Derivatives of trig functions

Answer Key - Grade 12 Math Worksheet - Challenge Me

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

1. 1.

   d(t) = 3 + 2 sin(πt/6), the tide's depth in m, t in hours. Bo watches the tide with the model shown. What is d′(4), in m per hour, to the nearest tenth?

   -0.5
2. 2.

   Esme times a bouncing spring with the model shown. What is y′(0.5), in cm per second, to the nearest tenth?

   4.4
3. 3.

   y(t) = 30 + 5 sin(πt/4), a swing seat's height above a mark in cm, t in seconds. Jun times a swing with the model shown. What is y′(1), in cm per second, to the nearest tenth?

   2.8
4. 4.

   Kira uses the chain rule on h(x) = cos(x²). Which is h′(x)?

   • h′(x) = −sin(x²)
   • h′(x) = 2x cos(x²)
   • h′(x) = 2x sin(x²)
   • h′(x) = −2x sin(x²)
5. 5.

   Luz uses the chain rule on h(x) = tan(5x). Which is h′(x)?

   • h′(x) = sec²(5x)
   • h′(x) = −5 sec²(5x)
   • h′(x) = 5 sec²(5x)
   • h′(x) = 5 tan(5x)
6. 6.

   Milo uses the chain rule on h(x) = sin³ x. Which is h′(x)?

   • h′(x) = 3 sin² x cos x
   • h′(x) = 3 sin² x
   • h′(x) = 3 cos² x
   • h′(x) = −3 sin² x cos x
7. 7.

   Pia looks for horizontal tangents of f(x) = x + 2 cos x for 0 ≤ x < 2π. Where is f′(x) = 0?

   • x = π/6 and x = 5π/6
   • x = π/6
   • x = 7π/6 and x = 11π/6
   • x = π/3 and x = 5π/3
8. 8.

   Ren looks for horizontal tangents of f(x) = x − 2 sin x for 0 ≤ x < 2π. Where is f′(x) = 0?

   • x = 2π/3 and x = 4π/3
   • x = π/6 and x = 5π/6
   • x = π/3 and x = 5π/3
   • x = π/3
9. 9.

   Tala looks for horizontal tangents of f(x) = √3 x + 2 cos x for 0 ≤ x < 2π. Where is f′(x) = 0?

   • x = π/3
   • x = π/3 and x = 2π/3
   • x = 4π/3 and x = 5π/3
   • x = π/6 and x = 11π/6
10. 10.

    Zee looks for horizontal tangents of f(x) = 2 sin x + √2 x for 0 ≤ x < 2π. Where is f′(x) = 0?

    • x = 3π/4 and x = 5π/4
    • x = 5π/4 and x = 7π/4
    • x = π/4 and x = 7π/4
    • x = 3π/4