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Calculus 12 Name ______ LG 1 – 3 Worksheet Package, Lecture notes of Calculus

Colegio de San Juan de Letran Calamba (CSJLC)Calculus

Find the first derivative of each function: a)!!y = 3x2 !5! ... d) How can you use a graphing calculator to check your answers to these types of questions?

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1
Calculus 12 Name _____________
LG 1 – 3 Worksheet Package
Part A:
1. Find the first derivative of each function:
a)!! y=3x2!5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y=8x!2
c)!! f(x)=6x2!3x+2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y=!x2+6
e)!!g(x)=
!
x3+6x!3!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f)!!h(x)=5
!
2x4+6x2!3
!
4
g)!!k(x)=1
4x8!2
3x6+2
5x4!3
4
!!!!!!!!!!!!!!!!!h)!! y=6
!
3!8
!
2+24
2. Given y, find
dy
dx
:
a)!! y=4x3!2x+6!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!b)!! y=1
5x5+2
3x3!1
2x2+1
c)!! y=x4!
!
4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!d)!! y=
!
3x3!3
!
x
3. Solve:
a)!!if !p=4q3+2q2!5!!!!!!!! find !! dp
dq
!=
b)!!if !g(t)=4t3!3t2+6t!!!!! find!g'(t)=
c)!!if !y=2x7!5x+3!!!!!!!!!!! find !y'=
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12
pf13
pf14
pf15

Partial preview of the text

Download Calculus 12 Name ______ LG 1 – 3 Worksheet Package and more Lecture notes Calculus in PDF only on Docsity!

Calculus 12 Name _____________

LG 1 – 3 Worksheet Package

Part A:

  1. Find the first derivative of each function: a )!! y = 3 x^2! 5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = 8 x! 2 c )!! f ( x ) = 6 x^2! 3 x + 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y =! x^2 + 6 e )!! g ( x ) =! x 3 + 6 x! 3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! h ( x ) = 5! 2 x 4 + 6 x 2 ! 3! 4 g )!! k ( x ) =

x 8 !

x 6

x 4 !

!!!!!!!!!!!!!!!!! h )!! y = 6! 3 ! 8! 2

  • 24
  1. Given y, find dy dx

a )!! y = 4 x 3 ! 2 x + 6 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y =

x 5

x 3 !

x 2

  • 1 c )!! y = x 4 !! 4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y =! 3 x 3 ! 3! x
  1. Solve: a )!! if! p = 4 q^3 + 2 q^2! 5 !!!!!!!! find !! dp dq

b )!! if! g ( t ) = 4 t 3 ! 3 t 2

  • 6 t !!!!! find! g '( t ) = c )!! if! y = 2 x^7! 5 x + 3 !!!!!!!!!!! find! y ' =
  1. If y = 2 x^3! 3 x + 7 find:

a )!!! y '! at! x =! 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!!! y '! at! ( 1 ,! 5 )

c )!!! f ( 0 )! and! f '( 0 ) !!!!! d ) How can you use a graphing calculator to check your answers to these types of questions?

  1. Find y ' if: a )!! y = ax^3 + bx^2 + d !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = ax^4! ax^2 + bx c )!! y = 4 ax^5 + kx^3! Cx + D !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = D^2 x^3 + 5 M^3 x^2! 7 !!!!!!
  2. Find dy dp if: a )!! y = 4 p 3 ! 2 p 2
  • 6 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y =! 5 p 4
  • 6 p!

c )!! y = 4 mp 4

  • 16 p 2 ! 6 c !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = 6 a 4
  • 8 a 3 ! 2 p 2
  1. If y = 6 x^5! 2 x^2 + 9 x! 3 find: a )!! y '!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y ''!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! c )!! y ''' d )!! dy dx !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! e )!! d 2 y dx^2 !!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! d 3 y dx^3 g )!! dy dx

2 !!!!!!!!!!!!!!!!!!!!!!!!!!!! h )!! dy dx

3 !!!!!!!!!!!!!!!!!!!!!!!!! i )!! y (^8 )

Part B:

  1. Simplify each expression then find y '.

a )!! y = ( 2 x + 1 ) ( 3 x! 5 )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = ( 2 x! 3 )

2

c )!! y = ( 4 x )

