Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![**Estimate \( f(5.07) \) for \( f \) as in the figure.**
The image shows a graph of a function \( y = f(x) \) with a blue curve and a pink tangent line. The curve is shown to pass through the point (4, 2). The tangent line touches the curve at this point and extends through the point (10, 4).
**Detailed Explanation:**
- **Function \( y = f(x) \):** This is represented by the blue curve. It appears to be a concave upward shape, indicating that the function might have a minimum point near (4, 2).
- **Tangent Line:** Illustrated in pink, the tangent line touches the curve at the point (4, 2) and extends towards (10, 4). The slope of this line can be calculated using the two points it passes through, (4, 2) and (10, 4).
- **Calculation of the Slope:**
\[
\text{Slope} = \frac{4 - 2}{10 - 4} = \frac{2}{6} = \frac{1}{3}
\]
- **Equation of the Tangent Line:**
Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \):
\[
y - 2 = \frac{1}{3}(x - 4)
\]
Simplifying, we find:
\[
y = \frac{1}{3}x + \frac{2}{3}
\]
- **Estimate \( f(5.07) \):**
Substitute \( x = 5.07 \) into the equation of the tangent line:
\[
y = \frac{1}{3}(5.07) + \frac{2}{3} = 1.69 + 0.67 = 2.36
\]
Thus, the estimated value of \( f(5.07) \) is approximately 2.360.
*(Give your answer to three decimal places.)*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd493979f-2be9-4ddd-9226-95237bfc98b0%2Fb3407f4b-2505-4af8-96bf-1314c76a7a75%2Fln6nc1j_processed.png&w=3840&q=75)
Transcribed Image Text:**Estimate \( f(5.07) \) for \( f \) as in the figure.**
The image shows a graph of a function \( y = f(x) \) with a blue curve and a pink tangent line. The curve is shown to pass through the point (4, 2). The tangent line touches the curve at this point and extends through the point (10, 4).
**Detailed Explanation:**
- **Function \( y = f(x) \):** This is represented by the blue curve. It appears to be a concave upward shape, indicating that the function might have a minimum point near (4, 2).
- **Tangent Line:** Illustrated in pink, the tangent line touches the curve at the point (4, 2) and extends towards (10, 4). The slope of this line can be calculated using the two points it passes through, (4, 2) and (10, 4).
- **Calculation of the Slope:**
\[
\text{Slope} = \frac{4 - 2}{10 - 4} = \frac{2}{6} = \frac{1}{3}
\]
- **Equation of the Tangent Line:**
Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \):
\[
y - 2 = \frac{1}{3}(x - 4)
\]
Simplifying, we find:
\[
y = \frac{1}{3}x + \frac{2}{3}
\]
- **Estimate \( f(5.07) \):**
Substitute \( x = 5.07 \) into the equation of the tangent line:
\[
y = \frac{1}{3}(5.07) + \frac{2}{3} = 1.69 + 0.67 = 2.36
\]
Thus, the estimated value of \( f(5.07) \) is approximately 2.360.
*(Give your answer to three decimal places.)*
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