1. Monday, Sept 28, 2015MAT 145

2. Monday, Sept 28, 2015MAT 145

3. Monday, Sept 28, 2015MAT 145

4. Monday, Sept 28, 2015MAT 145

5. Monday, Sept 28, 2015MAT 145

6. Monday, Sept 28, 2015MAT 145

7. Monday, Sept 28, 2015MAT 145

8. Monday, Sept 28, 2015MAT 145

9. Monday, Sept 28, 2015MAT 145

10. Monday, Sept 28, 2015MAT 145

11. Monday, Sept 28, 2015MAT 145

12. Monday, Sept 28, 2015MAT 145

13. Monday, Sept 28, 2015MAT 145 THE CHAIN RULE WORDS BY: JOHN A. CARTER TUNE: "CLEMENTINE" Here's a function in a function And your job here is to find The derivative of the whole thing With respect to x inside. Call the outside f of u And call the inside u of x. Differentiate to find df/du And multiply by du/dx. Use the chain rule. Use the chain rule whene'er you find The derivative of a function compositionally defined.

14. Monday, Sept 28, 2015MAT 145

15. Monday, Sept 28, 2015MAT 145 The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100?

16. Monday, Sept 28, 2015MAT 145 The derivative in action! S(t) represents the distance traveled by some object, where t is in minutes and S is in feet. What is the meaning of S’(12)=100? From the description of the context, the “rate units” are: feet per minute. The value 12 is an input variable, so we are looking at the precise instant that 12 minutes of travel has occurred, since some designated starting time when t = 0. S’ indicates rate of change of S, indicating we have information about how S is changing with respect to t, in feet per minute. The value 100 specifies the rate: 100 feet per minute. Putting it all together: At precisely 12 minutes into the trip, the object’s position is increasing at the rate of 100 feet per minute.

17. Monday, Sept 28, 2015MAT 145 The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5?

18. Monday, Sept 28, 2015MAT 145 The derivative in action! C(p) represents the total daily cost of operating a hospital, where p is the number of patients and C is in thousands of dollars. What is the meaning of C’(90)=4.5? From the description of the context, the “rate units” are: thousands of dollars per patient. The value 90 is an input variable, so we are looking at the precise instant when 90 patients are in the hospital. C’ indicates rate of change of C, indicating we have information about how C is changing with respect to p, in thousands of dollars per patient. The value 4.5 specifies the rate: 4.5 thousand dollars ($4500) per patient. Putting it all together: At precisely the instant that 90 patients are in the hospital, the cost per patient is increasing at the rate of $4500 per patient.

19. Monday, Sept 28, 2015MAT 145 The derivative in action! V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π?

20. Monday, Sept 28, 2015MAT 145 The derivative in action! V(r) represents the volume of a sphere, where r is the radius of the sphere in cm. What is the meaning of V ’(3)=36π? From the description of the context, the “rate units” are: cubic cm of volume per cm of radius. The value 3 is an input variable, so we are looking at the precise instant when the sphere’s radius is 3 cm long. V’ indicates rate of change of V, indicating we have information about how V is changing with respect to r, in cubic cm per cm. The value 36π specifies the rate: 36π cubic cm of volume per 1 cm of radius length. Putting it all together: At precisely the instant that the sphere has a radius length of 3 cm, the sphere’s volume is increasing at the rate of 36π cubic cm per cm of radius length.

21. Monday, Sept 28, 2015MAT 145