Problem 30 - Calculus an Applied Approach

Question

Use the limit definition to find the derivative of the function. $$ h(t)=6-\frac{1}{2} t $$

Step-by-Step Solution

Answer

The derivative of the function \(h(t) = 6 - \frac{1}{2} t\) is \(h'(t) = -\frac{1}{2}\).

Step 1: Write out the function and the definition of its derivative

The given function is \(h(t) = 6-\frac{1}{2} t\). According to the definition of the derivative, \[h'(t) = \lim_{{h \rightarrow 0}} \frac{{h(t + h) - h(t)}}{h}\]

Step 2: Substitute in the function

Replace \(h(t + h)\) and \(h(t)\) with their respective values from the function. Therefore, the above equation becomes \[h'(t) = \lim_{{h \rightarrow 0}} \frac{{(6 - \frac{1}{2}(t + h)) - (6 - \frac{1}{2}t)}}{h}\]

Step 3: Simplify the limit

Simplify the expression inside the limit to get \[h'(t) = \lim_{{h \rightarrow 0}} -\frac{1}{2}\]. Since there is no variable h on which the limit depends on, we can say this limit is just -1/2.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that examines how a function changes. This involves calculating the derivative, which gives us the slope of the tangent line at any point on the function's graph. By knowing the rate at which the function changes, we can understand the behavior of the function more deeply.

For linear functions, such as the one given in the exercise, the derivative is a constant. This is because linear functions have a constant rate of change. Differentiating involves applying rules or definitions, like the limit definition of the derivative, to find this rate.

• Derivative: Measures how a function's output value changes as its input changes.
• For linear functions, this rate of change is constant across the entire graph.

Overall, differentiation helps us examine and predict how different mathematical models behave under various conditions.

Limit Process

The limit process is a critical mathematical tool in calculus used to define derivatives rigorously. In the context of differentiation, it describes how we find the derivative of a function using the limit definition.

To locate the derivative using the limit process, consider the formula:\[ h'(t) = \lim_{{h \rightarrow 0}} \frac{{h(t + h) - h(t)}}{h} \]This formula outlines a process:

• Compute \(h(t + h)\) and \(h(t)\) and subtract them. This gives the markup of the y-values over an interval \(h\).
• Divide this by \(h\) to find the average rate of change over that interval.
• Take the limit as \(h\) approaches zero to find the instantaneous rate of change, which is the derivative.

This method works because it carefully analyzes the behavior of the function as the change in input shrinks, giving precise information about the function's slope at any point.

Linear Function Derivative

A linear function, like the one presented in the exercise, is a simple but important type of function. Linear functions are expressed in the form \(y = mx + c\), where \(m\) represents the slope and \(c\) the y-intercept.

The derivative of a linear function directly corresponds to its slope, which is constant. In the exercise, the function is given as \(h(t) = 6 - \frac{1}{2} t\). Here, after applying the limit definition of the derivative, we find:

• The slope \(m = -\frac{1}{2}\).
• This value remains the same everywhere on the graph, indicating a uniform rate of change.

Thus, finding the derivative of a linear function using the limit process emphasizes the concept that linear functions have a consistent rate of change, a fundamental attribute that differentiates them from non-linear functions.