Problem 11
Question
Find the average rate of change of \(g(t)=\frac{t}{t^{2}+2}+3 t\) over the intervals \([-1,1],[0,2]\), \([1,1+p]\)
Step-by-Step Solution
Verified Answer
The solution will be the average rate of change of \( g(t) \) over the intervals \([-1,1]\), \([0,2]\), \([1,1+p]\) as computed in Step 2.
1Step 1: Find the function values
The first step is to find the function values at the boundaries of each interval. The formula for the function is given by \(g(t)=\frac{t}{t^{2}+2}+3 t \). Compute \( g(-1) \), \( g(1) \), \( g(0) \), \( g(2) \), and \( g(1+p) \) as follows: Substitute the corresponding values into the function formula and simplify.
2Step 2: Find the average rate of change over each given interval
Once the function values at the boundaries have been computed, the next step is to calculate the average rate of change over each of the intervals \([-1,1]\), \([0,2]\) and \([1,1+p]\). Use the formula \((f(b) - f(a))/(b - a)\) for each interval [a, b], where f(a) and f(b) are the function values calculated in Step 1. Simplify the results.
3Step 3: Interpret the results
The average rate of change computed in the previous step represents how fast the function \( g(t) \) is changing on average over each interval.
Key Concepts
Function ValuesIntervalsCalculus ProblemStep by Step Solution
Function Values
When working with a function like \( g(t) = \frac{t}{t^2 + 2} + 3t \), understanding function values is key to solving calculus problems involving average rates of change.
The notion of function values refers to the output of the function for specific input values, which are determined by substituting numbers into the function.
For example:
The notion of function values refers to the output of the function for specific input values, which are determined by substituting numbers into the function.
For example:
- Evaluating at \( t=-1 \), you calculate \( g(-1) \).
- For \( t=1 \), it’s \( g(1) \).
- Similarly, compute \( g(0) \), \( g(2) \), and \( g(1+p) \).
Intervals
An interval defines the range of inputs for which we'll calculate an attribute of the function, like the average rate of change.
Intervals are expressed as pairs of numbers, such as \([-1,1]\), \([0,2]\), or \([1,1+p]\). These numbers indicate the start and end of the interval.
Understanding intervals helps you see over what range you're evaluating changes in the function.
Intervals are expressed as pairs of numbers, such as \([-1,1]\), \([0,2]\), or \([1,1+p]\). These numbers indicate the start and end of the interval.
Understanding intervals helps you see over what range you're evaluating changes in the function.
- For \([-1,1]\), we're looking at how the function behaves and changes from \( t=-1 \) to \( t=1 \).
- The interval \([0,2]\) considers the change from \( t=0 \) through \( t=2 \).
- \([1,1+p]\) includes a variable endpoint, allowing exploration of function behavior when the interval end is more flexible.
Calculus Problem
The problem of finding the average rate of change fits within the domain of calculus.
Calculus often deals with how functions change, and here, we're exploring the average rate which tells us how the function changes, on average, over specified intervals.
The average rate of change equation \[\frac{f(b) - f(a)}{b - a}\]deals with two aspects:
Calculus often deals with how functions change, and here, we're exploring the average rate which tells us how the function changes, on average, over specified intervals.
The average rate of change equation \[\frac{f(b) - f(a)}{b - a}\]deals with two aspects:
- \( f(b) \) and \( f(a) \) are the function values at the endpoints \( b \) and \( a \).
- \( (b - a) \) represents the length of the interval over which these changes are considered.
Step by Step Solution
Breaking down the problem into a step by step solution makes it more manageable and easier to understand.
Here’s the approach you take:- **Step 1**: Start by calculating the specific function values \( g(-1) \), \( g(1) \), \( g(0) \), etc., using substitutions. - **Step 2**: Apply these calculated values to the average rate of change formula for each interval. For instance, to find the change over \([-1, 1]\), compute: \[ \frac{g(1) - g(-1)}{1 - (-1)} \] - **Step 3**: Interpret these results to understand how the function changes over each interval. These easy steps allow you to systematically solve calculus problems involving average rates of change, ensuring a clear path from problem to solution.
Here’s the approach you take:- **Step 1**: Start by calculating the specific function values \( g(-1) \), \( g(1) \), \( g(0) \), etc., using substitutions. - **Step 2**: Apply these calculated values to the average rate of change formula for each interval. For instance, to find the change over \([-1, 1]\), compute: \[ \frac{g(1) - g(-1)}{1 - (-1)} \] - **Step 3**: Interpret these results to understand how the function changes over each interval. These easy steps allow you to systematically solve calculus problems involving average rates of change, ensuring a clear path from problem to solution.
Other exercises in this chapter
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