and points of inflection.
5. Interval(s) of increase:
a. (-2,-1)È(2,¥)
b. (-¥,-2)È (-1,2)
c. (-¥,0)È (3,¥)
d. (0, 3)
e. none of these
6. Interval(s) of decrease:
a.
b.
c.
d.
e.
(-2,-1)È(2,¥)
(-¥,-2)È (-1,2)
(-¥,0)È (3,¥)
(0, 3)
none of these
7. Interval(s) of concave up:
a.
b.
c.
d.
e.
(-2,-1)È(2,¥)
(-¥,-2)È (-1,2)
(-¥,0)È (3,¥)
(0, 3)
none of these
8. Interval(s) of concave down:
a.
b.
c.
d.
e.
(-2,-1)È(2,¥)
(-¥,-2)È (-1,2)
(-¥,0)È (3,¥)
(0, 3)
none of these
9. Local maximum:
a.
b.
c.
d.
e.
(-1, f (-1))
(0, f (0))
(3, f (3))
(-2, f (-2))
(2, f (2))
Local minimum:
10.
a.
b.
c.
d.
e.
(-1, f (-1))
(0, f (0))
(3, f (3))
(-2, f (-2))
(2, f (2))
11.
Points of inflection:
a. (-1, f (-1))
b. (-2, f (-2))
c. (2, f (2))
d. all of these
e. none of these