Final answer:
To get the equation of the tangent line to the curve y = e^(-x^2) at the point (2, 1/e^4), calculate the derivative to find the slope, substitute the x-value of the point into the derivative to get the slope at that point, then use the point-slope form to write the equation.
Step-by-step explanation:
To find an equation of the tangent line to the graph of y = e^(-x^2) at the point (2, 1/e^4), you need to follow these steps:
- Calculate the derivative of the function to get the slope of the tangent line. The derivative of y = e^(-x^2) with respect to x is found using the chain rule and is y' = -2x · e^(-x^2).
- Calculate the slope at the given point by substituting x = 2 into the derivative. The slope at x = 2 is -2 · 2 · e^(-4), which simplifies to -4/e^4.
- Use the point-slope form of a line to express the equation of the tangent line. The equation is y - (1/e^4) = (-4/e^4)(x - 2).