IVT
What two conditions must be true to use IVT?
Function must be continuous on the interval [a,b].
What does EVT guarantee?
A continuous function on a closed interval must have an absolute max and min.
True/false: A horizontal tangent must exist between two equal endpoints.
True.
State the MVT conclusion.
There exists a point c where f'(c) = average rate of change.
Linear approximation uses what line?
The tangent Line.
If f(2) = -3 and f(6)= 5, what does IVT guarantee?
There is some value on the interval [-3,5] where f(c) = 0.
Why must the interval be closed?
Open intervals might approach extrema but never reach them.
If all conditions are met, what does Rolle’s guarantee?
There exists a point c where f'(c) = 0.
Give one condition MVT shares with Rolle’s.
Continuous on [a,b] and differentiable on (a,b).
When are linear approximations usually accurate?
When x is close to a.
True or false: IVT can tell you exactly what number c is.
False; it only guarantees its existence.
For f(x) = x³ - 3x on [-2,2], how many guaranteed extrema exist?
Exactly one absolute max and one absolute min.
State the three conditions for Rolle’s Theorem.
Continuous on [a,b], differentiable on (a,b), and f(a) = f(b).
A function is differentiable everywhere. Does that guarantee MVT works on every interval?
No; still must be continuous on a closed interval.
Why can linearization underestimate some functions?
If the function is concave up at a, the tangent line lies below the curve.