3.5 Derivatives of Trig Functions, p. 141

1. AP Calculus AB/BC 3.5 Derivatives of Trig Functions, p. 141

2. slope Consider the function We could make a graph of the slope: Now we connect the dots! The resulting curve is a cosine curve.

3. slope We can do the same thing for The resulting curve is a sine curve that has been reflected about the x-axis.

4. Example 1 A weight hanging from a spring is stretched 9 units beyond its rest position (s = 0) and released at time t = 0 to bob up and down. Its position at any later time t is What are its velocity and acceleration at time t? Velocity = s′ = −9 sin t Acceleration = v' = s'' = −9 cos t

5. Definition: Jerk, page 144 Jerk is the derivative of acceleration. If s(t) is the position function at time t, the body’s jerk at time t is Example 2 (a) Find the jerk caused by the constant acceleration of gravity (g = −32 ft/sec2). j(t) = 0 (b) Find the jerk of the simple harmonic motion of Example 1 s = 9 cos t. j(t) =

6. We can find the derivative of tangent x by using the quotient rule.

7. Derivatives of the remaining trig functions can be determined the same way.

8. Example 3 First, use the quotient rule. u' = −csc2x u = cot x v' = −csc2x v = 1 + cot x

9. Example 4 Show that the graphs of y = sec x and y = cos x have horizontal tangents at x = 0. Remember, the slopes of horizontal tangents is zero. So, solve sec x tan x = 0 When x = 0, tan x = 0 Therefore, sec x tan x = 0 sin x = 0 When x = 0,

10. Example 5, #13 p. 146 s = 2 + 3 sin t is the position function of a body moving in simple harmonic motion (s in meters, t in seconds) (a) Find the body’s velocity, speed, and acceleration at time t. (b) Find the body’s velocity, speed, and acceleration at time t = π/4. │3 cos t│ (a) v(t) = and speed = │s′(t)│ = 3 cos t, s′(t) = (b) v(π/4) = 3 cos π/4 =

11. Example , #23 p. 146 Find the tangent and normal lines to y = x2 sin x at x = 3. Use the product rule: p