3.1 Power Rule - Calculus

3.1 Power Rule

Write your questions here!

Notation

()

POWER RULE

=

Find the derivative of the following.

()

=

37

-

45

-

1 3

3

+

2

-

3

+

7

NOTES

= 2-3 + 4 +

Rewrite and then take the derivative.

= 37 - + 252

()

=

1

+

4 2

-

1 (3)2

()

=

-162+5-1 2

Evaluate

()

=

1 2

4

-

4-2

+

Find (3)

Find =4

1 = + 4

Higher Order Derivatives

() = 7 - 24 + 52 - 3 + 9 () = () = () = (4)() =

= + -2 = 2 2 =

Find Derivative on the Calculator

()

=

1 2

2

-

1

(4) =

() =

() = 1 + csc () = 2 =

Derivative means...

Slope at a point

Given

=

1 2

4

-

+

2

find

the slope at = 2

Slope of the tangent line

Write the equation of the line

tangent to

=

1 2

4

-

+

2

at = 2

Instantaneous rate of change

What is the instantaneous rate of change at 3 seconds? () = -4.92 + 40 + 6

Normal Line

Write the equation of the normal line at = 3 and then graph it! () = 3 - 42 + + 3

SUMMARY:

Derivative Rules

Constant Rule = 0

Power Rule

=

-1

Now, summarize your notes

here!

Constant Multiple Rule () =

Sum/Difference Rule

( ? ) = ?

3.1 Power Rule

Find the derivative of the following.

1. () = 23 - 4 + 5

2. = 3100 - 28 - 7

PRACTICE

3.

()

=

5-2

-

1 2

4

4.

()

=

6 3

+

62/3

-

41/2

+

2

5.

()

=

1 3

+

12

6.

=

3 -2

-

1 (6)2

7. () = + 33 + 2

8. = 32 + 847

9.

()

=

1

+

3 6

10.

()

=

1

+

3 52

13.

()

=

4 3

3

11. () = -162 + 40 + 5

12. = 2 -

14.

()

=

23+4-5

15.

()

=

63+42-9 3

Find the derivatives of the following.

16. () = 37 - 43 + 5 + 7

17. = 4 +

() = () =

=

() = (4)() =

2 2 =

18.

=

1 3

-

1 2

4

+

2

=

=

=

Given () = - + ,

()

=

+

,

and

() = , find the following.

19. (2) =

20. (-3) =

21. 2(4) =

22. Find the slope of () at = 3. 23. At what value of x is () = 0 ? 24. What is the slope of the tangent line of () at the point (16, 4) ?

Find the equation for the slope of the line tangent to the given function.

25. () = 2 - 2

26.

=

-23

+

1 2

2

-

7

+

5

27.

()

=

1 2

-

1 2

Is the slope of the tangent line positive, negative, or zero at the given point?

28.

()

=

43-162 2

at = 2

29. = 24 + 53 at = -2

30. () = 335 - 4-1 at = 8

Write the equation of the tangent line and the normal line at the point given.

31. () = 3 + 4 at = 4

32.

=

2+3-4 2

at = 8

The function is graphed below. Write the equation of the tangent line at the given point and graph it.

33. () = -2 - 2 - 1 at = -2

34.

=

3 2

-

2 2

-

at = 1

The function is graphed below. Write the equation of the normal line at the given point and graph it.

35. () = -3 + 22 - 2 at = 2

36. = 2 + 6 + 9 at = -2

You will need to use a graphing calculator for 37-42

Use the graph to find the derivative of the function at the given value. Round to nearest thousandth.

37.

()

=

2+1 -2

at

=6

38. = at = -1

39.

()

=

2

sin

at

=

2

Write the equation of the tangent line at the point given and sketch the graph. Round to nearest thousandth.

40. () = -3 + 4 at = 5

41. = ln() + 4 at =

42.

()

=

csc

+

1

at

=

4

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download