Introduction to Maximun and Minimun.pptx

Derivatives are a fundamental concept in calculus used to measure how a function changes as its input changes. In simple terms, the derivative represents the rate of change or the slope of a curve at a specific point.

They are widely used in science and engineering to analyze motion, optimize systems, and model real-world situations. Common rules include the power rule, product rule, quotient rule, and chain rule, which simplify the process of differentiation.

Understanding derivatives helps describe how quantities vary over time — for example, how velocity is the derivative of position, and acceleration is the derivative of velocity.

1. Maximums and minimums

2. Steps Derive Equate to “0” and solve Replace in Find out if it is Max or Min .     Differentiating allows us to find the slope

3. Step 1. Derive       Earring:  

4. Step 2. Equate to “0” and solve Earring:     Simple equation        

5. Step 3. Replace in   Replace the X in the function:             Y coordinate is found when X equals 1 Critical point: (1,-3)

6. Step 4. Find out if it is Max or Min . There are 2 methods to find out if it is MAX or MIN     Method 1: m=0 Step 1. If we know that the slope is 0 when X equals 1, we must choose one number to the right and another to the left       Note: When the slope is negative, it is because on the left the function is DECREASING

7. Step 4. Find out if it is Max or Min . Now we choose one on the right     Method 1: m=0       GROWING   DECREASING This means that this point (1,-3) IS A MINIMUM.

8. Step 4. Find out if it is Max or Min . Method 2: m=0   second derivative is applied , which allows us to know if the critical points are MAX or Min.       We replace X   Note: Whenever the second derivative is positive it means there is a MINIMUM ; If it is negative it means the point is a MAXIMUM Minimum

9. EXERCISE 5  

10. Step 1. Derive       Earring:    

11. Step 2. Equate to “0” and solve   )         EITHER        

12. Step 3. Replace in   Replace the X in the function:           First point: (0,1)     Second point: (-2.5)

13. Step 4. Find out if it is Max or Min .               Positive =Minimal     Negative = Maximum

14. EXERCISE  

15. Step 1. Derive     Step 2. Equate to “0” and solve        

16. Step 3. Replace in   Replace the X in the function:           First point: (2,-19)     Second point: (-1.8)

17. Step 4. Find out if it is Max or Min . Positive =Minimal Negative = Maximum     First point: (2,-19) Second point: (-1.8)          

18. Step 1. Derive     Step 2. Equate to “0” and solve                

19. Step 3. Replace in   Replace the X in the function:             First point: (0, )       Second point: (-2,-2)     Third point: (1, )  

20. Step 4. Find out if it is Max or Min . First point: (0, )   Second point: (-2,-2) Third point: (1, )             Negative = Maximum   Positive =Minimal     Positive =Minimal