Question

find the equation of the tangent line to the curve y = x - 2 cos x at the point (0, -2). y =

Explanation:

Step1: Find the derivative of the function

The derivative of $y = x-2\cos x$ is $y'=1 + 2\sin x$ using the sum - rule and the derivatives of $x$ (which is 1) and $\cos x$ (which is $-\sin x$).

Step2: Evaluate the derivative at the given x - value

Substitute $x = 0$ into $y'$. So $y'(0)=1+2\sin(0)=1 + 2\times0=1$. The value of the derivative at $x = 0$ is the slope $m$ of the tangent line.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(0,-2)$ and $m = 1$. Substituting these values gives $y-(-2)=1\times(x - 0)$.

Step4: Simplify the equation

$y + 2=x$, which simplifies to $y=x - 2$.

Answer:

$y=x - 2$