.

0.010.005

sin(0.01)

= 0

cos(A+B)=cos(A)cos(B)-sin(A)sin(B) If A = B = x cos(2x)=1-sin2x-sin2x Replacing x with x/2 gives

0 0

1 cos( ) cos( ) 1lim limx x

x xx x

2 2cos(2 ) cos sinx x x

cos(2x)=1-sin2x-sin2x

2sin2(x/2)= 1-cos(x)

0

1 coslimx

xx

2

0

2sin2lim

x

x

x

0

2sin sin2 2lim

x

x x

x

0

sin22(

12) lim sin

22

x

xx

x

(1)(1)(0) 0

= 00 0

1 cos( ) cos( ) 1lim limx x

x xx x

.

0.00.1

2 0

cos(2 ) 1lim

2t

tt

.

0.00.1

2 0

2 cos(2 ) 1lim

sin(2 ) 2t

t tt t

0

sin( )lim 1h

hh

0

sin(7 )limx

xx

0 7 0

sin(7 ) sin(7 )lim l m7

7i

x x

x xx x

0

sin( )lim7h

hh

7(1) 7

0

00

limsin(7 ) / 7sin(7 )lim

sin(3 ) limsin(3 )?

/ 3?x

xx

x xxx x x

0

0

limsin(7 ) /(7 )

limsin(3 ) /(3 )37 x

x

x x

x x

73

.

0.50.1

0

sin(4 )lim

sin(8 )x

xx

Equation of Lines

Write the equation of a line that passes through (-3, 1) with a slope of – ½ .

or or

( 3)

10.5

y

x

0.5(1 3)xy

( 3 10.5 )y x

Passes through (0, 1) with a slope of -3. What is the missing blue number?

0.00.1

13

y

x

( _1 _)3y x

Write the equation of the line Write the equation of the line tangent to y = x + sin(x) when x tangent to y = x + sin(x) when x

= 0= 0given the slope there is 2.given the slope there is 2.

A.A. y = 2x + 1y = 2x + 1

B.B. y = 2x + 0.5y = 2x + 0.5

C.C. y = 2xy = 2x

Find the slope of the tangent Find the slope of the tangent line of f(x) = 2x + 3 when x = line of f(x) = 2x + 3 when x =

1.1.1. Calculate f(1+h) – f(1)1. Calculate f(1+h) – f(1)

f(1+h) = 2(1+h) + 3f(1+h) = 2(1+h) + 3

f(1) = 5 f(1) = 5

f(1+h) – f(1) = 2 + 2h + 3 – 5 f(1+h) – f(1) = 2 + 2h + 3 – 5 =2h=2h

2. Divide by h and get 22. Divide by h and get 2

3. Let h go to 0 and get 23. Let h go to 0 and get 2

0

1( ) ( )lim

1h

h

h

f f

Find the slope of the tangent Find the slope of the tangent line of f(x) = xline of f(x) = x22 when x = x. when x = x.

1. Calculate f(x+h) – f(x)1. Calculate f(x+h) – f(x)

f(x+h) = xf(x+h) = x22 + 2xh + h + 2xh + h22

f(x) = xf(x) = x22

f(x+h) – f(x) = 2xh + hf(x+h) – f(x) = 2xh + h2 2 ..

2. Divide by h and get 2x + 2. Divide by h and get 2x + hh

3. Let h go to 03. Let h go to 0

0limslop

(e

) ( )h

f fx h x

h

2

0 0li

22m lim

h h

xh hx h

h

Find the slope of f(x)=xFind the slope of f(x)=x22

A.A. 2x+h2x+h

B.B. 2x2x

C.C. xx22

0limslop

(e

) ( )h

f fx h x

h

2

0

2 2 2 2

0lim l

( ) ( )i

2m

h h

x h x x xh h x

h h2

0 0li

22m lim

h h

xh hx h

h

Find the slope of the tangent Find the slope of the tangent line of f(x) = xline of f(x) = x22 when x = x. when x = x.

1. Calculate f(x+h) – f(x)1. Calculate f(x+h) – f(x)

f(x+h) = xf(x+h) = x22 + 2xh + h + 2xh + h22

f(x) = xf(x) = x22

f(x+h) – f(x) = 2xh + hf(x+h) – f(x) = 2xh + h2 2 ..

2. Divide by h and get 2x + 2. Divide by h and get 2x + hh

3. Let h go to 0 and get 2x3. Let h go to 0 and get 2x

0limslop

(e

) ( )h

f fx h x

h

Finding the slope of the Finding the slope of the tangent line of f(x) = xtangent line of f(x) = x22, f(x+h) , f(x+h)

- f(x) =- f(x) =A.A. (x+h)(x+h)22 – x – x22

B.B. xx22 + h + h22 – x – x22

C.C. (x+h)(x – h)(x+h)(x – h)

(x+h)(x+h)22 – x – x22 = =A.A. xx2 2 + 2xh + h+ 2xh + h22

B.B. hh22

C.C. 2xh2xh + h+ h22

==

A.A. 2x2x

B.B. 2x + h2x + h22

C.C. 2xh2xh

0limslop

(e

) ( )h

f fx h x

h

22xh + h

h0limh

3

21. lim

3x

xEvaluate

x

16

42. lim

16x

xEvaluate

x

2

21

23. lim

1t

t tEvaluate

t

TheoremsTheorems

1. (f + g) ' (x) = f ' (x) + g ' (x), and 1. (f + g) ' (x) = f ' (x) + g ' (x), and

2. (f - g) ' (x) = f ' (x) - g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x)

