A car accelerates from a stop to 45 m/sec in 4 sec. Explain why the car must have been accelerating at exactly 11.25 m/sec at some moment. 2 Do Now.

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Presentation transcript:

A car accelerates from a stop to 45 m/sec in 4 sec. Explain why the car must have been accelerating at exactly m/sec at some moment. 2 Do Now

Mean Value Theorem, Antiderivatives Section 4.2b

Mean Value Theorem If is continuous at every point of the closed interval and differentiable at every point of its interior, then there is at least one point c in at which There is at least one point where the instantaneous slope is equal to the average slope…

Let me put it another way… The MVT says that somewhere between points A and B on a differentiable curve, there is at least one tangent line parallel to chord AB: 0 A B abc Slope Tangent and chord are parallel

A car accelerates from a stop to 45 m/sec in 4 sec. Explain why the car must have been accelerating at exactly m/sec at some moment. 2 Do Now The average acceleration of the car over the interval is 45/4 = m/sec. By the MVT, the instantaneous acceleration of the car must have been that same value at least once during the interval… 2

More Practice with the MVT For each of the following, (a) show that the given function satisfies the hypotheses of the MVT on the given interval. (b) Find each value of c as a solution of the MVT.  The function is continuous on [0, 2] and differentiable on (0, 2) and differentiable on (0, 2) Can we interpret this result with a graph???

More Practice with the MVT For each of the following, (a) show that the given function satisfies the hypotheses of the MVT on the given interval. (b) Find each value of c as a solution of the MVT.  The function is continuous on [2, 4] and differentiable on (2, 4) and differentiable on (2, 4) The graph???

A corollary of the MVT Functions with are Constant If at each point of an interval I, then there is a constant for which for all in I.

Which leads to another corollary Functions with Same Derivative Differ by a Constant If at each point of an interval I, then there is a constant such that for all in I. Let’s see this one graphically… WWWWhat if the derivative of two functions is 2x???

Applying this new tool Find the function whose derivative is and whose graph passes through the point. Use the given point to identify C: has the same derivative as So must equal (C a constant)

Definition: Antiderivative A function is an antiderivative of a function if for all in the domain of. The process of finding an antiderivative is called antidifferentiation.

More Practice Problems Find all possible functions with the given derivative.

More Practice Problems Find all possible functions with the given derivative.

More Practice Problems Find the function with the given derivative whose graph passes through the point P. All possible functions with this derivative: Solve for C: Final answer:

More Practice Problems Find the function with the given derivative whose graph passes through the point P. Final answer: All possible functions with this derivative: Solve for C: