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Grade 12 · Math · Calculus

12th grade Average value and motion with integrals worksheet

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Average value and motion with integrals

Grade 12 Math Worksheet - Challenge Me

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  1. The average value of f(x) = 3x² on 0 ≤ x ≤ b is 16. Ari solves for b > 0. What is b?

  2. The average value of f(x) = 3x² on 0 ≤ x ≤ b is 25. Bo solves for b > 0. What is b?

  3. The average value of f(x) = 3x² on 0 ≤ x ≤ b is 36. Esme solves for b > 0. What is b?

  4. A sled has velocity v(t) = (t − 1)(t − 4) m/s for 0 ≤ t ≤ 5. Kira finds the total distance. What is it, to the nearest hundredth?

  5. A toy cart has velocity v(t) = (t − 0.5)(t − 3) m/s for 0 ≤ t ≤ 4. Milo finds the total distance. What is it, to the nearest hundredth?

  6. A robot has velocity v(t) = (t − 1)(t − 5) m/s for 0 ≤ t ≤ 7. Noah finds the total distance. What is it, to the nearest hundredth?

  7. Pia wants the total distance for v(t) = t − 2 on 0 ≤ t ≤ 5. Which expression gives it?

    • ∫ from 0 to 2 of (t − 2) dt + ∫ from 2 to 5 of (t − 2) dt
    • ∫ from 0 to 2 of (2 − t) dt + ∫ from 2 to 5 of (t − 2) dt
    • |∫ from 0 to 5 of (t − 2) dt|
    • ∫ from 0 to 5 of (t − 2) dt

  8. Ren wants the total distance for v(t) = 3 − t on 1 ≤ t ≤ 6. Which expression gives it?

    • ∫ from 1 to 6 of (3 − t) dt
    • |∫ from 1 to 6 of (3 − t) dt|
    • ∫ from 1 to 3 of (3 − t) dt + ∫ from 3 to 6 of (3 − t) dt
    • ∫ from 1 to 3 of (3 − t) dt + ∫ from 3 to 6 of (t − 3) dt

  9. Tala wants the total distance for v(t) = 2t − 6 on 0 ≤ t ≤ 4. Which expression gives it?

    • ∫ from 0 to 3 of (6 − 2t) dt + ∫ from 3 to 4 of (2t − 6) dt
    • ∫ from 0 to 3 of (2t − 6) dt + ∫ from 3 to 4 of (2t − 6) dt
    • |∫ from 0 to 4 of (2t − 6) dt|
    • ∫ from 0 to 4 of (2t − 6) dt

  10. Zee wants the total distance for v(t) = t − 4 on 1 ≤ t ≤ 7. Which expression gives it?

    • |∫ from 1 to 7 of (t − 4) dt|
    • ∫ from 1 to 4 of (4 − t) dt + ∫ from 4 to 7 of (t − 4) dt
    • ∫ from 1 to 7 of (t − 4) dt
    • ∫ from 1 to 4 of (t − 4) dt + ∫ from 4 to 7 of (t − 4) dt
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12th grade Average value and motion with integrals worksheet – printable
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About this worksheet

Average value and motion with integrals. Every version has 10 questions, and the answer key prints on the second page.

What your child will learn

  • I can find the average value of a function on an interval.
  • I can find displacement and total distance from a velocity.
  • I can work back from an average to the interval, and write the integral for distance.

Standard: none (beyond CCSS)

Words to know

average value: (1/(b − a)) ∫ from a to b of f(x) dx.

displacement: ∫ v dt: the change in position.

Common mistakes

Gave the integral instead of the average. The average value is the integral divided by the interval's width.

Averaged the two end values. The average value uses every value, through the integral.

Gave the final velocity instead of the distance. Distance is the integral of speed over time.

Questions parents ask

What is the most common mistake with average value and motion with integrals?

Gave the integral instead of the average. The average value is the integral divided by the interval's width.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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