Question

find an equation of the tangent line to the graph of the function at the given point,use a graphing utility to graph the function and its tangent line at the point, anduse the tangent feature of a graphing utility to confirm your results.

Answer 1

The equation of the tangent line may be identified using the first derivative of the function which gives us its slope.

`\begin{gathered} y=\cos3x \\ \\ y^(\prime)=-3\sin3x \\ \\ m=-3\sin3x \\ \\ m=-3\sin3((\pi)/(4)) \\ \\ m=-3((√(2))/(2)) \\ \\ m=-(3√(2))/(2) \end{gathered}`

The tangent line passes through (/4, -√2/2) so we can solve for the y-intercept, b.

`\begin{gathered} y=mx+b \\ \\ -(√(2))/(2)=-(3√(2))/(2)((\pi)/(4))+b \\ \\ -(√(2))/(2)=-(3\pi√(2))/(8)+b \\ \\ b=-(√(2))/(2)+(3\pi√(2))/(8) \\ \\ b=(-4√(2))/(8)+(3\pi√(2))/(8) \\ \\ b=((3\pi-4)√(2))/(8) \end{gathered}`

So the equation of the tangent line is:

`y=-(3√(2))/(2)x+((3\pi-4)√(2))/(8)`