Problem 28 - Algebra and Trigonometry

Question

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(t)=\sqrt{t} ; \quad t=a, t=a+h $$

Step-by-Step Solution

Answer

The average rate of change is \( \frac{\sqrt{a + h} - \sqrt{a}}{h} \).

Step 1: Understand the function and variables

We are given the function \( f(t) = \sqrt{t} \), where \( t \) is a variable. We need to find the average rate of change of this function between \( t = a \) and \( t = a + h \). The average rate of change is essentially the change in the function's value divided by the change in \( t \).

Step 2: Formula for average rate of change

The formula for the average rate of change of a function \( f(t) \) between \( t = a \) and \( t = a + h \) is given by:\[\frac{f(a + h) - f(a)}{a + h - a} = \frac{f(a + h) - f(a)}{h}\]

Step 3: Evaluate the function at \( t = a \)

The function \( f(t) = \sqrt{t} \) evaluated at \( t = a \) is \( f(a) = \sqrt{a} \).

Step 4: Evaluate the function at \( t = a + h \)

The function \( f(t) = \sqrt{t} \) evaluated at \( t = a + h \) is \( f(a + h) = \sqrt{a + h} \).

Step 5: Substitute into the average rate of change formula

Using the previously evaluated expressions, substitute into the average rate of change formula:\[\frac{\sqrt{a + h} - \sqrt{a}}{h}\]

Step 6: Simplified expression for the average rate of change

The average rate of change of the function \( f(t) = \sqrt{t} \) from \( t = a \) to \( t = a + h \) is:\[\frac{\sqrt{a + h} - \sqrt{a}}{h}\] This is the required expression for the average rate of change.

Key Concepts

Functions

Functions are fundamental building blocks in mathematics, used to describe relationships between variables. In simple terms, a function takes an input and produces an output. Functions are usually defined using a specific rule or formula. For instance, a function can be shown as \( f(t) \) where \( f \) is the function and \( t \) is the input. By applying the rule of the function to the input, you get the output.

• Functions can model real-world situations, like calculating the area of a circle or predicting stock prices.
• The variable \( t \) is often referred to as the independent variable. It's what you plug into the function to get something out.

One of the key properties of functions is that they give exactly one output for each input value. This consistency makes them very useful in both math and everyday situations.
In the exercise, the function \( f(t) = \sqrt{t} \) is in focus. Here, for every input \( t \), the function calculates the square root of \( t \). By understanding the basics of functions, you're off to a great start in math!

Square Root Function

The square root function is a special type of function that assigns to each number its square root. Formally, for the function \( f(t) = \sqrt{t} \), the output is the number that, when squared, equals \( t \). This function only works for non-negative numbers, because the square root of a negative number is not a real number.

• It's important to note that each input \( t \) must be greater than or equal to zero.
• When \( t = 0 \), the square root is 0.

The graph of a square root function is a curve that starts at the origin \( (0, 0) \) and rises slowly to the right.
In our exercise, the square root function \( f(t) = \sqrt{t} \) is used to find how values change on average between two points. By understanding how square root functions behave, we can better analyze changes over intervals, which is key for understanding the average rate of change.

Difference Quotient

The difference quotient is a crucial concept for determining the average rate of change of a function between two points. In simple terms, it measures how much the function's output changes, relative to how much the input changes. It's given by the formula:\[\frac{f(a + h) - f(a)}{h}\]where \( a \) and \( a + h \) are points in the domain of the function.

• \( f(a) \) and \( f(a+h) \) are the function's outputs at points \( a \) and \( a + h \).
• \( h \) is the distance or step between the two inputs \( a \) and \( a + h \).

This quotient allows us to approximate how quickly or slowly the function is changing. It's especially useful for understanding more complex functions, like the square root function.
In our exercise, this concept is applied to the square root function \( f(t) = \sqrt{t} \), and helps us find the average rate of change from \( t = a \) to \( t = a + h \). The expression \( \frac{\sqrt{a + h} - \sqrt{a}}{h} \) represents how much the square root function changes over the interval, divided by the length of the interval. Understanding the difference quotient is foundational to calculus and can give you great insights into function behavior.