Problem 30 - College Algebra and 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 down the function and limit definition of derivative

We first write down the given function \(h(t)=6-\frac{1}{2}t\). The limit definition of a derivative is \(h'(t) = \lim_{h \rightarrow 0} \frac{f(t + h) - f(t)}{h}\). We identify the function \(f(t) = 6 - \frac{1}{2}t\) in the definition.

Step 2: Substitute the function into the definition

Using \(f(t) = 6 - \frac{1}{2}t\), we substitute into the definition to get \(h'(t) = \lim_{h \rightarrow 0} \frac{(6 - \frac{1}{2}(t+h)) - (6 - \frac{1}{2}t)}{h}\).

Step 3: Simplify the expression

Simplify the above expression to get \(h'(t) = \lim_{h \rightarrow 0} -\frac{1}{2}h/h\). This further simplifies to \(h'(t) = \lim_{h \rightarrow 0} -\frac{1}{2}\).

Step 4: Apply the Limit

As constant limits exist and are equal to the constant itself, the derivative evaluates to \(h'(t) = -\frac{1}{2}\).