1. a. f(1) = -1 and f(2) = 2. As the sign changes ⇒ there is a root between x = 1 and x = 2.

b. x = 1.71667 (to 5 decimal places.)

2.a. f(1) is negative and f(2) is positive. As the sign changes ⇒ there is a root between x = 1 and x = 2.

b. x = 1.763223

3. Put f(x) = 68x0.75 − 700 + 2x = 0 so then f '(x) = 51x-0.25 + 2

If x0 = 16 then f(x0) = -124 and f '(x0) = 27.5 so x1 = 16 + 124 /27.5 = 20.51

4. a. Derivative = 2ex − 2

b. x1 = 0.8730

x2 = 0.8579

c. At x0 = 0, the tangent of f is parallel to the x-axis and does not intersect with the x-axis and hence x1 cannot be obtained