The mass of the part of a metal rod that lies between its left end and a point x meters to the right is 4x2 kg. (See this example.) (a) Find the linear density when x is 4 m. kg/m (b) Find the linear density when x is 6 m. kg/m (c) Find the linear density when x is 9 m. kg/m

A) 32 kg/m

B) 48 kg/m

C) 72 kg/m

Step-by-step explanation:

We are given mass is; m = 4x²

Linear density is the derivative of mass with respect to the position x. So linear density is;

ρ = dm/dx

Differentiating 4x²,we have; ρ = 8x

A)at x = 4; ρ = 8(4) = 32 kg/m

B) at x = 6; ρ = 8(6) = 48 kg/m

C) at x = 9; ρ = 8(9) = 72 kg/m

1. 3.4/4

2. 0.8/7

3. 0.5 x 3

4. 0.9 x 6

5. 10.8

6. 2.35

7. 0.567

8. 9.35

9. 0.02

10. 0.06

11. 0.16

12. 0.42

13. 0.63

14. 5.6

15. 8.1

16. 9.9

17. 0.28

18. 0.48

19. 0.27

20. 0.49

21. 0.45

22. 0.72

23. 0.9

24. 0.88

25. 0.75

26. 0.16

27. 0.328

28. 0.495

29. 0.72

30. 1.068

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.