Worksheet

Average value and motion with integrals

Grade 12 Math Worksheet - Challenge Me

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Circle the best answer, or write your answer on the line.

  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. The average value of f(x) = 3x² on 0 ≤ x ≤ b is 49. Jun solves for b > 0. What is b?

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

  6. 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?

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

    • |∫ from 0 to 5 of (t − 2) dt|
    • ∫ from 0 to 2 of (t − 2) dt + ∫ from 2 to 5 of (t − 2) dt
    • ∫ from 0 to 5 of (t − 2) dt
    • ∫ from 0 to 2 of (2 − t) dt + ∫ from 2 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 3 of (3 − t) dt + ∫ from 3 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 (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 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

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

    • ∫ from 1 to 4 of (t − 4) dt + ∫ from 4 to 7 of (t − 4) dt
    • ∫ from 1 to 7 of (t − 4) dt
    • |∫ from 1 to 7 of (t − 4) dt|
    • ∫ from 1 to 4 of (4 − t) dt + ∫ from 4 to 7 of (t − 4) dt
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Average value and motion with integrals

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

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

    4

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

    5

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

    6

  4. The average value of f(x) = 3x² on 0 ≤ x ≤ b is 49. Jun solves for b > 0. What is b?

    7

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

    15.33

  6. 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?

    4.54

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

    • |∫ from 0 to 5 of (t − 2) dt|
    • ∫ from 0 to 2 of (t − 2) dt + ∫ from 2 to 5 of (t − 2) dt
    • ∫ from 0 to 5 of (t − 2) dt
    • ∫ from 0 to 2 of (2 − t) dt + ∫ from 2 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 3 of (3 − t) dt + ∫ from 3 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 (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 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

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

    • ∫ from 1 to 4 of (t − 4) dt + ∫ from 4 to 7 of (t − 4) dt
    • ∫ from 1 to 7 of (t − 4) dt
    • |∫ from 1 to 7 of (t − 4) dt|
    • ∫ from 1 to 4 of (4 − t) dt + ∫ from 4 to 7 of (t − 4) dt