How do you find the tangent line approximation to $f (x) = cos (x)$ $f(x)=cos(x)$ at $x = \frac{π}{4}$ $x=pi/4$ ?

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Answer 1 · 2014-09-24T04:01:46

In this problem we need to follow 5 steps.

1) Find the value of y given the x-value of $\frac{π}{4}$ $pi/4$ . This gives you the point of tangency.

2) Find the derivative of $f (x)$ $f(x)$

3) Substitute in the x-value of $\frac{π}{4}$ $pi/4$ to find the numeric slope.

4) Substitute the slope and the $x$ $x$ and $y$ $y$ values to find the y-intercept.

5) Use the y-intercept and slope to find the equation of the of the tangent line.

$f (x) = cos (x)$ $f(x)=cos(x)$

$f (\frac{π}{4}) = cos (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ $f(pi/4)=cos(pi/4)=sqrt(2)/2$

Point of tangency: $(\frac{π}{4}, \frac{\sqrt{2}}{2})$ $(pi/4,sqrt(2)/2)$

$f'(x)=-sin(x)$

$f'(pi/4)=-sin(pi/4)=-sqrt(2)/2 => slope or m$

$y=mx+b$

$sqrt(2)/2=(-sqrt(2)/2)(pi/4)+b$

$sqrt(2)/2=((-sqrt(2)pi)/8)+b$

$((sqrt(2)pi)/8)+sqrt(2)/2=b$

$1.2625=b$

$y=mx+b$

$y=(-sqrt(2))/2x+1.2625 =>$ Equation of the tangent line

How do you find the tangent line approximation to f(x)=cos(x)f(x)=cos(x)f(x)=cos(x) at x=pi/4x=π4x=pi/4 ?