3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )! y = x^2 ( x^3! (^6) )

e )!! y =(! x )

3 ! 3! x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = x^2! 5 x + 4 x! 1

  1. Rewrite each rational expression using exponents to remove quotients and then find the first derivative. a )!! y =

x 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!!^ y^ =^!^

x 3 c )!! y =

x^4

x^2

x !! 7 x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = 4 x 3 !!!

x^2 !+! 7 x ! 5 !!!

x!^4

  1. Rewrite each expression using exponents to remove radicals and quotients, then find the first derivative. a )!! y = 10 x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = 4 3 x^2 c )!! y =

(^5) x 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = x !+! 6 x^3 e )!! y =

x

(^3) x

x 4 3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! e )!! y =

(^5) x

2 x ! 3 3

  1. Use the PRODUCT RULE to find the first derivative. DO NOT simplify answer. a )!! y = ( 3 x + 1 )( 2 x! 5 )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = 3 x^2 ( 8 x! 3 ) c )!! y = ( 2 x + 1 )( 4 x 2 ! 4 x + 1 )!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = ( 3 x 3 ! 2 x 2 )( 3 x 3 + 2 x 2 )
  2. Find dy dx at the given value of x. Do NOT simplify before evaluating. a )!! y = ( 2 + 7 x )( x! 3 )!;! x = 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = ( 1 + 2 x )( 1! 2 x )!;! x =

c )!! y = ( x^4! 4 )( x^4 + 4 )!;! x = 1

  1. Use the QUOTIENT RULE to find the first derivative. DO NOT simplify answer. a )!! y = x^2 2 x + 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = 4 x^2 1! 6 x^3 c )!! y = x 2 ! 4 x x + 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = x 2 ! 9 x 2 + 9 e )!! y = x^3 8! x^3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = 4! x^2 3 x

Part C:

  1. Use the CHAIN RULE to find the first derivative. DO NOT simplify answers. a )!! y = 6 x 2 ( ) 5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y =! 3 x 4 ( ) 5
  • 6 x 2 ! 7 x c )!! y = p 2 (!^3 p^ +^1 ) 4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = x 2 (!^1 ) 3

(^2 x^!^1 )

4 e )!! y = 6 x (^ x^2 +^1 ) 4 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!^ f^ )!!^ y^ =^2 x

( ) 1 2 g )!! y = (^) ( 3 t^4! 2 t ) 1 (^4) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! h )!! y = 5 x + 7 i )!! y =

4 + t^2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! j )!! y = 1 + u 1 3

6 k )!! y = (^) ( 1 + 3 u ) 6 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! l )!! y = 1 +

(^3) x

6

m )!! y =(! x )

3

  • 2! 2 x + 6! x !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! n )! y = (^) ( 2 x^3 + x ) 4 ! o )!! y = 6! x (^ x^3!^ !) 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!^ p )!!^ y^ =^4 x 2

(^2 x^!^5 )

3

  1. Find the first derivative of each expression below. DO NOT simplify your answer.

a )!! y =! x + ( 5! x )

3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = (^) ( 1! x + 2 x^2! 3 x^3 ) 4 c )!! y = (^) ( ( 2 x )^4 + ( 16! x )^3 ) 2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y =

(^2 x^!^1 )

2

(^ x^!^2 )

3

e )!! y = ( 2 x! 1 )

! 3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = ! x (^ x^3!^ !) 2

g )!! y = x ( 1! 2 x )

5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! h )!! y = x^2! 1 x^2 + 1

2

  1. Differentiate each expression below. DO NOT simplify.

a )!! f ( x ) =! x! ( 2! x )

3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! g ( x ) =! x + ! x

2 c )!! h ( x ) = (^) ( 2 x^2! 3 x + (^5) ) ! 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! f ( x ) = x !!!

x!^3

4 e )!! k ( x ) =! x 2 +! 2 ( ) ! 3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! p ( x ) =! 2 x 2 (!^3 x^ +^1 ) ! 4

  1. Given y = 3 x x + 2 find dy dx by: a) using the Quotient Rule b) using the Product Rule c) show the results in (a) and (b) are identical.

