AP Calculus AB FIVES Sheet
F = Fun at Home, I = Intellectual, V = Variety, E = Endeavors, S = Specimens
#29 Monday 10/7__________________________________________
F
Worksheet 24
I
Complete the table below….Then MEMORIZE all values!
V
E
S
Find the x and y intercepts of the line that is tangent to the curve y = x 3 at the point when x = −2 .
Linearization (Linear Approximation) and Differentials
2
Ex. 1 Approximate ( 3.0301) using a tangent line approximation.
Ex. 2 Approximate using linearization: 3 10
Ex. 3 Use a tangent line approximation to estimate f ( 9.246 ) for f ( x ) = x .
#30 Tuesday 10/8_________________________________________
F
Worksheet 25
I
Pg. 126 #62 and #63
⎛π⎞
V
If f x = cos 3x , then f ' ⎜ ⎟ =
⎝ 9⎠
()
( )
E
Derivative and Sine and Cosine Functions -
S
Find
(
d
sin
dx
) = cos
⋅
(
d
d
and
cos
dx
dx
) = − sin
⋅
d
dx
.
Ex. 1 a) y = sin 6x
Ex. 2 a) y =
cos x
x2
b) sin ( x 2 + 2x − 1)
c) y = sin 2 x
b) y = sin x cos x
c) y = x 2 sin x
#31 Wednesday 10/9 – Delayed Start______________________________________
F
Worksheet 26
I
1) Expand the Logarithm
a) log ( 8xy 4 )
b) ln
x2 + 5
x
⎛ 3x − 5 ⎞
2
c) ln ⎜
⎝ 2x ⎟⎠
2) Condense the Logarithm as a single logarithm
a) 5 ln x − 2 ln x − ln y
V
E
b) 5 ln x + 4 ln y
Find an equation of the tangent line to the graph of the function f ( x ) = sin 2x + 2 cos x at the point
(π , −2 ) .
Derivatives of Trig Functions – We will be finding and proving the derivatives of the remaining
trigonometric functions.
S
Find
.
Ex. 1 a) y = tan ( 5x )
b) y =
( )
tan x
x
d) y = cot 3 ( 4x )
c) y = csc x 2
Ex. 2 Find the equation of the tangent line to the function y = sec 2 ( 2x ) when x =
π
.
8
#32 Thursday 10/10______________________________________
F
Worksheet 27
I
Find an equation of the tangent line to the graph of the given function at the indicated x-value.
π
x =1
1) f x = 3x 2 − ln x,
2) f x = ln 1 + sin x , x =
4
x2
3) f x = ln e ,
x = −2
()
()
V
E
S
()
( )
Find the second derivative of f ( x ) = xe x
Derivatives of ln and e
Ex. 1 Evaluate each derivative.
a)
( )
d
! ln 2x #
$
dx "
b)
(
)
d !
ln x 2 + 1 #$
dx "
c)
(
)
d
! ln x ln x #
$
dx "
d)
( )
3
d !
ln x #
$&
dx "%
Ex. 2 Evaluate each derivative.
a)
d ! x2 #
e
dx " $
b)
d ! 2 x 2 %3x #
e
$
dx "
c)
d sin x
!e #
$
dx "
d)
d
! xe x #
dx " $
#33 Friday 10/11________________________________________
F
Worksheet 28
⎧ 3x + 2, x < 1
I
Determine if f ( x ) = ⎨
V
E
S
If f ( x ) = 1 + sin x , approximate f ( 0.1) .
Limits and Derivatives Review
None
2
⎩⎪ x , x ≥ 1
is continuous. Verify using the 3-part definition of continuity.
#34 Monday 10/14___________________________________________
F
Worksheet 29
d
If f x = x 2 + 2x, then
f ln x =
I
dx
( ( ))
()
()
2
()
V
If f x = e x , then f ' x =
E
S
Special Techniques for Derivatives of functions with variables as exponents
Ex. 1 Find the derivative of the function y = x x
Ex. 2 Find the derivative of function y = 8 x
#35 Tuesday 10/15__________________________________________
F
Worksheet 30
I
The line x + y = k is tangent to the graph of y = x 2 + 3x + 1 . What is the value of k?
