Let f (x) = sin (5x). Use a tangent line approximation to estimate f (3).
3 is close to π
Following the hint, let's draw the tangent line at x = π, which is close to 3. Then
Δ x = 3 - π.
If we're drawing a tangent line at π to approximate f (3), then yes, Δ x will be negative.
We'll wait to round until the very end of the problem. Now let's figure out the other values we need.
yold = f (π) = sin(5π) = 0.
f '(x) = 5cos (5x)
the slope of the tangent line is
f '(π) = 5cos (5π) = -5.
Now we can do the estimation, leaving all rounding until the end: