Slope of a skateboard: 1·4 1·3 1·0 0·8 0·6 0 –1·9 –1·6 –0·2 –1·4 0·3 0·9 1·7 2 1 How steep is a hill?

Slide 3:

… the less accurate the slope The longer the skateboard ….

Slide 4:

and looking at the slope at one spot… P Zooming in on part of the hill

Slide 5:

P We get a more accurate measure … if we move the wheels closer together

Slide 6:

P We get a more accurate measure … if we move the wheels closer together

Slide 7:

P Definition of the slope of a hill at a point: The slope of a hill at point P is the LIMIT of the slopes of skateboards with one wheel at point P, as the other wheel approaches the wheel at P.

Differentiation from first principle :

Differentiation from first principle

Slide 9:

P y x (x, y) Graph y = f(x) y = f (x)

Slide 10:

P Now take a point a distance from P: x y (x + x, y + y) (x, y) y x y = f (x)

Slide 11:

y = f (x) P What is the slope of this line? y2 – y1 x2 – x1 y y x x

Slide 12:

P What happens as x gets smaller? x y

Slide 13:

P What happens as x gets smaller? x y

Slide 14:

y P What happens as x gets smaller? x

Slide 15:

x P What happens as x gets smaller? y

Slide 16:

x P What happens as x gets smaller? y

Slide 17:

P Slope becomes nearer to the slope of the tangent What happens as x gets smaller?

Slide 18:

P The slope of the tangent to the curve f ' (x)

Slide 19:

y + ∆y = x2 + 2x∆x + ∆x2 Differentiate x2 with respect to x from first principles. Let y = x2 Let x change to x + Dx, with a corresponding change in y to y + Dy Subtract Divide by Dx y + ∆y = (x + ∆x)2 = (x + ∆x)(x + ∆x) = x2 + x∆x + x∆x + ∆x2 y = x2 = 2x∆x + ∆x2 = 2x + ∆x ∆y Cancel 20

= 2x∆x + ∆x2 + x∆x = (x + ∆x)(x + ∆x) – 3(x + ∆x) + x∆x – 3∆x y + ∆y = x2 + 2x∆x + ∆x2 – 3x – 3∆x (b) Differentiate x2 – 3x with respect to x from first principles. Let y = x2 – 3x Let x change to x + Dx, with a corresponding change in y to y + Dy Subtract Divide by Dx y + ∆y = (x + ∆x)2 – 3(x + ∆x) = x2 + ∆x2 y = x2 – 3x = 2x + ∆x ∆y Cancel – 3x Each term inside the brackets changes sign 20 – 3∆x – 3

Slide 22:

y + ∆y = (x + ∆x)2 + 3(x + ∆x) y + ∆y = x2 + 2x∆x + ∆x2 + 3x + 3∆x (b) Differentiate x2 + 3x with respect to x from first principles. Let y = x2 + 3x Let x change to x + Dx, with a corresponding change in y to y + Dy Subtract Divide by Dx = (x + ∆x)(x + ∆x) + 3(x + ∆x) = x2 + x∆x + x∆x + ∆x2 + 3x + 3∆x y = x2 + 3x = 2x∆x + ∆x2 = 2x + ∆x ∆y + 3∆x Cancel + 3 20

Slide 23:

– ∆x + x∆x = 3(x + ∆x)(x + ∆x) – (x + ∆x) y + ∆y = 3x2 (b) Differentiate 3x2 – x from first principles with respect to x. Let y = 3x2 – x Let x change to x + Dx, with a corresponding change in y to y + Dy Subtract Divide by Dx y + ∆y = 3(x + ∆x) 2 – (x + ∆x) + ∆x2) y = 3x2 – x = 6x∆x – 3∆x2 = 6x – 3∆x ∆y Cancel = 3 (x2 + x∆x – x – ∆x Each term inside the brackets changes sign + 2x∆x – 1 1 20 + 6x∆x + 3∆x2 – x – ∆x

Differentiation from first principle :

Differentiation from first principle

Slide 25:

y = f (x) P f (x) x (x, f (x)) Graph y = f(x)

Slide 26:

y = f (x) P Now take a point a distance from P: f (x) x h (x, f (x)) f (x + h) (x + h, f (x + h))

Slide 27:

y = f (x) P What is the slope of this line? f (x) x h f (x + h) y2 – y1 x2 – x1

Slide 28:

x2 – x1 ( ) P What is the slope of this line? f (x) x h f (x + h) f (x + h) – f (x) y2 – y1 h + – x x What happens as h gets smaller?

Slide 29:

P What happens as h gets smaller? f (x) h f (x + h) f (x + h) – f (x) h x

Slide 30:

P What happens as h gets smaller? f (x) h f (x + h) f (x + h) – f (x) h x

Slide 31:

P What happens as h gets smaller? f (x) h f (x + h) f (x + h) – f (x) h x

Slide 32:

P What happens as h gets smaller? f (x) h f (x + h) f (x + h) – f (x) h x

Slide 33:

P What is the slope of this line? f (x) h f (x + h) f (x + h) – f (x) h x

Slide 34:

P What happens as h gets smaller? f (x) f (x + h) – f (x) h x Slope becomes nearer to the slope of the tangent

Slide 35:

P The slope of the tangent to the curve f (x) f (x + h) – f (x) h x f ' (x)

Slide 36:

f (x) = x 2
f (x + h) = (x + h)2 = x 2 + 2xh + h 2 = 2x + 0 = 2x Find the derivative of f(x) = x2

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.

By: satheesan (68 month(s) ago)

your ppt presentation is very useful. thank you so much. can you please send the full file. myemail is sathesan@hotmail.com