1. (f + g) ' (x) = f ' (x) + g ' (x) 1. (f + g) ' (x) = f ' (x) + g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x) 2. (f - g) ' (x) = f ' (x) - g ' (x)

If f(x) = 3If f(x) = 322 x + 7, find f ’ x + 7, find f ’ (x)(x)

f ’ (x) = 9 + 0 = 9f ’ (x) = 9 + 0 = 9

If f(x) = x - 7, find f ’ (x)If f(x) = x - 7, find f ’ (x)

f ’ (x) = - 0 = f ’ (x) = - 0 =

55 5

If f(x) = -2 x + 7, find f ’ (x)If f(x) = -2 x + 7, find f ’ (x)

-2.0-2.0

0.10.1

If f(x) = thenIf f(x) = then f’(x) = f’(x) =

Proof : f’(x) = Lim [f(x+h)-f(x)]/h = Proof : f’(x) = Lim [f(x+h)-f(x)]/h =

x1

2 x

If f(x) = then f’(x) = If f(x) = then f’(x) =

A.A. ..

B.B. ..

C.C. ..

D.D. ..

x

0limh

x h x

h

0limx

x h x

h

0limh

x x

h

x h x

h

f’(x) = = f’(x) = =

A.A. ..

B.B. ..

C.C. ..

D.D. ..

0limh

x h x

h

0limh

x

h x h x

0limh

x x

h

0

lim( )h

x h x

h x h x

x h x

x h x

0limh

x h x

h x h x

f’(x) = = f’(x) = =

A.A. ..

B.B. ..

C.C. ..

0lim

( )h

h

h x h x

0limh

x x

h

0lim

( )h

x h x

h x h x

0

*1limh

h

h x h x

f’(x) = = f’(x) = =

A.A. ..

B.B. 00

C.C. ..

0

*1lim

( )h

h

h x h x

1

x

x

1

2 x

g(x) = 1/x, find g’(x)g(x) = 1/x, find g’(x)

g(x+h) = 1/(x+h)g(x+h) = 1/(x+h) g(x) = 1/xg(x) = 1/x

g’(x) = g’(x) =

1 1 ( )1

( )

x x hx h xh x x h

1 1x h xh

( )

( )

x x h

hx x h

1

2

1

x

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-1)(n-1)

If f(x) = xIf f(x) = x44 then f ' (x) = 4 xthen f ' (x) = 4 x33

If If 2

3( )g x

x 23x

2 2 3'( ) (3 ) ' 3( ) ' 3( 2 )g x x x x 3

3

66x

x

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n xn-1 n-1

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4 xf ' (x) = 4 x3 3 + . . . .+ . . . .

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3

f ’ (1) = 4 + 9 – 4 – 3 = 6f ’ (1) = 4 + 9 – 4 – 3 = 6

3y

If f(x) = xIf f(x) = xnn then f ' (x) = n x then f ' (x) = n x (n-1)(n-1)

If f(x) = If f(x) = xx44 then f ' (x) = 4then f ' (x) = 4 x x33

If f(x) = If f(x) = 44 then f ' (x) = 0then f ' (x) = 0 If If ( ) 3g x x

1

23x1 1 1

2 2 21

'( ) (3 ) ' 3( ) ' 3( )2

g x x x x

1

23 3

2 2x

x

If f(x) = then f ‘(x) =If f(x) = then f ‘(x) =x

1 1

2 21

'( ) ( ) '2

f x x x

1

2 x

Find the equation of the line Find the equation of the line tangent to g when x = 1. tangent to g when x = 1.

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

g ' (x) = 3 xg ' (x) = 3 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

g (1) =g (1) =

g ' (1) =g ' (1) =

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g (1)find g (1)

0.00.0

0.10.1

If g(x) = xIf g(x) = x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4find g’ (1)find g’ (1)

-4.0-4.0

0.10.1

Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.

g(1) = 0g(1) = 0

g ' (1) = – 4g ' (1) = – 4

14

0

x

y

4(0 1)xy

( 1)4y x

Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f (1) = 1 + 3 – 2 – 3 + 4 = 3f (1) = 1 + 3 – 2 – 3 + 4 = 3

f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6

Find the equation of the line Find the equation of the line tangent to f when x = 1. tangent to f when x = 1.

f(1) = 1 + 3 – 2 – 3 + 4 = 3f(1) = 1 + 3 – 2 – 3 + 4 = 3

f ' (1) = 4 + 9 – 4 – 3 = 6 f ' (1) = 4 + 9 – 4 – 3 = 6

61

3Y

X

Write the equation of the Write the equation of the tangent line to f when x = 0. tangent line to f when x = 0.

If f(x) = xIf f(x) = x44 + 3 x+ 3 x33 - 2 x - 2 x22 - 3 x + 4 - 3 x + 4

f ' (x) = 4xf ' (x) = 4x33 + 9 x+ 9 x22 - 4 x – 3 + 0 - 4 x – 3 + 0

f (0) = write downf (0) = write down

f '(0) = for last questionf '(0) = for last question

Write the equation of the line Write the equation of the line tangent to f(x) when x = 0.tangent to f(x) when x = 0.

A.A. y - 4 = -3xy - 4 = -3x

B.B. y - 4 = 3xy - 4 = 3x

C.C. y - 3 = -4xy - 3 = -4x

D.D. y - 4 = -3x + 2y - 4 = -3x + 2

http://www.youtube.com/watch?v=P9dpTTpjymE Derive Derive

http://www.9news.com/video/player.aspx?aid=52138&bw= Kids= Kids

http://math.georgiasouthern.edu/~bmclean/java/p6.html Secant Lines Secant Lines