For the function $$ \mathrm{f}(\mathrm{x})=\mathrm{x}^{2} $$, compute the following average rates of change. (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with f . What do you think is happening to the secant lines? (g) To what number are the slopes of the secant lines getting closer? to $$ \mathrm{x}=1.1 $$ is $$

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For the function $$ \mathrm{f}(\mathrm{x})=\mathrm{x}^{2} $$, compute the following average rates of change. (a) From 1 to 2 (b) From 1 to 1.5 (c) From 1 to 1.1 (d) From 1 to 1.01 (e) From 1 to 1.001 (f) Use a graphing utility to graph each of the secant lines along with f . What do you think is happening to the secant lines? (g) To what number are the slopes of the secant lines getting closer? to $$ \mathrm{x}=1.1 $$ is $$ <.1 $$. (Type an integer or a decimal.) (d) The average rate of change of $$ f $$ from $$ x=1 $$ to $$ x=1.01 $$ is 2.01 . (Type an integer or a decimal.) (e) The average rate of change of $$ f $$ from $$ x=1 $$ to $$ x=1.001 $$ is 2.001 (Type an integer or a decimal.) (f) Which statement best describes what is happening to the secant lines? The slopes of the secant lines are getting smaller and smaller and the secant lines are beginning to look more and more like the graph of $$ f $$ at the point where $$ x=1 $$. The slopes of the secant lines are getting smaller and smaller and the secant lines are beginning to look more and more like the tangent line to the graph of $$ f $$ at the point where $$ x=1 $$. (g) The slopes of the secant lines are approaching $$ \square $$ (Type an integer.)

UpStudy AI · Accepted Answer

To solve the problem, we will compute the average rates of change of the function $$ f(x) = x^2 $$ over the specified intervals. The average rate of change of a function $$ f $$ from $$ x = a $$ to $$ x = b $$ is given by the formula:

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

### Step-by-step calculations:

1. **Calculate $$ f(1) $$**:
   $$
   f(1) = 1^2 = 1
   $$

2. **(a) From 1 to 2**:
   $$
   f(2) = 2^2 = 4
   $$
   $$
   \text{Average Rate of Change} = \frac{4 - 1}{2 - 1} = \frac{3}{1} = 3
   $$

3. **(b) From 1 to 1.5**:
   $$
   f(1.5) = (1.5)^2 = 2.25
   $$
   $$
   \text{Average Rate of Change} = \frac{2.25 - 1}{1.5 - 1} = \frac{1.25}{0.5} = 2.5
   $$

4. **(c) From 1 to 1.1**:
   $$
   f(1.1) = (1.1)^2 = 1.21
   $$
   $$
   \text{Average Rate of Change} = \frac{1.21 - 1}{1.1 - 1} = \frac{0.21}{0.1} = 2.1
   $$

5. **(d) From 1 to 1.01**:
   $$
   f(1.01) = (1.01)^2 = 1.0201
   $$
   $$
   \text{Average Rate of Change} = \frac{1.0201 - 1}{1.01 - 1} = \frac{0.0201}{0.01} = 2.01
   $$

6. **(e) From 1 to 1.001**:
   $$
   f(1.001) = (1.001)^2 = 1.002001
   $$
   $$
   \text{Average Rate of Change} = \frac{1.002001 - 1}{1.001 - 1} = \frac{0.002001}{0.001} = 2.001
   $$

### Summary of Results:
- (a) Average rate of change from 1 to 2: **3**
- (b) Average rate of change from 1 to 1.5: **2.5**
- (c) Average rate of change from 1 to 1.1: **2.1**
- (d) Average rate of change from 1 to 1.01: **2.01**
- (e) Average rate of change from 1 to 1.001: **2.001**

### (f) Graphing Utility:
Using a graphing utility, we can graph the function $$ f(x) = x^2 $$ and the secant lines for the intervals calculated. The slopes of the secant lines will appear to approach the slope of the tangent line at $$ x = 1 $$.

### (g) Approaching Number:
The slope of the tangent line at $$ x = 1 $$ can be found by taking the derivative of $$ f(x) $$:
$$
f'(x) = 2x
$$
At $$ x = 1 $$:
$$
f'(1) = 2 \cdot 1 = 2
$$

Thus, the slopes of the secant lines are approaching **2**.

### Final Answers:
- (a) 3
- (b) 2.5
- (c) 2.1
- (d) 2.01
- (e) 2.001
- (f) The slopes of the secant lines are getting smaller and smaller and the secant lines are beginning to look more and more like the tangent line to the graph of $$ f $$ at the point where $$ x=1 $$.
- (g) The slopes of the secant lines are approaching **2**.
- (a) 3 - (b) 2.5 - (c) 2.1 - (d) 2.01 - (e) 2.001 - (f) The slopes of the secant lines are getting smaller and smaller and the secant lines are beginning to look more and more like the tangent line to the graph of $$ f $$ at the point where $$ x=1 $$. - (g) The slopes of the secant lines are approaching **2**.

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