# Holiii

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DEFINITION OF THE DERIVATIVE
EXERCISES

Find an equation for the derivative. (See examples 1 and 2)
1. y=2x2+1

2. y=-x2

3. y=-2x2-1.

4. y=x-1.

5. y=2-x+2

6.fx=-x+1, &x<0x, &x≥0

7. fx=-1, & x<-1x, & -1≤x≤11, x>1

8. fx=x+2, &x<0 2, 0≤x≤2x, x>29. fx=-x2, &x<0x, &0≤x2x+2, &x≥1<1

10. fx=-23, &x≤-1x2-1, &-1<x≤1x, &x>1

Find the derivative of the following functions. (See examples 3 and 4)11. fx=2x+1
12. fx=-x+3
13. gx=4x2
14. fx=x2-1
15. fx=3x3
16. fx=x3+1
17. ft=3t
18. gx=1x2
19. hx=x+2
20. fx=1-x

Given the following position functions, findthe equation of the velocity and acceleration functions. (See examples 5 and 6)
21. st=t
22. st=-2t
23. st=-4.9t2
24. st=3t-4.9t2
25. st=t3-1
26. st=t36
27. st=-1t
28.st=1t+1
29. st=3t, 0 ≤ t<30, 3≤t≤5-3t, 5<t≤8
30. st=0, t<0t, 0≤t≤2-16t2,2<t≤4

Use the notations of the derivative. (See example 7)
31. The displacement of a particle on a vibrating string is given by the equation, where s is measured in centimeters and t inseconds: st=10+14sin⁡(10πt). a) Express the instantaneous velocity of the particle using the different notations of the derivative. b) In what units is the velocity of the particle?
32. The cost (indollars) of producing x units of a certain item is: Cx=5000+10x+0.05x2. a) Express the instantaneous rate of change of C with respect to x (also called marginal cost) using the different notations ofthe derivative. b) What are the units of marginal cost?
33. On a college campus of 5000 students, one student returned from vacation with a contagious flu virus. The spread of the virus is given...