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5.2…and...5.3. All mixed together…. Rolles Theorem. If f is continuous on [ a,b ] and differentiable on ( a,b ) and if f(a)=f(b ), then f ‘ (c)=0 for at least one number c in (a,b). Mean Value Theorem (MVT).
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5.2…and...5.3 All mixed together….
Rolles Theorem If f is continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b), then f ‘(c)=0 for at least one number c in (a,b).
Mean Value Theorem (MVT) If y = f(x) is continuous on [a,b] and differentiable on (a,b), then there is at least one point c in (a,b) such that
Tangent parallel to chord. Slope of tangent: Slope of chord:
Inflection Point A point where the graph of a function has a tangent line and where the concavity changes is a point of inflection.
is positive is negative is zero is positive is negative is zero First derivative: Curve is rising. Curve is falling. Possible local maximum or minimum. Second derivative: Curve is concave up. Curve is concave down. Possible inflection point (where concavity changes).
Example: Graph
rising, concave down local max falling, inflection point local min rising, concave up Make a summary table: p
Antiderivative A function is an antiderivative of a function if for all x in the domain of f. The process of finding an antiderivative is antidifferentiation. The process of finding the original function from the derivative is so important that it has a name: You will hear much more about antiderivatives in the future. This section is just an introduction.
Functions with the same derivative differ by a constant. These two functions have the same slope at any value of x.
could be or could vary by some constant . Example 6: Find the function whose derivative is and whose graph passes through . so:
Example 6: Find the function whose derivative is and whose graph passes through . so: Notice that we had to have initial values to determine the value of C.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward. (We let down be positive.) Since acceleration is the derivative of velocity, velocity must be the antiderivative of acceleration.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward. The power rule in reverse: Increase the exponent by one and multiply by the reciprocal of the new exponent. Since velocity is the derivative of position, position must be the antiderivative of velocity.
Example 7b: Find the velocity and position equations for a downward acceleration of 9.8 m/sec2 and an initial velocity of 1 m/sec downward. The initial position is zero at time zero. p