Question

Use the definition of the derivative to compute the derivative of the given function. $$f(t)=4-3 t$$

Short answer

The derivative of the function is -3.

Step-by-step solution

Recall the definition of derivative

The derivative of a function at any point is defined as the limit of the difference quotient as the difference in the input approaches zero. Mathematically, for a function \(f(t)\), the derivative \(f'(t)\) is given by: \[f'(t) = \lim_{h \to 0} \frac{f(t + h) - f(t)}{h}\]

Substitute the function into the difference quotient

Substitute \(f(t) = 4 - 3t\) into the difference quotient. We need to find \(f(t + h)\) first:\[f(t + h) = 4 - 3(t + h) = 4 - 3t - 3h\]Now, substitute \(f(t + h)\) and \(f(t)\) into the difference quotient:\[\frac{f(t + h) - f(t)}{h} = \frac{(4 - 3t - 3h) - (4 - 3t)}{h} = \frac{-3h}{h}\]

Simplify the difference quotient

Simplify the expression:\[\frac{-3h}{h} = -3\]

Compute the limit

Take the limit as \(h\) approaches 0:\[f'(t) = \lim_{h \to 0} (-3) = -3\]

Conclusion

The derivative of the function \(f(t) = 4 - 3t\) is \(-3\). This means that the function decreases at a constant rate of 3 for each unit increase in \(t\).

Key concepts

Limit

The limit is a key concept in calculus that describes the value a function approaches as the input gets closer to a particular point. It allows us to understand behavior at points that may not be immediately clear from simple substitution. For derivatives, limits help us explore how a function changes infinitesimally. Understanding the limit involves looking at the difference between two values on a function as they get very close to each other. When we compute derivatives, we look at the limit of the difference quotient as the change in input, typically represented by 'h', approaches zero.This method helps us find the instantaneous rate of change of the function, which isn't possible by just observing the function at individual points. In the exercise, the limit of the difference quotient \[ \lim_{h \to 0} \frac{f(t + h) - f(t)}{h} \] is calculated. It provides the slope of the tangent line to the function at any point, revealing how the function behaves in that vicinity.

Difference Quotient

The difference quotient is a crucial expression in calculus used to find the derivative. It approximates the slope of the secant line between two points on a function. This approximation becomes exact as those points move infinitely close to one another. Here's how it works:

• The difference quotient is given by the formula: \( \frac{f(t + h) - f(t)}{h} \).
• It calculates the average rate of change of the function over the interval \( [t, t + h] \).
• As \( h \) approaches zero, the difference quotient approaches the derivative, which represents the exact slope of the tangent line at the point \( t \).

In our exercise, substituting the function \( f(t) = 4 - 3t \) into the difference quotient allows us to simplify and find \(-3\) as its derivative, which is the consistent rate of change for the entire function.

Function

A function is a foundational concept in mathematics, representing a relationship that uniquely associates members of one set with members of another set. In context, the function \( f(t) = 4 - 3t \) describes a linear relationship where the input \( t \) is transformed to produce an output. Key aspects of functions:

• The input \( t \) and output \( f(t) \) are the critical components of any function.
• Each input corresponds to exactly one output, making the relationship reliable and predictable.
• In calculus, functions are used to model real-world processes and analyze changes.

For this exercise, the given function \( f(t) = 4 - 3t \) is particularly simple, which makes it a linear function. Its graph is a straight line, and its derivative \(-3\) indicates a constant slope, signifying a uniform rate of decrease as \( t \) increases.