Question

Find an equation of the tangent line to the following curve at the given point.

y = e^3x cos pi x, (0,1)

Answer 1

Answer:y = 3x + 1

the equation of the tangent line to the curve y = e^3x cos pi x at the point (0,1) is y = 3x + 1.

Explanation:
we need to find the slope of the tangent line at that point, which is the derivative of the function at that point
y' = (e^3x)(-sin pi x)(pi) + (cos pi x)(3e^3x)
= e^3x (3cos pi x - pi sin pi x)
Now we can evaluate this derivative at x = 0 to find the slope of the tangent line at the point (0,1):

y'(0) = e^0 (3cos 0 - pi sin 0) = 3
y - 1 = 3(x - 0)
y = 3x + 1

Therefore, the equation of the tangent line to the curve y = e^3x cos pi x at the point (0,1) is y = 3x + 1.