1
that has a slope 1.
x −1
V
Find an equation for each normal to the curve y =
E
L’Hopital’s Rule and Review for our next Derivatives Quiz – Trigonometric, Logarithm and
Exponentials
Review
S
#36 Wednesday 10/16__________________________________________
F
Finish Worksheet 31 - Study for Tomorrow’s quiz. Know your derivative rules!!
I
None
V
None
E
Review
S
Review Problems
#37 Thursday 10/17_______________________________________
F
Worksheet 32
I
None
V
Find an equation of the tangent line to the curve of f x = xe x − e x when x = 1 .
()
E
S
Test #5 - Derivatives Quiz #2
Good luck on today’s quiz!
#38 Friday 10/18________________________________________
F
Pg. 146 (1-15 odd, 23, 33, 35, 45, 47)
I
Pg. 159 (41, 43, 49)
V
Pg. 147 #28
E
Implicit Differentiation S
Ex. 1 Find
dy
for x 2 + y 2 = 1 .
dx
Ex. 2 Find
dy
for 3x 2 + 5y = 2x
dx
Ex. 3 Find
dy
for x 2 + xy − y 3 = xy 2
dx
Ex. 4 Find
dy
for sin ( xy ) = 2x + 5
dx
#39 Monday 10/21___________________________________________
F
Worksheet 33
I
Find the tangent and normal to the ellipse x 2 − xy + y 2 = 7 at the point ( −1, 2 ) .
V
E
S
Find the slope of the tangent line and the normal line to the curve y = x x + 1 at the point where x = 1.
Implicit Differentiation
None
#40 Tuesday 10/22__________________________________________
F
Worksheet 34
I
None
( )
V
What is the slope of the line tangent to the curve y = arctan 4x at the point at which
E
Derivatives of Inverse Trigonometric Functions
?
S
Evaluate each of the following
1.
( )
d
⎡arcsin 2x ⎤
⎦
dx ⎣
2.
( )
d
⎡arctan 3x ⎤
⎦
dx ⎣
3.
d ⎡
arcsin x ⎤
⎦
dx ⎣
4.
d
⎡ arcsec e2 x ⎤
⎦
dx ⎣
5.
⎛ 3⎞ ⎤
d ⎡
⎢arccos ⎜ ⎟ ⎥
dx ⎣
⎝ t⎠⎦
#41 Wednesday 10/23 – Delayed Start__________________________________________
F
Worksheet 35
I
Find a function that has x 2 − 3x + 2 as its derivative.
V
If f ( x ) = e x ⋅ sin x , find f "( x ) .
E
Derivative of Inverse Trigonometric Functions/Review
S
Review for our Quiz on Thursday ! All Rules of Derivatives – Power Rule, Product Rule, Quotient
Rule, Package Rule, Trigonometric Derivatives, Logarithmic and Exponential Derivatives, Implicit
Differentiation and Inverse Trigonometric Functions
#42 Thursday 10/24__________________________
F
Whew! Enjoy your evening without Calculus homework.
I
None
V
None
E
All Derivatives Test
S
Good luck on today’s test! Keep up the good work!
#43 Friday 10/25_________________________________________
F
Worksheet 36
5
I
1) Find f ' ( x ) if f ( x ) = ( 3x 2 + 4 )
2) Find f ' ( x ) if f ( x ) =
V
E
S
(x
3
)
−3
3
If ( x + 2 ) = 25 and x > 0, what is the value of x 2 ?
Basic Integration: The process of integration is essentially “backwards derivatives”.
Integrate
1⎞
⎛
⎛1 2
⎞
1. ∫ x 7 − 6x + 8 dx
2. ∫ ⎜ x 3 − 3 ⎟ dx
3. ∫ ⎜ − 3 + 2x ⎟ dx
⎝
⎠
⎝
⎠
x
5 x
1
3
4. ∫ 5 dz
5. ∫ ( x − 2 ) dx
6. ∫ x −1 dx
4
z
2
(
)