calculus explained with examplesDescripción completa
dddd
differential calculus
calculusDescripción completa
differential calculus
differential calculusFull description
Clyde E. Love and RainvilleFull description
:)
Radius of Curvature
Differential Calculus by Shanti Narayan for B.A. & B.Sc. Students কলকাতা বিশ্ববিদ্যালয় সিলেবাস নির্দেশিত পুস্তক 10th Revised Edition 1962 AD 424 pages 16.10 MB PDF
Dif Forms
Descrição completa
Differential Calculus - PART 3 - Maxima-minima-timeratesFull description
Calculus Of Variations
Differential CalculusFull description
Descripción: Calculus Of Variations
its all about mathematicsDescripción completa
Descripción: calculus
makalah differentialDeskripsi lengkap
CalculusFull description
Math HL IB limits past paper
Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during those years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffu
Derivatives Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit
f’(x) = lim f(x+h) f(x) –
h
h
provided this limit exists. If this limit exists for each x in an open interval I, then we say that f is differentiable on I.
Definition Example: f(x) = x2 + 2x + 8 2
lim h
0
f(x+h) h – f(x) =
lim h
0
2
[(x+h) + 2(x+h)+8]-[x + 2x+8] h
= lim
[(x2+2xh+h2)+2x+2h+8]-[x2+2x+8] h
= lim
[2xh + h2 + 2h] h
= lim
2x + h + 2
h
h
h
0
0
0
2x + 2
=
Definition Example: f(x) = x2 + 9x + 1 2
lim h
0
f(x+h) h – f(x) =
lim h
0
2
[(x+h) + 9(x+h)+1]-[x + 9x + 1] h
= lim
[(x2+2xh+h2)+9x+9h+1]-[x2+9x+1] h
= lim
[2xh + h2 + 9h] h
= lim
2x + h + 9
h
h
h
0
0
0
2x + 9
=
One-sided Derivative Definition. Let y = f(x) be a function and let a be in the domain of f. The right-hand derivative of f at x = a is the limit
f’+(x) =lim f(a+h) f(a) –
h +
h
and the left-hand derivative of f at x = a is the limit
f’-(x) = lim f(a+h) f(a) –
h -
h
Definition The function f is differentiable on the interval I if •
when I has a right-hand endpoint a, then the left hand derivative of f exists at x = a, when I has a left-hand endpoint b, then the righthand derivative of f exists at x = b, and •
•
f is differentiable at all other points of I.
Definition Example: Calculate the derivative of f(x) = |x| at
x = 0.
lim h
0+
f(0+h) – f(0) h
0+h – = lim h h
h = lim = 1 h h
lim h
0-
f(0+h) – f(0) h
0+
0+
Since h>0
0+h – = lim h h
0-
h = lim h h
0-
= -1
Since h<0
Definition 2
f(x) = |x| 1.5
1
Since the right-hand and the left-hand derivatives are different, the function f is not differentiable at x = 0.
0.5
-2
-1
1
2
Notations If y = f (x) , then all of the following are equivalent notations for the derivative f’(x) = y’ =
df dy d = = (f(x)) = Df(x) = dx dx dx
Dy lim D Dx x
0+
Notations If y = f (x) , all of the following are equivalent notations for the derivative evaluated at x = a f’(a) = y’|x=a =
|
df dx
x=a
=
dy dx
|
x=a
=
d (f(a)) = Df(a) dx
Interpretations If y = f (x) , then m = f ’(a) is the slope of the tangent = f (x)is at x=aby line to y line and the equation of the tangent given y=f(a)+f ’(a)(x-a)
Example: let f(x) = x2
f’(x) = 2x Then 2 = f ’(1) is the slope of the tangent line to x 2 at x= 1 and the equation of the tangent line is given by y=1+2(x -1) y=2x - 1
Interpretations Suppose we want to find the equation of the line tangent to x 2 when a = 1.5. Then 3 = f ’(1.5) is the slope of the tangent line at x= 1.5 and the equation of the tangent line is given by y=f (1.5)+ f ’(1.5)(x-1.5)
y = (1.5)2 + f ’(1.5)(x-
1.5) y = 2.25 + 3(x-1.5) y = 2.25 + 3x - 4.5 y = 3x - 2.25
Interpretations 4
f(x) = x2 f(x) = 3x - 2.25 3
f(x) = 2x-1 2
1
0.5
1
1.5
2
Interpretations If f ’(a) is the instantaneous rate of change of f(x) at x=a
If f ’(x) is the position of an object at time x , then f ’(a) is the velocity of the object at x=a
Properties and Formulas d (xn) = nxn-1 dx Example: f(x) = x2
Alternative Solutions For product rule, we could simplify the polynomial by expanding the factors and then take the derivative Example: f(x) = x2 (2x2 + 4x + 2)
= 2x4 + 4x3 + 2x2 f’(x) = 8x3 + 12x2 + 4x
Alternative Solutions Example: f(x) = (x+2)(4x3 + 3x + 1)
Alternative Solutions For quotient rule, recall that fractional functions can be expressed as a product by taking the negative of the denominator’s exponent Example: f(x) =
Alternative Solutions By simplifying, the srcinal polynomial, we will always arrive at the answer through the simplest solution x2 + 2x + 1 x+1 = (x+1)(x+1) x+1
