Question
Estimate the rate of change of y=2x^2+3 at x=0.5 using a graph.
Answers
GPT-4o mini
To estimate the rate of change of the function \( y = 2x^2 + 3 \) at \( x = 0.5 \) using a graph, follow these steps:
1. **Calculate the function value**: First, evaluate the function at \( x = 0.5 \):
\[
y(0.5) = 2(0.5)^2 + 3 = 2(0.25) + 3 = 0.5 + 3 = 3.5
\]
2. **Calculate values around \( x = 0.5 \)**: Pick points around \( x = 0.5 \) to calculate the slope of the secant line. For example, let’s choose \( x = 0.4 \) and \( x = 0.6 \).
- At \( x = 0.4 \):
\[
y(0.4) = 2(0.4)^2 + 3 = 2(0.16) + 3 = 0.32 + 3 = 3.32
\]
- At \( x = 0.6 \):
\[
y(0.6) = 2(0.6)^2 + 3 = 2(0.36) + 3 = 0.72 + 3 = 3.72
\]
3. **Calculate the slopes**: Now calculate the slopes of the secant lines between these points:
- **Slope from \( x = 0.4 \) to \( x = 0.5 \)**:
\[
\text{slope} = \frac{y(0.5) - y(0.4)}{0.5 - 0.4} = \frac{3.5 - 3.32}{0.1} = \frac{0.18}{0.1} = 1.8
\]
- **Slope from \( x = 0.5 \) to \( x = 0.6 \)**:
\[
\text{slope} = \frac{y(0.6) - y(0.5)}{0.6 - 0.5} = \frac{3.72 - 3.5}{0.1} = \frac{0.22}{0.1} = 2.2
\]
4. **Estimate the rate of change**: The derivative at \( x = 0.5 \) can be approximated by finding the average of the two slopes:
\[
\text{Estimated rate of change} \approx \frac{1.8 + 2.2}{2} = \frac{4.0}{2} = 2.0
\]
Thus, the estimated rate of change of \( y = 2x^2 + 3 \) at \( x = 0.5 \) is approximately **2.0**.
To further validate this, you could also find the derivative of the function explicitly:
\[
\frac{dy}{dx} = 4x
\]
Evaluate at \( x = 0.5 \):
\[
\frac{dy}{dx} \Big|_{x=0.5} = 4(0.5) = 2
\]
This confirms our estimate.
1. **Calculate the function value**: First, evaluate the function at \( x = 0.5 \):
\[
y(0.5) = 2(0.5)^2 + 3 = 2(0.25) + 3 = 0.5 + 3 = 3.5
\]
2. **Calculate values around \( x = 0.5 \)**: Pick points around \( x = 0.5 \) to calculate the slope of the secant line. For example, let’s choose \( x = 0.4 \) and \( x = 0.6 \).
- At \( x = 0.4 \):
\[
y(0.4) = 2(0.4)^2 + 3 = 2(0.16) + 3 = 0.32 + 3 = 3.32
\]
- At \( x = 0.6 \):
\[
y(0.6) = 2(0.6)^2 + 3 = 2(0.36) + 3 = 0.72 + 3 = 3.72
\]
3. **Calculate the slopes**: Now calculate the slopes of the secant lines between these points:
- **Slope from \( x = 0.4 \) to \( x = 0.5 \)**:
\[
\text{slope} = \frac{y(0.5) - y(0.4)}{0.5 - 0.4} = \frac{3.5 - 3.32}{0.1} = \frac{0.18}{0.1} = 1.8
\]
- **Slope from \( x = 0.5 \) to \( x = 0.6 \)**:
\[
\text{slope} = \frac{y(0.6) - y(0.5)}{0.6 - 0.5} = \frac{3.72 - 3.5}{0.1} = \frac{0.22}{0.1} = 2.2
\]
4. **Estimate the rate of change**: The derivative at \( x = 0.5 \) can be approximated by finding the average of the two slopes:
\[
\text{Estimated rate of change} \approx \frac{1.8 + 2.2}{2} = \frac{4.0}{2} = 2.0
\]
Thus, the estimated rate of change of \( y = 2x^2 + 3 \) at \( x = 0.5 \) is approximately **2.0**.
To further validate this, you could also find the derivative of the function explicitly:
\[
\frac{dy}{dx} = 4x
\]
Evaluate at \( x = 0.5 \):
\[
\frac{dy}{dx} \Big|_{x=0.5} = 4(0.5) = 2
\]
This confirms our estimate.