Part D

  1. If y = 2 x 3 + 5 x! 7 find: a) the rate of change of y with respect to x. b) the rate of change of y with respect to x at x = 1. c) the rate of change of y with respect to x at (2, 19).
  2. If y = 4 x^2! 6 x! 3 find: a) y ' b) dy dx c) an expression that calculates the slope of the tangent line at any point. d) the slope of the tangent line at x = 1 e) the slope of the tangent line at (2, - 1)

Part E

  1. Use IMPLICIT DIFFERENTIATION to find dy dx in terms of x and y. a )!! 4 x^2 + y^2 = 8 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!!! 3 x! 4 y^2 = 2 c )!! x 2
  • 5 y 2
  • y = 10 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!!! xy 2 = 4 e )!! x^2 + 2 xy! y^2 = 13 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!!! y^3 + y = 4 x g )!! y ( x 2
  • 3 ) = y 4
  • 1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! h )!!! xy 3
  • x 3 y = 2

Part F:

  1. Each position function below describes motion in a straight line. Find the velocity and acceleration as functions of time ( t ). a )!! s ( t ) = 5 t^2! 2 t + 7 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! s ( t ) = 4 t^4!

t^2 + 3 c )!! s ( t ) = 6 t! 8 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! s ( t ) = t! 8 +

t

e )!! s ( t ) = t ( t! 3 )

2 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! s ( t ) = t + 4 t t + 2 Part G:

  1. If x^2 + y^2 = 8 and dx dt = 3 , find dy dt at (-2, 2).
  2. If x 2 + y 2 = z 2 and dx dt

dy dt =! 1 , x = 1 and y = - 3, find dz dt

  1. If A is the area of a circle of radius r , find dA dt in terms of dr dt
  1. The area of a circular oil slick on the surface of the sea is increasing at the rate of 150 m^2 / s. How fast is the radius changing when: a) the radius 25 m. b) the area is 1000 m^2
  2. How fast is the side of a square shrinking when the length of the side is 2 m and the area is decreasing at 0.25 m^2 / s ?.
  3. The hypotenuse of a right triangle is of fixed length but the lengths of the other two sides x and y depends on time. How fast is y changing when dx dt = 4 and x = 8 if the length of the hypotenuse is 17?
  4. A spherical balloon is inflated so that the volume is increasing at the rate of 5 m^3 / min. a) at what rate is the diameter increasing when the radius is 6 m? b) at what rate is the diameter increasing when the volume is 36 m 3 ?
  1. Find dy dx in each case where A, B, m and n are constants:

a )!! y = cos ( Ax + B )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y = A cos n^ Bx

c )!! y = sin m^ ( xn^ )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! d )!! y = Axn^ sin m^ Bx

  1. Find dy dx in each case: a )!! y = 2 tan x! tan 2 x !!!!!!!!!!!!!!!!!!! b )!! y = 3 sec 2 x 2 ( +^1 )!!!!!!!!!!!!!!!!!!! c )!!^ y^ =^3 sec^5 x d )!! y =!! x 2
  • sec 2 x !!!!!!!!!!!!!!!!!!!!! e )!! y = x^2 tan x !!!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = tan( x 2 )! tan 2 x g )!! y = x csc x !!!!!!!!!!!!!!!!!!!!!!!!!! h )!! y = x 2 tan

x

'!!!!!!!!!!!!!!!!!!!!!!!! i )!!^ y^ =^ sin^ (t an^ x )

  1. Use the derivatives of sin x and cos x to verify the derivatives of cot x and csc x as given on the Calculus 12 Formula Sheet.
  1. Find dy dx in each case. Watch for the need for Implicit Differentiation! a )!! y = cot 2 x + csc 2 x !!!!!!!!!!!!!!!!!!!!! b )!! y = 2 x^3 cot x !!!!!!!!!!!!!!!!!! c )!! y = (^) ( x + csc x^2 ) d )!! y =! 2 + csc^2 x !!!!!!!!!!!!!!!!!!!!!!! e )!! y = cot x 1 + csc 2 x !!!!!!!!!!!!!!!!! f )!! y = x !csc x

g )!! y = sin ( xy )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! h )!! y = cot ( x + y )!!!!!!!!!!!!!!!!! i )!! y = (c ot x + sin x )

