Final answer:
To find the equation for the line tangent to f when t = 3, we need to find the derivative of f(t) using the product rule and evaluate it at t = 3. Substituting the given values, the equation for the line tangent to f at t = 3 is y = -2t + 18.
Step-by-step explanation:
To find the equation for the line tangent to f when t = 3, we need to determine the derivative of f(t) = tg(t) and evaluate it at t = 3. First, we find the derivative of f(t) using the product rule: the derivative of f(t) is f'(t) = g(t) + tg'(t). Next, we substitute t = 3 into both g(t) and g'(t) to find their values. Given that g(3) = 4 and g'(3) = -2, we can evaluate f'(t) at t = 3 to find that f'(3) = 4 + (3)(-2) = -2. The equation for the line tangent to f at t = 3 is therefore y = f(3) + f'(3)(t-3). Since f(3) = tg(3) = 3(4) = 12, the equation simplifies to y = 12 - 2(t-3) or y = 12 - 2t + 6. This can be further simplified to y = -2t + 18.