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8-145.

The function below represents the velocity in kilometers per hour of the Starship Energize over a ‑second period during which the warp-drive was engaged.

  1. What was the average velocity of the starship during this interval?

    You were given a velocity function. How will you find distance information?

  2. At what time, if any, was the ship’s velocity equal to its average velocity? Does the Mean Value Theorem apply in this case? Explain.

    The MVT states that if a function is both continuous and differentiable on a closed interval, then the average value will equal the actual value somewhere.

    The Mean Value Theorem

    The Mean Value Theorem for Integrals

    If   is continuous on , then there exists at least one point  in  such that                     .

    First quadrant, bell curve labeled, f of x, left end point on the y axis, labeled, a, right end point labeled, b, dashed horizontal segment, about 1 fourth up from x axis to peak, labeled average, & shaded rectangle between, A & b, segment & x axis, & circled points where the curve intersects the dashed segment, corresponding to tick marks on x axis, labeled, c subscript 1 & c subscript 2.

    The Mean Value Theorem for Derivatives

    If  is continuous on  and differentiable on , then there exists at least one point  in  such that .

    Increasing curve labeled, capital F of x, starting at the origin, labeled a on the x axis, changing from concave up to concave down, in center of quadrant, ending at point corresponding to tick mark on x axis labeled, b, with dashed segment labeled, m = A R O C, from origin to end point, & 2 dashed parallel lines, one tangent before point of inflection, corresponding to tick mark on x axis labeled, c subscript 1, & one tangent after point of inflection, corresponding to tick mark on x axis labeled, c subscript 2.

  3. What was the average acceleration of the starship during this interval?

    Compare and contrast parts (a) and (c). Both are looking for averages. But, based on the function you were given, strategies to find averages are different.

  4. At what time, if any, was the ship’s acceleration equal to this average acceleration? Does the Mean Value Theorem apply in this case? Explain.

    MVT also states that if a function is both continuous and differentiable on a closed interval, then its average rate of change (AROC) will equal its actual rate of change (IROC) somewhere.