Math 1431
2 cos x 1
lim⇡
x! 2LAB
5x session
A) 94
B)1
Quiz 62
A)- 5⇡
C)0
B) 2⇡
5
3
D) 49
C) 3⇡
9
D) 4⇡
9
Example 1:
x2 function
5x + 6
Give the expression for derivative
of
given
f (x) =
f (x) = 2x3
p
x
2
x
How to write a thesis
Daria Kurzanova
March,2015
lim⇡
x! 2
2 cos x
5x
1
D) 49
3⇡
9
D) 4⇡
9
x2
5x + 6
f Example
(x) = 2:
x 2
1
The following limit represents the derivative of a function f(x) at a number c. Determine f(x) and c.
(1 + h)5
lim
h!0
h
1
f (x) = 2x3
p
x
2
4x3 + 5x
x
Example 3:
(1 + h)5
lim
Find derivative of the given function
1
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
h
h!0
f (x) =
f (x) =
4x3 + 5x
How to write a thesis
Daria Kurzanova
March,2015
1
lim⇡
x! 2
A) 94
B)1
C)0
B)
5
C)
9
D) 4⇡
9
f (x) =
f (x) = 2x3
p
x2
5x + 6
x 2
x
(1 + h)5
lim
h!0
h
f (x) =
1
D) 49
Example 4:
Continuity
2 and Differentiability
2⇡
3⇡
A)- 5⇡
2 cos x
5x
1
4x3 + 5x
Give the point
function
is3 not differentiable
3 where
3 the following
2
2
(a + b) = a + 3a b + 3ab + b
f (x) = |x 6|
1
How to write a thesis
Example 5:
Daria
Kurzanova
Kurzanova
Find the equation of a tangent Daria
line for function
f(x) at point -4,
given that f(x) goes through the point (-4,5) and f’(-4)=6
March,2015
How to
write a thesis
March,2015
How to write a thesis
Daria Kurzanova
2 cos x 1
2lim
cos x 1
limMarch,2015
x! ⇡2
5x
⇡
Daria Kurzanova
x! 2
5x
9
B)1 C)0 4 D) 49
9A) 4
A) 4 March,2015
B)1 C)0 D) 9
2
2⇡
3⇡
4⇡
A)2 5⇡ 2⇡B) 5
3⇡C) 9
4⇡D) 9
A)- 5⇡ B) 5
C) 9
D) 9
2 cos x
1
lim⇡
2
x! 2 2 x5x
5x + 6
2 cos x 1
f (x)x = 5x + 6
f4 (x) =
lim⇡
x 2
A) 94 x!B)1
C)0
D)
x
2
5x
2
p
9
3
p
f
(x)
=
2x
x
3
f (x)
x
D) 49= 2x
5
2
2⇡
3⇡
4⇡
(1
+
h)
1
5
A)- 5⇡ B) 5
C) 9
D) 9(1
+ h)
1
lim
lim h!0
h
4⇡
h!0
h
C) 3⇡
D)
9
9
3
x2 5x + 6
f
(x)
=
3 4x + 5x
f (x) = 4x2 + 5x
f (x) =
x
5x + 6
x 2
f (x) = 3p 3
2
2
3
2
3#b) 3=xa +
(a
+
3a
b
+
3ab
+
b
2
2
3
Question
f (x)(a=+2x
b) = a x+ 3a b + 3ab + b
Find
the
derivative of the following function at point5 2
(1 + h)
1
f (x)f (x)
= |x= |x6|2 6| 5
lim
=2 5x
3x 1
(13x
++h)
f (x)f (x)
= 5x
+
h!0
h
lim
h!0
h C)10x + 3 D)26
A)10x
B)23
3
f (x) = 4x + 5x
x
3
3
2
2
3
(a
+
2b) =
3 a + 3a b + 3ab + b
a b + 3ab + b
f (x) = |x 6|
f (x) = 5x2 + 3x
Question
A)10x
B)23 C)10x + 3 D)26
C)10x
+ 3 #D)26
Give the point where the following function is not differentiable
(
(
x
if x  1
x
if x  1
f (x) =
f (x) =
x + 5 if x > 1.
x + 5 if x > 1.
2
A)5
B)0
C)1
D)3
1
1
1
1
lim
f (x) = |x 6|
x! ⇡2 x
5x if x  1
f
(x)
=
7 5x2 + 3x
fQuiz
(x)9 =
x + 5 if x > 1.
A) 4 B)1
C)0 D) 49
Example
6:
A)10x
B)23 C)10x + 3 D)26
FindA)5
derivatives
B)0 C)1 D)3
2
4
(
f (x)
4x5B) 2⇡
3x
+C)
5x3⇡
+ 10D) 4⇡
A)- =
5⇡
5
9
9
x
if x  1
2=
f (x)
x
5x + 6
f (x) =
x + 5 if x > 1.
x
p
f (x) =A)5
2x3
x
B)0
C)1 D)3
1h)5
5
4
(1
+
f (x) = 4x
3x + 5x +
lim10
h!0
h
f (x) = x13
f (x) =
2
1
4x3 + 5x
(a + b)3 = a3 + 3a2 b + 3ab2 + b3
f (x) = |x 6|
1
f (x) = 5x2 + 3x
A)10x B)23 C)10x + 3 D)26
(
x
if x  1
f (x) =
x + 5 if x > 1.
