(21) 3.* Tangent Line and Vertical Tangent Line
‫كيف توجد معادلة المماس عند‬
How to find the equation of the tangent at
1
Find
2
Then the point of the tangent is
3
Find
4
Find
5
The slope of the tangent is
‫اوجد‬
1
‫تكون النقطة المطلوب عندها المماس هي‬
2
‫اوجد‬
3
Or
‫اوجد‬
4
Or
‫فيكون الناتج ميل المماس‬
5
‫ فيكون‬normal line ‫إذا كان المطلوب هو‬
6
‫عوض في القانون التالي‬
‫لتحصل على معادلة المماس‬
6
Or
The slope of the normal line
6
Use the formula
Where
The graph has horizontal tangent line at
1
2
3
‫ضع البسط يساوي صفر و حل المعادلة‬
continuous at
The graph has Vertical tangent line If
1
2
3
‫ضع المقام يساوي صفر و حل المعادلة‬
continuous at
If
(21) 3.* tangent line and vertical tangent line (A)
Example 1
P2
34 March 23,2002
Find an equation of the line normal to the graph of
at
Solution
Example 2
Let
54 November 16, 2009 A
. Find the x- coordinate(s) of the point(s) on the graph of
(a) the tangent line is horizontal
at which
(b) the tangent line is vertical
Solution
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
P3
Example 3
36 April 19, 2003
A
Show that
has a horizontal tangent in
Solution
Example 4
Find the points on the graph of
where
33 October 25,2001 A
at which the tangent line is perpendicular to the line
Solution
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
P4
04 August 7, 2011
Example 5
(4 Points ) Find the
where
coordinates of the points on the graph
(a) the tangent line is horizontal
(b) the tangent line is vertical
Solution
60 October 31 , 2011
Example 6
( 4 points ) Let
.
Find all the point on the graph of at which
(a) the tangent line is horizontal
(b) the tangent line is vertical .
Solution
H∙ ∙L
,
V∙ ∙L
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
Example 7
P5
52 April 9, 2009 A
Show that the graph of
has a vertical tangent line
Solution
∙
∙
Example 8
23 April 27,2000
Let
Find all x at which the tangent line to the
graph of f is Parallel to the line
Solution
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
P6
Example 9
Show that the curve
has no tangent line with slope 2
Solution
the curve
Example 10
No solution
has no tangent line with slope 2
Find equations of both lines that are tangent to the curve
and parallel to the line
Solution
At
At
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
P7
Homework
40 October 28, 2004 A
1
Find the point on the graph of
2
Suppose
Show that the graph of
at which the tangent line is horizontal
has a horizontal tangent line
56 July 10, 2010
3
Let
. Find the x- coordinate of the points on the graph of
a) the tangent line is horizontal
b) the tangent line is vertical
at which :
(4 points)
57 November 8, 2010
4
Find the x- coordinate of the points at which
has
a) a vertical tangent line,
b) a horizontal tangent line.
5
(4 pts.)
14 March 28, 1996
Find the -coordinates of the points on the graph of
the tangent line is parallel to the line
and
at which
is in the interval
61 24 March 2012
6
[4 pts.]
Find the
coordinates of the points on the graph of the function
where : (a) the tangent line is horizontal
(b) the tangent line is vertical
02 May 26 , 2012
7
Show that
has a horizontal tangent line
64 March 9, 2013
8
[ 5 + 5 = 10 pts. ] Find the
coordinate of the points where the curve :
a) has a horizontal tangent line .
b) has a vertical tangent line .
HOSSAM GHANEM
86881166
‫حسام غانم‬
(21) 3.* tangent line and vertical tangent line (A)
P8
Homework
47 November 10, 2007 A
1
Let
Find the point on the graph of f at which the tangent line is vertical
2
32 August 02, 2008
Find the
of the point where the tangent line to the curve
is vertical.
3
Show that f has vertical tangent at
4
Let
graph of
5
Show that
. Find all values of a such that the tangent Line to the
is parallel to the line
at
has a corner at
Given
6
7
;
Use the intermediate value theorem to show that
there is a real number c between
Let
point on the graph of
and
such that
Use the intermediate value theorem to show that there is a
at which the tangent line is horizontal
58 7April 2011
[ 3 pts. ] At what points on the curve
8
is the tangent line parallel to the line
?
37 June 6, 2010
Let
9
(a) Find the
coordinate(s), if any , of the point(s) on the curve
the tangent line is vertical
(b) Find the
coordinate(s), if any , of the point(s) on the curve
the tangent line is horizontal
HOSSAM GHANEM
86881166
where
where
‫حسام غانم‬