2 Part I:

  1. Find the derivative of each function: a )!! f ( x ) = ln( x! 2 )!!!!!!!!!!!!!!!!!!!!!! b )!! g ( x ) = 3 ln( 4! 3 x )!!!!!!!!!!!!!!!!!!!!! c )!! k ( x ) = ln( x^2 + 5 ) d )!! h ( x ) = ln x 2 + ln 5 !!!!!!!!!!!!!!!!!!! e )!! p ( x ) = ln x +

x

'!!!!!!!!!!!!!!!!!!!^ f^ )!! h ( x )^ =^ x 2 ln x

g )!! f ( x ) = ( ln x )

4

!!!!!!!!!!!!!!!!!!!!!!!!! h )!! f ( x ) = ln( x^4 )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! i )!! g ( x ) = ( x ln x )

4

j )!! h ( x ) = ( ln x + x )

3

!!!!!!!!!!!!!!!!!!!! k )!! m ( x ) = (s in x ) (l n x )!!!!!!!!!!!!!!!!!!!!! l )!! p ( x ) = ln (s in x )

m )!! f ( x ) = ln x 3 + ln x

!!!!!!!!!!!!!!!!!!!!!! n )!! w ( x ) = ln (s in x + cos x )!!!!!!!!!!!!!!! o )!! r ( x ) = cos

2

(l n^ x )!!!!!!!!!

  1. Differentiate each functions: a )!! f ( x ) = 5 e 2 x !!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! h ( x ) = 2 e x^2! x !!!!!!!!!!!!!!!!!!!!!!!!!! c )!! k ( x ) = 3 e 2 sin x d )!! p ( x ) = x 2 e x !!!!!!!!!!!!!!!!!!!!!!!!!!! e )!! q ( x ) = x 2 e ! 3 x !!!!!!!!!!!!!!!!!!!!!!!!!! f )!! m ( x ) = e 2 x ! e ! 2 x ( ) 2 g )!! g ( x ) = x! e x !!!!!!!!!!!!!!!!!!!!!! h )!! f ( x ) = ln! + e 2 x ( )!!!!!!!!!!!!!!!!!!!!! i )!! m ( x )^ =^ e^2 x 1 + e^2 x j )!! g ( x ) = ex^ ln x !!!!!!!!!!!!!!!!!!!!!!!!!!!! k )!! w ( x ) = ln (^) ( e x^ + e!^ x )!!!!!!!!!!!!!!!!!!!!!!!!!! l )!! r ( x ) = e x ln x
  2. If y defined implicitly as a function of x by the given equation, find dy dx

a )!! x + ylnx = 2 !!!!!!!!!!!!!!!!!!!!!!!!! b )!! y! e xy = 5 !!!!!!!!!!!!!!!!!!!!!!! c )!! e sin 2 y

  • 2 x = 4 y
  1. Find dy dx

a )!! y = x^!^ !!!!!!!!!!!!!!!!!!!!!!!!!!! b )!! y =! x^ !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! c )!! y = e^!^ x d )!! y = 2 x^ !!!!!!!!!!!!!!!!!!!!!!!!!!! e )!! y = ex^! xe^ !!!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = x^!^ 2 g )!! y = 10 x 2 !!!!!!!!!!!!!!!!!!!!!!!!! h )!! y = 2 sin^ x^ !!!!!!!!!!!!!!!!!!!!!!!!!!!!! i )!! y = 3 x (^2) + 3 x

  1. Find dy dx

a )!! y = ln x^2! 1 !!!!!!!!!!!!!!!!!!!!!!!! b )!! y = ln x^3! 7 x + 1 !!!!!!!!!!!!!!!!!!!!!! c )!!! y = (^) (l n x ) 3 d )!! y = ln tan x !!!!!!!!!!!!!!!!!!!!!!! e )!! y = cos x ln cos x !!!!!!!!!!!!!!!!!!!!!!!!! f )!! y = sin (^) (l n x ) Part J:

  1. Find dy dx

a )!! y = 5 ( 3 x! 1 )

3 !!!!!!!!!!!!!!!!!!!!! b )!! y = Ax^3! 2 Bx^2 + C^4 !!!!!!!!!!!!!!!!!! c )!! y = 5 x^3!

x

2 +^

x ! 5