The Fundamental Theorem of Calculus

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   Esme defines g(x) = ∫ from x to 9 of (t² + 1) dt, with x at the bottom. What is g′(3)?

   -10
2. 2.

   Jun defines g(x) = ∫ from x to 9 of (t² + 4) dt, with x at the bottom. What is g′(2)?

   -8
3. 3.

   Kira defines g(x) = ∫ from x to 9 of (t² + 2) dt, with x at the bottom. What is g′(6)?

   -38
4. 4.

   Luz defines g(x) = ∫ from x to 9 of (t² + 6) dt, with x at the bottom. What is g′(1)?

   -7
5. 5.

   Milo's rain barrel holds 20 liters. Water flows in at r(t) = 2t + 3 liters per minute. How much is in it after 4 minutes?

   48
6. 6.

   A pool slide's tank holds 50 gallons, and Noah fills it at r(t) = 4t + 1 gallons per minute. How much does it hold after 3 minutes?

   71
7. 7.

   Zee's sand pile has 100 kg, and sand is added at r(t) = 2t + 5 kg per hour. How much is there after 6 hours?

   166
8. 8.

   If F′ = f, what does ∫ from a to b of f(x) dx equal, for Tala's notes?

   • f(b) − f(a)
   • F(b) − F(a)
   • F(a) − F(b)
9. 9.

   Bo knows F(x) = x⁴/4 has F′(x) = x³. What is ∫ from 1 to 2 of x³ dx?

   • f(2) − f(1) = 7
   • F(2) − F(1) = 15/4
   • F(1) − F(2) = −15/4
10. 10.

    What is the derivative of ∫ from 3 to x of sin t dt, for Ari's notes?

    • sin x
    • sin x − sin 3
    • cos x