ap unit 2 review part 1 (2a-2e) -...

Unit2ReviewPart1(2A-2E) Name: Date: Period:

Hahn2014-20151

2A: Definition of Derivative Choose 2 problems in section 2C to prove the derivative using the derivative formula and taking the limit as h approaches 0. 1. Problem # 2. Problem # 2B: Tangent and Normal Lines Find the tangent line to the following functions: 3. 𝑦 = 𝑥! + 𝑥; 𝑥 = −1 4. 𝑦 = 𝑥! + 2𝑥 𝑥 + 1 ; 𝑥 = 1 5. 𝑦 = 𝑥! + 9; 𝑥 = 0 6. 𝑦 = sin 𝑥 ; 𝑥 = 1


Hahn2014-20152

2C: Derivative Rules Find !"

!":

Rahn © 2006

Differentiation Quiz

Use the rules for differentiation to find (no t’s in your answers.)dydx

1. y x x � �4 3 23

2. y u u v v x �, sin , 3 1

3. y x x xx � � � �log cos ( )3 10 3 5 4

4. y t x t sin , cos

5. y x �tan ( )3 2 4

6. x xy y4 45 6 212� �

7.� �

yx

�

5

42 3

8. y x x � �( ) ( )4 3 110

9. � � � �3 82 1 3 43 4

x xy

x� �

�

10. � �e y xx3 2 sec

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.


Hahn2014-20153

2C Answers:

Rahn © 2006

1. y x x � �4 3 23

2. y u u v v x �, sin , 3 1

3. y x x xx � � � �log cos ( )3 10 3 5 4

4. y t x t sin , cos

5. y x �tan ( )3 2 4

6. x xy y4 45 6 212� �

7.� �

yx

�

5

42 3

8. y x x � �( ) ( )4 3 110

9. � � � �3 82 1 3 43 4

x xy

x� �

�

10. � �e y xx3 2 sec

1. 212 3x �

2.2 3

3

3 cos( 1)2 sin( 1)x x

x�

�

3. 31 10 ln10 sin 12(3 5)ln10

x x xx

� � � �

4.21

xx

�

�

5. 2 26 tan (2 4) sec (2 4)x x� �

6.3

3

5 424 5y xy x��

7. 2 42 4

3030 ( 4)( 4)

xx x orx

� ��

�

8.9

9

( 4) (10(3 1) 3( 4)( 4) (33 2)x x xx x� � � �

� �

9. 2 6(2 1) (3 4) (60 3)simplify firstx x x� � �

10.2 3

3

2sec tan 3 x

x

x x yee

�


Hahn2014-20154

(2D)

MasterMathMentor.com - 50 - Stu Schwartz

Differentiation of Trigonometric Functions - HomeworkTake the derivatives of the following functions. Identify the form of the problem and rewrite with parentheses.

1. y x= sin3 2. y x x= sin 3. y x=$"

&

'(

)

*+cos

2

4. y

x

x=

sin5.

y

x

x=

sin6. y x x= 3 2

sin

7. y x x= "cos sin2 3 8. y x= cos4 4 9. y x x= +sin cos

2 2

10. y x= +sin 2 11. y x= "tan 3 1 12. y x x= " +! "sec2

2 3

13)

yx

=&

'()

*+cot

4

214)

y

x

x=

+

sin

cos12

15) y x= ! "sin cos

Find the equation of the tangent line to the following curves at the indicated point. Confirm by calculator.

16) y x x= ! "sin cos , at 0 0 17)

y

x

x= ! "

20 0

cos, at 18)

y x x x= +! "$&

'(

)

*+sin sin cos , at

41



1. y x= sin3 2. y x x= sin 3. y x=$"

&

'(

)

*+cos

2

4. y

x

x=

sin5.

y

x

x=

sin6. y x x= 3 2

sin


2 2


2 3

13)

yx

=&

'()

*+cot

4

214)

y

x

x=

+

sin

cos12

15) y x= ! "sin cos


16) y x x= ! "sin cos , at 0 0 17)

y

x

x= ! "

20 0

cos, at 18)

y x x x= +! "$&

'(

)

*+sin sin cos , at

41



1. y x= sin3 2. y x x= sin 3. y x=$"

&

'(

)

*+cos

2

4. y

x

x=

sin5.

y

x

x=

sin6. y x x= 3 2

sin


2 2


2 3

13)

yx

=&

'()

*+cot

4

214)

y

x

x=

+

sin

cos12

15) y x= ! "sin cos


16) y x x= ! "sin cos , at 0 0 17)

y

x

x= ! "

20 0

cos, at 18)

y x x x= +! "$&

'(

)

*+sin sin cos , at

41

7.

8. 9.


Hahn2014-20155

2E: Chain Rule (Composition)

Given the following functions of x, write the equation for the derivative function. Remember to apply the constant, linear, product, quotient, and most importantly, the chain rule. Please do NOT simplify your answers. 1. ( ) ( )xxf 3sin= ( ) =xf ' 2. ( ) ( )( )43sin xxg = ( ) =xg' 3. ( ) ( )( ) xxxh 103sin 4 += ( ) =xh' 4. ( ) ( )( )[ ]24 103sin xxxi += ( ) =xi'

5. ( )x

xxj 13 −= ( ) =xj'

6. ( )x

xxk 13 −= ( ) =xk '

7. ( ) ( )( )[ ]24 103sin13 xxx

xxl +⋅−= ( )=xl'

ap unit 2 review part 1 (2a-2e) -...

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