Figure 6 shows the geometry behind the derivative formula (sin )= cos . Segments BA and BD

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Figure 6 shows the geometry behind the derivative formula (sin θ)= cos θ. Segments BA and BD are parallel to the x- and y-axes. Let Δ sin θ = sin(θ + h) − sin θ. Verify the following statements:

(a) Δ sin θ = BC
(b) ∠BDA = θ Hint: O̅A̅ ⊥ AD.
(c) BD = (cos θ)AD
Now explain the following intuitive argument: If h is small, then BC ≈ BD and AD ≈ h, so Δ sin θ ≈ (cos θ)h and (sin θ)'= cos θ.

y h D B A FIGURE 6 1 X

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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