3.2 Rolle’s Theorem and the Mean Value Theorem

Objective: Understand and use Rolle’s Theorem and the Mean Value Theorem.

Miss BattagliaAP Calculus AB/BC

Sketch a coordinate plane on a piece of paper. Label the points (1,3) and (5,3). Using a pencil, draw the graph of a differentiable function of f that starts at (1,3) and ends at (5,3). Is there at least one point on the graph for which the derivative is zero? Would it be possible to draw the graph so that there isn’t a point for which the derivative is zero?

Exploration

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 in c in (a,b) such that f’(c)=0.

Thm 3.3 Rolle’s Theorem

Find the two x-intercepts of f(x)=x2-3x+2 and show that f’(x)=0 at some point between the two x-intercepts.

Illustrating Rolle’s Theorem

Let f(x)=x4-2x2. Find all values of c in the interval (-2,2) such that f’(c)=0.

Illustrating Rolle’s Theorem

If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that

Thm 3.4 The Mean Value Theorem (MVT)

Requirements: function must be continuous and differentiable (regular smooth function no gaps or cusps)

The secant line connecting points (a,f(a)) and (b,f(b)) has a slope given by the slope formula

The derivative at a point is the same thing as the slope of the tangent line at that point, so the theorem just says that there must be at least one point between a and b where the slope of the tangent is the same as the slope of the secant line from a to b.

Now for the plain English version…

Given f(x)=5-(4/x), find all values of c in the open interval (1,4) such that

Finding a Tangent Line

Two stationary patrol cars equipped with radar are 5 miles apart on a highway (please draw!). As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit (of 55 mph) at some time during the 4 minutes.

Finding an Instantaneous Rate of Change

Page 171 #65-68 Read 3.2 Page 176 #1-3, 11, 15, 17, 18, 23, 30, 43, 44, 45, 54, 60, 77-80

Classwork/Homework