Figure 6 shows the geometry behind the derivative formula (sin ) cos Segments BA and BD are parallel to the x and y axes Let sin sin( h) sin Verify the following statements (a) sin BC (b) BDA Hint OA AD (c) BD (cos )AD Now explain the following intuitive argument If h is small, then BC BD and AD h,...

a We note that sin0 h is the ycoordinate of the point C a ...

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

Question:

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