Final answer:
To find the derivative of the function f(t) = 7t - 2t², we use the definition of derivative and apply the limit process to get the derivative, which is f'(t) = 7 - 4t.
Step-by-step explanation:
Finding the Derivative Using the Definition
To find the derivative of the function f(t) = 7t - 2t² using the definition of the derivative, we need to apply the limit process. The definition of the derivative is:
f'(t) = lim_(h→0) [(f(t+h) - f(t))/h]
Let's apply this to our function:
1. First, calculate f(t+h):
f(t+h) = 7(t+h) - 2(t+h)²
= 7t + 7h - 2(t² + 2th + h²)
= 7t + 7h - 2t² - 4th - 2h²
2. Next, form the difference quotient:
[f(t+h) - f(t))] / h =
[(7t + 7h - 2t² - 4th - 2h²) - (7t - 2t²)] / h
= (7h - 4th - 2h²) / h
= 7 - 4t - 2h
3. Finally, take the limit as h approaches 0:
f'(t) = lim_(h→0) [7 - 4t - 2h]
= 7 - 4t
The derivative f'(t), which represents the rate of change of the function, is 7 - 4t.