Find dx dy if Find an equation to the tangent line to the graph of We are given the graphs of f and g below. Find ( ) ( ) 1 / f g o , i.e. the derivative of f g o at 1. Find the differential of ( ) 3 1 2 2 / x x y + + = . Find ( ) x T y = , an equation of the tangent line to the graph of Two cars start moving from the same point. One travels east at a constant rate of 60 miles per hour and the other travels north at a constant rate of 70 miles per hour. Find the rate at which the distance between the cars is changing 1 hour later. You may round your answer to two digits after the decimal sign. we can draw the following figure. Find an equation of the tangent line to the graph of ( ) ( ) 2 2 2 2 2 25 2 y x y x - = + Use Newton’s method to find the solution of 0 6 5 2 3 = - + x x by using the following steps Find, ( ) 0 f where ( ) 6 5 2 3 - + = x x x f Find, ( ) 1 f where ( ) 6 5 2 3 - + = x x x f Choose a guess point 0 x Perform the iterations, until the values stabilize to the 8-dgits after the decimal sign. 1. A graph of the function f f is given below Fill in the following table (of course approximate) 2. For the function f x( ) = 1 x fx () = 1 x find lim x→4 f x( )− f 4( ) x − 4 lim x®4 fx () -f4 () x-4 and an equation of the tangent line to the graph of the above function at 4, 1 2 ⎛ ⎝⎜ ⎞ ⎠⎟ 4, 1 2 æ è ç ö ø ÷ In the questions 3-6 , find dy dx dy dx 3. y = x3 − x2 − x53 y=x 3 -x 2 -x 5 3 4. y = x3 ln 1+ x2( ) y=x 3 ln1+x 2 () You may use d dx ln u( ) = 1 u du dx d dx lnu () = 1 u du dx 5. y = sinx 2+ cosx y= sinx 2+cosx 6. y = 1+ ex( )sinx y=1+e x () sinx Rewrite this equation as: y = e ln 1+ex( )sinx y=e ln1+e x () sinx Practice Question In 15–18, find the slopes and equations of the lines tangent to y = f (x) at the given points. 15. f (x) = x2 + 8 at (1, 9) and (-2, 12). ...

1. Find
dx
dy
if
Find an equation to the tangent line to the graph of
We are given the graphs of
f
and
g
below. Find
(
)
(
)
1
/
f
g
o
, i.e. the derivative of
f
g

2. o
at 1.
Find the differential of
(
)
3
1
2
2
/
x
x
y
+
+
=
. Find
(
)
x
T
y
=
, an equation of the tangent line to the graph of
Two cars start moving from the same point. One travels east at a
constant rate of 60 miles per hour and the other travels north at
a constant rate of 70 miles per hour. Find the rate at which the

3. distance between the cars is changing 1 hour later. You may
round your answer to two digits after the decimal sign.
we can draw the following figure.
Find an equation of the tangent line to the graph of
(
)
(
)
2
2
2
2
2
25
2
y
x
y
x
-
=
+
Use Newton’s method to find the solution of
0
6
5
2
3
=

4. -
+
x
x
by using the following steps
Find,
(
)
0
f
where
(
)
6
5
2
3
-
+
=
x
x
x
f
Find,
(
)
1
f

5. where
(
)
6
5
2
3
-
+
=
x
x
x
f
Choose a guess point
0
x
Perform the iterations, until the values stabilize to the 8-dgits
after the decimal sign.

6. 1.
A graph of the function
f
f
is given below
Fill in the following table (of course approximate)
2.
For the function
f x( ) = 1
x
fx
()
=
1
x
find
lim
x→4
f x( )− f 4( )

7. x − 4
lim
x®4
fx
()
-f4
()
x-4
and an equation of the tangent line to
the graph of the above
function at
4,
1
2
⎛
⎝⎜
⎞
⎠⎟
4,
1
2
æ
è
ç
ö
ø
÷
In the questions 3-6 , find

8. dy
dx
dy
dx
3.
y = x3 − x2 − x53
y=x
3
-x
2
-x
5
3
4.
y = x3 ln 1+ x2( )
y=x
3
ln1+x
2
()
You may use
d

9. dx
ln u( ) = 1
u
du
dx
d
dx
lnu
()
=
1
u
du
dx
5.
y =
sinx
2+ cosx
y=
sinx
2+cosx
6.
y = 1+ ex( )sinx
y=1+e
x

10. ()
sinx
Rewrite this equation as:
y = e
ln 1+ex( )sinx
y=e
ln1+e
x
()
sinx
Practice Question
In 15–18, find the slopes and equations of the lines tangent to y
= f (x) at the given points.
15. f (x) = x2 + 8 at (1, 9) and (-2, 12).