Final answer:
To find the equation of the tangent line to the curve y=3ˣ at t=25, we need to calculate the derivative of the function y=3ˣ and evaluate it at t=25. The slope of the tangent line can be found by evaluating the derivative at t=25 and the equation of the tangent line can be determined using the point-slope form.
Step-by-step explanation:
To find the tangent line to the curve y=3ˣ at t=25, we need to calculate the derivative of the function y=3ˣ and evaluate it at t=25. The derivative of y=3ˣ is dy/dx = 3ˣ * ln(3). Evaluating this derivative at t=25 gives dy/dx = 3^25 * ln(3). This is the slope of the tangent line.
To find the equation of the tangent line, we use the point-slope form of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope. Since we know the point of tangency is (25, 3^25), we can plug in the values into the equation to get the final result.