1 / 11

3.5 Derivatives of Trig Functions, p. 141

AP Calculus AB/BC. 3.5 Derivatives of Trig Functions, p. 141. slope. Consider the function. We could make a graph of the slope:. Now we connect the dots!. The resulting curve is a cosine curve. slope. We can do the same thing for.

akasma
Download Presentation

3.5 Derivatives of Trig Functions, p. 141

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related