Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ f(t)=t^{3}-12 t $$

Step-by-Step Solution

Answer

\(f'(t) = 3t^2 - 12\)

Step 1: Apply the Limit Definition

According to the limit definition of a derivative, \(f'(a)=\lim\_{h \to 0} \frac {f(a+h) - f(a)}{h}.\) Let's apply this to our function \(f(t)=t^{3}-12t.\) This would give us: \[f'(t) = \lim\_{h \to 0} \frac {((t+h)^3 - 12(t+h)) - (t^3 -12t)}{h}.\] This simplifies to : \[f'(t) = \lim\_{h \to 0} \frac {(t^3 + 3t^2h + 3th^2 + h^3 -12t -12h) - (t^3 -12t)}{h}.\]

Step 2: Simplify the Terms

Simplify further by cancelling out same terms in the numerator part of the expression to get: \[f'(t) = \lim\_{h \to 0} \frac {3t^2h + 3th^2 + h^3 - 12h}{h}.\] We then factor out an 'h' from the numerator to simplify the expression, this gives us: \[f'(t) = \lim\_{h \to 0} h(3t^2 + 3th + h - 12).\]

Step 3: Evaluate the Limit

Since there's h in both the numerator and denominator, we can cancel out h. Then we can find the limit as h approaches 0, we end up with \[f'(t) = \lim\_{h \to 0} (3t^2 + 3th + h - 12),\] which when h approaches 0, simplifies to: \[f'(t) = 3t^2 - 12.\]