Calculus!!!

1. 3.2 Rolle’s Theorem and the Mean Value Theorem Calculus!!! We

2. Rolle’s Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) f’(c) = 0. f’(c) means slope of tangent line = 0. Where are the horiz. tangent lines located? f(a) = f(b) c c a b

3. Ex. Find the two x-intercepts of f(x) = x2 – 3x + 2 and show that f’(x) = 0 at some point between the two intercepts. f(x) = x2 – 3x + 2 0 = (x – 2)(x – 1) x-int. are 1 and 2 f’(x) = 2x - 3 0 = 2x - 3 x = 3/2 Rolles Theorem is satisfied as there is a point at x = 3/2 where f’(x) = 0.

4. Let f(x) = x4 – 2x2 . Find all c in the interval (-2, 2) such that f’(x) = 0. Since f(-2) and f(2) = 8, we can use Rolle’s Theorem. f’(x) = 4x3 – 4x = 0 8 4x(x2 – 1) = 0 x = -1, 0, and 1 Thus, in the interval (-2, 2), the derivative is zero at each of these three x-values.

5. The Mean Value Theorem If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then a number c in (a,b) (b,f(b)) secant line represents slope of the secant line. (a,f(a)) c a b

6. Given f(x) = 5 – 4/x, find all c in the interval (1,4) such that the slope of the secant line = the slope of the tangent line. ? But in the interval of (1,4), only 2 works, so c = 2.