Final answer:
To find the equation of the tangent line to the curve y = 4ex cos(x) at the point (0, 4), find the derivative of the function and evaluate it at x = 0. The equation of the tangent line is y = 4x + 4.
Step-by-step explanation:
To find the equation of the tangent line to the curve y = 4ex cos(x) at the point (0, 4), we need to find the derivative of the function and evaluate it at x = 0. The derivative of y = 4ex cos(x) can be found using the product rule and chain rule:
y' = (4ex (-sin(x))) + (4eˣ cos(x))
Simplifying, y' = 4eˣ(cos(x) - sin(x))
Now we can substitute x = 0 into the derivative to find the slope of the tangent line at (0, 4):
y'(0) = 4e⁰(cos(0) - sin(0)) = 4(1)(1 - 0) = 4
So the slope of the tangent line is 4.
Using the point-slope form of a line,
y - y1 = m(x - x1)
where
(x1, y1) = (0, 4)
m = 4
the equation of the tangent line is y - 4 = 4(x - 0),
which simplifies to y = 4x + 4.