Final answer:
To find the equation of the tangent line to the curve y = e^x sin(x) at the point (0,0), we calculate the derivative, evaluate it at x=0 to obtain the slope, and then use the point-slope form to get the equation y = x.
Step-by-step explanation:
To find an equation for the line that is tangent to the curve y = e^x sin(x) at the point (0,0), we first need to calculate the derivative of the function to find the slope of the tangent line at that point. The slope of a curve at a certain point is the same as the slope of the line tangent to the curve at that point.
The derivative of y = e^x sin(x) is obtained using the product rule, as follows:
y' = e^x sin(x) + e^x cos(x).
Now, we evaluate the derivative at x = 0 to find the slope of the tangent line:
y'(0) = e^0 sin(0) + e^0 cos(0) = 0 + 1 = 1.
The slope of the tangent line at x = 0 is thus 1. The tangent line passes through the point (0,0), so the equation of the tangent line is in the form y = mx + b. Since we have a slope (m) of 1 and the point (0,0), the y-intercept (b) is 0.
Therefore, the equation of the tangent line is y = x.