Differensial And Integral
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DIFFERENTIAL AND INTEGRAL
Symbol of DIFFERENTIAL r = f(t) = at2 + bt + c dr / dt 2. f ’ (t) 3. lim Δ t → 0 Δr Δt 1.
DIFFERENTIAL n F(t) = mt n-1 F’(t) = n.mt F(t) = m F’(t) = 0 n and m are constant
Example F(t)= 4t5 + ⅓t3 + 5t + 9 The differensial is: 5.4t5-1 + 3.⅓t3-1 + 1.5t1-1 + 0 The final answer is: 20t4 + t2 + 5 + 0
exercises Find the first differensial of these equations 1. F(x)=
6x4 + ⅓x2 + 4x 3 ½ -2 2.v(x) = 4x + 4x + 5x 3.r(t) = t5 + t4 + 5t + 6 4.s(t)= 2t3 i + 4t2 j + 4t k 5.a(t)= (2t3 + 5t2 ) i + (6t-2) j
Symbol of INTEGRAL
f(t) = at
b
∫ f(t) dt =
1 .at b+1
C= constanta
b+1
+c
example
f(t) = 5t + 6t + t + 8 9
3
the integral is:
5/9+1.t
9+1
1/1+1.t
+ 6/3+1.t
1+1
+ 8t + c
3+1
+
The final answer is
= 1/2.t + 6/4.t + 8t + c 10
4
+ 1/2.t
2
exercises Find the first integral of these equations 1. F(x)=
6x4 + 3x2 + 4x 3 2 2.v(x) = 4x + 4x + 5 3.r(t) = 8t3 + t2 + 5t + 6 4.s(t)= 2t3 i + 6t2 j + 4t k 5.a(t)= (9t3 + 5t4 ) i + (6t2) j
APPLICATION DIFFERENTIAL AND INTEGRAL IN KINEMATICS MOTION
DIFFERENTIAL Position
r(t) Velocity v = Δr / Δ t = r ‘ ( t ) Acceleration
a = Δv / Δt = v ’ ( t ) Lim Δt → 0
example r(t) = ( t5
+ t4 + 5t + 6 ) i V(t) = r ’(t) V(t) = ( 5t4 + 4t3 + 5 ) i a(t) = V ‘ (t) a(t) = ( 20t3 + 12t2 ) i
Integral r = position v = velocity a = acceleration
∫ a dt = v ( t ) ∫ v dt = r ( t )
Symbol of DIFFERENTIAL r = f(t) = at2 + bt + c dr / dt 2. f ’ (t) 3. lim Δ t → 0 Δr Δt 1.
DIFFERENTIAL n F(t) = mt n-1 F’(t) = n.mt F(t) = m F’(t) = 0 n and m are constant
Example F(t)= 4t5 + ⅓t3 + 5t + 9 The differensial is: 5.4t5-1 + 3.⅓t3-1 + 1.5t1-1 + 0 The final answer is: 20t4 + t2 + 5 + 0
exercises Find the first differensial of these equations 1. F(x)=
6x4 + ⅓x2 + 4x 3 ½ -2 2.v(x) = 4x + 4x + 5x 3.r(t) = t5 + t4 + 5t + 6 4.s(t)= 2t3 i + 4t2 j + 4t k 5.a(t)= (2t3 + 5t2 ) i + (6t-2) j
Symbol of INTEGRAL
f(t) = at
b
∫ f(t) dt =
1 .at b+1
C= constanta
b+1
+c
example
f(t) = 5t + 6t + t + 8 9
3
the integral is:
5/9+1.t
9+1
1/1+1.t
+ 6/3+1.t
1+1
+ 8t + c
3+1
+
The final answer is
= 1/2.t + 6/4.t + 8t + c 10
4
+ 1/2.t
2
exercises Find the first integral of these equations 1. F(x)=
6x4 + 3x2 + 4x 3 2 2.v(x) = 4x + 4x + 5 3.r(t) = 8t3 + t2 + 5t + 6 4.s(t)= 2t3 i + 6t2 j + 4t k 5.a(t)= (9t3 + 5t4 ) i + (6t2) j
APPLICATION DIFFERENTIAL AND INTEGRAL IN KINEMATICS MOTION
DIFFERENTIAL Position
r(t) Velocity v = Δr / Δ t = r ‘ ( t ) Acceleration
a = Δv / Δt = v ’ ( t ) Lim Δt → 0
example r(t) = ( t5
+ t4 + 5t + 6 ) i V(t) = r ’(t) V(t) = ( 5t4 + 4t3 + 5 ) i a(t) = V ‘ (t) a(t) = ( 20t3 + 12t2 ) i
Integral r = position v = velocity a = acceleration
∫ a dt = v ( t ) ∫ v dt = r ( t )
