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- Continuous on the closed interval [a,b]
What are the requirements for the Mean Value Theorem?
What does the Mean Value Theorem guarantee?
That there is a C in the interval (a,b) such that f'(c)= f(b) - f(a)/ b - a
The relative minimums and maximums
What are relative extrema of f(x)?
When do critical values of f(x) occur?
When the derivative of f(x) is zero or undefined
When f(x) goes from increasing to decreasing
With Respect to f(x), when do relative maximums occur?
When f'(x) goes from positive to negative
With respect to the derivative of f(x), when do relative maximums occur?
when f(x) goes from decreasing to increasing
With respect to f(x), when do relative minimums occur?
When f'(x) goes from negative to positive
With respect to the derivative of f(x), when do relative minimums occur?
when f'(x) is positive
With respect to its derivative, when is f(x) increasing?
When f'(x) is negative
With respect to its derivative, when is f(x) decreasing?
f''(x) needs to change signs there
What are the two requirements for a point of inflection on f(x) to occur?
when f'(x) is increasing
In regards to f'(x), when is f(x) concave up?
when f''(x) is positive
In regards to f''(x), when is f(x) concave up?
When f'(x) is decreasing
In regards to f'(x), when is f(x) concave down?
When f"(x) is negative
In regards to f"(x), when is f(x) concave down?
6
Find c that satisfies the MVT: f(x) = 5x^2 - 4x - 10 [4,8]
π/2 and 3π/2
Find c that satisfies the MVT: f(x) = sinx [0,2π]
MVT doesn't hold true
Find c that satisfies the MVT f(x) = 1/(x + 3) [-5, 0]
How do you find absolute extrema?
Find the critical points, make a number line, determine the max or min inside the interval, then consider the endpoints and if they are max or min. Finally, make and x, y chart and compare the y-values
What is an absolute maximum?
The biggest y-value on the given interval
What does the Intermediate Value Theorem guarantee?
That there is a y-value between f(a) and f(b) where the corresponding x-value is between a and b
f(x) must be continuous on [a,b]
What is the restriction of the IVT?