A)5 B)0 C)1 D)3
5
4
f (x)
=
4x
3x
+ 5x + 10
Example
1 7:
f (x)
= x3 the following function. Find the equation of the normal line at the point (1, 3).
Consider
f (x) = x3
3x2 + 5x
1
f (x) =
1
x3
Example 8:
Find the points3 where 2the tangent line is horizontal
f (x) = x
3x + 5x
f (x) = 2x4 x3 + 10
1
f (x) = x3 3x2 + 5x
f (x) = 2x4 x3 + 10
f (x) = 3x2
f (x) = x3 3x2 + 5x
f (x)Example
= 2x4 9: x3 + 10
⇡
Find derivative at point 4
f (x) = 3x2
4 sin x
4 sin x
f (x) = 3x2
4 sin x
Example 10:
Find⇡ third derivative
4
f (x) = 8x3
15x2 + 3x + sin x
f (x) = x3 3x2 + 5x
f (x) = 2x4 x3 + 10
3 2 + 5x2
f (x)f =
3 3x + 5x2
(x)x3= x3x
4
3
4
f (x)f (x)
= 2x= 2xx + x
103 + 10
f (x) = x
3x + 5x
f (x) = 2x4 x3 + 10
f (x) = 3x2
2
f (x)f =
(x)3x=33x42sin x24 sin x
(x) =
=xx3 3 3x
3x2 ++25x
5x
ff(x)
f (x) = x
⇡
4
3x + 5x
(x)⇡4 =
=⇡2x
2x44 4 xx33++
10
2
ff(x)
10
3
f
(x)
=
3x
4 sin x
3
2 x + 10
f (x) =4 2x
f (x) = 8x3
f (x)Question
= 8x #15x
+ 3x2 + sin x
3
⇡
(x)
=
8x
15x
+following
3x + sinfunction
x
⇡ f
Find
derivative
of
the
at
point
2
⇡
2
2
2
(x) 4x
=33x
3x
4 sin x
ff2 (x)
f (x)
= =
+ 53sin4xsin x
f (x)
f⇡
(x) = 4x 2+ 5 sin x
f (x) = 3x
4 sin x
=
f (x) = 8x
f⇡ (x) =
2
16
A)0,
22
15x + 3x + sin x
15x + 3x + sin x
9
16
4x + 5 cos x
32
B) 5, 02 2 C)0,
9
32
2 2
⇡⇡ = 3
3 ⇡⇡ C)
fA)
(x)
Question
#
B)C)⇡5⇡5 x D)D)-⇡3⇡3
A)
B)f (x)
=
x4x
33 2
3+
3 5 cos
2
3
D)0, 13
5
2
⇡ = 35 of points
⇡
Findfx-coordinate
where⇡tangent line is⇡horizontal
(x)
15
5x
3
A)- x6 2 3 B) x464 C)-3x33
D)- 2x54 5
3
5
2
ff(x)
10
(x)
=158x
8x ++3x
3x
10
⇡=
⇡
⇡
⇡
5
A)A)B)C)x355 D)-D)B) x356 C)3 x6
3
x4
A)
B)-
99
A)0,
f (x)
A)0,
16
16
C)
40 C)0,3
5,5,
=B)
8x
+
B)
0 3x
C)0,
4
3
f (x) = 8x + 3x
A)0,
9
10
D)-
1
10D)0,
D)0,3 13
99
3232
9
1
C)0, D)0,
D)0,
1
32
3
3
B) 5, 0
Question
#
9 16
9
A)0,
B)
5,
0
C)0,
Find derivative
of the following function 32
16
2
(x)==x35 x35
ff(x)
1515
5
A)B)
6
A)-x 6 B)
x6
x
5 C)- 35
3D)- 54
5
C)D)x
x
x6
x5
x4
2
15x2 + 3x + sin x
4x3 + 5 cos x
⇡⇡4 2
A)3⇡
B)-3⇡
C)5⇡ 2 22 D)-3⇡ 2 2
2
3
4
⇡
⇡
4 A) ⇡ 3
B)D)-x⇡3
2 C) 5
3
2
3
3
f
(x)
=
8x
15x
+
3x
+
sin
f (x)⇡ = 8x
15x + 3x + sin x
⇡⇡⇡ f4(x) = 8x4 + 3x3 10
3 8x4 + 3x
2 3 10
222
f (x) =
3
f(x)
(x)A)0,
= 9 4x
4x3B)++5,
x
9
=
553cos
0cosx
C)0,
D)0, 1
f (x) = 8x
4 sin x