The problem is given with the function $f(x) = x^3 - 3x$.

1. Find the equation of the tangent line to the curve $y = f(x)$ at the point $(t, f(t))$.

2. Given that the tangent line from (1) passes through the point $(-1, a)$ for some real number $t$, find the range of values for $a$ such that there is exactly one real solution.

Solution Outline:

1. Step 1: Differentiate $f(x)$
   Calculate $f'(x)$ to find the slope of the tangent at any point $x = t$: $f'(x) = 3x^2 - 3$ So, at $x = t$, the slope is $f'(t) = 3t^2 - 3$.

2. Step 2: Equation of the Tangent Line
   Using the point-slope form of a line, the equation of the tangent at $(t, f(t))$ is: $y - f(t) = f'(t)(x - t)$ Substituting $f(t) = t^3 - 3t$ and $f'(t) = 3t^2 - 3$, we get: $y - (t^3 - 3t) = (3t^2 - 3)(x - t)$ Simplifying this equation will provide the tangent line's formula in terms of $t$.

3. Step 3: Condition for the Tangent Passing Through $(-1, a)$
   Substitute $(-1, a)$ into the tangent line equation and solve for $t$ in terms of $a$.

4. Step 4: Solve for $a$'s Range
   Determine the values of $a$ for which there exists exactly one real solution for $t$ by analyzing the resulting equation.

Would you like me to go through each calculation step-by-step or provide further details?

Follow-up Questions

1. What is the importance of finding the derivative in tangent line problems?
2. How do we interpret the condition of having "exactly one solution" in the context of this problem?
3. Why do we use point-slope form to find the equation of a tangent line?
4. What role does the parameter $t$ play in defining the tangent line and the solution range for $a$?
5. How can we determine the existence of multiple solutions in terms of the discriminant of a resulting equation?

Tip

In problems involving tangents, always start by differentiating to obtain the slope function, as it directly relates to the line's angle at any point on the curve.