Successfully reported this slideshow.
Upcoming SlideShare
×

Calculus 45S Slides March 17, 2008

1,209 views

Published on

Proofs of the power rule, constant multiplier rule, sum and difference rules.

• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Calculus 45S Slides March 17, 2008

1. 1. The Definition of a Derivative Mathematics is the science of pattern ... let's look for some patterns ... Let's use the definition to find some derivatives ...
2. 2. The Derivative of a Linear Function If then Proof:
3. 3. The Derivative of a Constant Function If then Proof: Example: Find the derivative of
4. 4. The Derivative of a Power Function Proof:
5. 5. Expand and simplify ... a+b
6. 6. The Binomial Theorem ... Algebraically Combinatorically Notice the patterns ... (1) The coefficient of the term is: (2) The exponent on a is given by: [n - (i - 1)] (3) The exponent on b is given by: i - 1 (4) This relation holds for each term in the expansion: [exponent on a] + [exponent on b] = n (5) The number of terms in any binomial expansion is: n + 1
7. 7. The Binomial Theorem ...
8. 8. The Derivative of a Power Function Proof:
9. 9. The Derivative of a Constant Times a Function Proof: Example: Find the derivative of
10. 10. The Derivative of a Sum of Functions Proof:
11. 11. The Derivative of a Difference of Functions Proof: