Determine whether rolles theorem can be applied to f on the

Question

1. In each case, find the linear approximation of f (x) at the indicated value of x₀.

(a) f (x) = √(x+1); x₀ = 0

(b) f (x) = (x+2)/(ln(x)+1); x₀ = 1

2. For each function, find the differential dy.

(a) y = xsin(6x) (b) y = tan^-1(x+2)

3. Use differentials to approximate ?y when x changes as indicated.

(a) y = 3x² -5x + 4; from x = 2 to x = 2.05

(b) y = 4e^-x2; from x = 1 to x = 1.01

4. Use differentials to estimate the change in the volume of a cube is the side of the cube is changed from 7 inches to 6.9 inches.

5. Determine whether Rolle's Theorem can be applied to f on the given interval. If Rolle's theorem can be applied, find all the values guaranteed by the theorem.

(a) f (x) = x^4/3 - 1, [-1, 1]

(b) f (x) = tan(x), [0, π]

6. Determine whether the Mean Value Theorem can be applied to f on the given interval. If the Mean Value Theorem can be applied, find all the values guaranteed by the theorem.

(a) f (x) = 1/x, [-2, 2]

(b) f (x) = 1- e^-x, [0, ln(3)].