`y = cos(3x), ((pi/4),-sqrt(2)/2)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point,...

`y = cos(3x), ((pi/4),-sqrt(2)/2)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of the graphing utility to confirm your results.

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1 Answer

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function y = cos (3x). The slope of the tangent to the graph of this function at `(pi/4, -sqrt2/2)` is `(y + sqrt 2/2)/(x - pi/4) = y'(pi/4)`

`y' = -3*sin(3*x)`

`y'(pi/4) = -3/sqrt 2`

The equation of the tangent is:

`(y + sqrt 2/2)/(x - pi/4) = -3/sqrt 2`

`y = (-3/sqrt 2)*(x - pi/4) + 1/sqrt 2`

The required equation of the tangent to the curve y = cos 3x at x = `pi/4` is `y = (-3/sqrt (2))*(x - pi/4) - 1/sqrt (2)`

The graph of the curve and the tangent is: