Question

f(b)-f(a)/b-a
Rate of change using the interval (-2,2)
Given f(x)=2^x^+1

Answer 1

The +1 is up with the x

Answer 2

To calculate the average rate of change of the function \( f(x) = 2^{x} + 1 \) over the interval \((-2, 2)\), we use the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Where:
- \( a = -2 \)
- \( b = 2 \)

**Step 1**: Calculate \( f(a) = f(-2) \)

\[
f(-2) = 2^{-2} + 1 = \frac{1}{4} + 1 = \frac{1}{4} + \frac{4}{4} = \frac{5}{4}
\]

**Step 2**: Calculate \( f(b) = f(2) \)

\[
f(2) = 2^{2} + 1 = 4 + 1 = 5
\]

**Step 3**: Substitute \( f(-2) \) and \( f(2) \) into the formula

\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{5 - \frac{5}{4}}{2 + 2}
\]

**Step 4**: Simplify the numerator

\[
5 - \frac{5}{4} = \frac{20}{4} - \frac{5}{4} = \frac{15}{4}
\]

**Step 5**: Evaluate the denominator

\[
2 + 2 = 4
\]

**Step 6**: Substitute into the average rate of change formula

\[
\text{Average Rate of Change} = \frac{\frac{15}{4}}{4} = \frac{15}{4} \times \frac{1}{4} = \frac{15}{16}
\]

Thus, the average rate of change of the function \( f(x) = 2^{x} + 1 \) over the interval \((-2, 2)\) is

\[
\boxed{\frac{15}{16}}.
\]

Answer 3

Thank you for the clarification. So the function you're referring to is \( f(x) = 2^{x + 1} \).

Now, let’s calculate the average rate of change of \( f(x) = 2^{x + 1} \) over the interval \((-2, 2)\).

### Step 1: Identify \(a\) and \(b\)
- \( a = -2 \)
- \( b = 2 \)

### Step 2: Calculate \( f(a) \) and \( f(b) \)

**Calculate \( f(-2) \):**
\[
f(-2) = 2^{-2 + 1} = 2^{-1} = \frac{1}{2}
\]

**Calculate \( f(2) \):**
\[
f(2) = 2^{2 + 1} = 2^{3} = 8
\]

### Step 3: Apply the average rate of change formula
The average rate of change is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
Substituting values into the formula, we get:
\[
\text{Average Rate of Change} = \frac{f(2) - f(-2)}{2 - (-2)} = \frac{8 - \frac{1}{2}}{2 + 2}
\]

### Step 4: Simplify the numerator
First, convert \(8\) to a fraction:
\[
8 = \frac{16}{2}
\]
Thus,
\[
f(2) - f(-2) = \frac{16}{2} - \frac{1}{2} = \frac{16 - 1}{2} = \frac{15}{2}
\]

### Step 5: Evaluate the denominator
\[
b - a = 2 - (-2) = 2 + 2 = 4
\]

### Step 6: Substitute into the average rate of change formula
\[
\text{Average Rate of Change} = \frac{\frac{15}{2}}{4} = \frac{15}{2} \times \frac{1}{4} = \frac{15}{8}
\]

Thus, the average rate of change of the function \( f(x) = 2^{x + 1} \) over the interval \((-2, 2)\) is

\[
\boxed{\frac{15}{8}}.
\]