Final answer:
To find the equation of the tangent line to the curve y=5-3x^3 at the point (3,-76), we can use the derivative of the function. The equation of the tangent line is y = -81x + 167.
Step-by-step explanation:
To find the equation of the tangent line to the curve y=5-3x^3 at the point (3,-76), we can use the derivative of the function.
The derivative of h(x) = 5 - 3x^3 is h'(x) = -9x^2.
Now, plug in x = 3 into h'(x) to find the slope of the tangent line: h'(3) = -9(3)^2 = -81.
Since the slope of the tangent line is -81, we can use the point-slope form of a linear equation to find the equation of the tangent line: y - y1 = m(x - x1), where (x1, y1) is the point of tangency.
Plugging in the values: y - (-76) = -81(x - 3)
Simplifying the equation: y = -81x + 243 - 76
Therefore, the equation of the tangent line to the curve y=5-3x^3 at the point (3,-76) is y = -81x + 167.