Could you show me how to do this?

9514 1404 393

  b ≈ 0.240450937397

Step-by-step explanation:

To find the y-intercept of the tangent line, you need to know ...

  a) the value of y at x=3

  b) the slope of the curve at x=3

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The slope is found by differentiating the function.

  y = ln(x·e^(cos(x)²)) = ln(x) +cos(x)²

Then the derivative is ...

  y' = 1/x +2(-sin(x)·cos(x)) = 1/x -sin(2x)

The y-intercept is ...

  b = y - x·y'

  b = ln(3) +cos(3)² -3(1/3 -sin(6))

The y-intercept is ...

  b = ln(3) +3·sin(6) +cos(3)² -1 ≈ 0.240450937397

example 2 – simplify: step 1: completely factor both the numerators and denominators of all fractions. step 2: change the division sign to a multiplication sign and flip (or reciprocate) the fraction after the division sign; essential you need to multiply by the reciprocal. step 3 : cancel or reduce the fractions.