a) Find the local extrema

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Presentation transcript:

a) Find the local extrema To help you get started, here is #1 on page192: a) Find the local extrema This point occurs at Notice also that f (x) is an upside down parabola. So this point is a max. 

a) Find the local extrema To help you get started, here is #1 on page192: a) Find the local extrema This point occurs at b) Find the intervals on which the function is increasing and c) Find the intervals on which the function is decreasing If we hadn’t already noticed that it is a parabola, we could have just tested the intervals of the derivative: when So f is increasing when So f is decreasing

#15 on page 192 on the interval [0, 1] Show that the function f satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b] We start by looking at this graph (again a parabola) on the interval [0, 1] It is continuous (no holes or breaks in the graph) and differentiable (no cusps or corner points) on the interval [0, 1] That’s all that it takes to satisfy the hypotheses of the MVT.

#15 on page 192 on the interval [0, 1] Find each value of c that satisfies the Mean Value Theorem on the given interval [a, b] We start by drawing a secant line over the interval [0, 1] This satisfies:  So the slope of the secant line is 3.  Now we have to find a number c between 0 and 1 such that…

#15 on page 192 on the interval [0, 1] Find each value of c that satisfies the Mean Value Theorem on the given interval [a, b] We start by drawing a secant line over the interval [0, 1] Now we have to find a number c between 0 and 1 such that…  In this case, we know that the right side is equal to 3. So… The derivative of f at x = c in this case is...  Solving gives us c =

#15 on page 192 on the interval [0, 1] Find each value of c that satisfies the Mean Value Theorem on the given interval [a, b] We start by drawing a secant line over the interval [0, 1] This means that at x = ½, the tangent line is parallel to the secant line that we’ve already drawn.    Let’s enlarge the graph so we can get a better look at the two parallel lines

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