2. We have used the definition of the derivative at a point. We can also write the derivative of f(x) as a new function called f'(x). For any point a = a the new function f' gives the slope of the line that is tangent to the original function f at a = a. Think of it this way: f(x) and f'(x) are two different pieces of information that you can glean from looking at the graph of y = f(x): f(a) is the height of the point (a, f(a)) f'(a) is the slope of the line tangent to the curve at the point (a, f(x)) Let's see how that works in practice. (a) The following is the graph of a function y = f(x). At the values r = -4,-3, -2,-1,0, 1, 2, 3 and 4, draw a tangent line to the function and visually estimate the slope of the line. Example: it looks like the tangent line at z = 2 has a slope of about 1.0, so I have filled in that value in the table. Use your estimates to complete this table: -4 -3 -1 0 1 2 3 4 1.0 Slope of tangent || -2 -1 (b) The values in the bottom line of the table are the values of f'(x). For example, the table shows that f'(2) = 1.0. Plot these points in the ry-plane, and sketch a line or curve through. them. This is the graph of y = f'(a). L₁_1Y=f(x) (e) Now, try the same exercise with this function g(x). Make a table of slope values (for the integers from -3 up to 4), plot them in the zy-plane, and sketch a line or curve through them to find the graph of y = g(x). Y₁ = g(x) -b - 3
2. We have used the definition of the derivative at a point. We can also write the derivative of f(x) as a new function called f'(x). For any point a = a the new function f' gives the slope of the line that is tangent to the original function f at a = a. Think of it this way: f(x) and f'(x) are two different pieces of information that you can glean from looking at the graph of y = f(x): f(a) is the height of the point (a, f(a)) f'(a) is the slope of the line tangent to the curve at the point (a, f(x)) Let's see how that works in practice. (a) The following is the graph of a function y = f(x). At the values r = -4,-3, -2,-1,0, 1, 2, 3 and 4, draw a tangent line to the function and visually estimate the slope of the line. Example: it looks like the tangent line at z = 2 has a slope of about 1.0, so I have filled in that value in the table. Use your estimates to complete this table: -4 -3 -1 0 1 2 3 4 1.0 Slope of tangent || -2 -1 (b) The values in the bottom line of the table are the values of f'(x). For example, the table shows that f'(2) = 1.0. Plot these points in the ry-plane, and sketch a line or curve through. them. This is the graph of y = f'(a). L₁_1Y=f(x) (e) Now, try the same exercise with this function g(x). Make a table of slope values (for the integers from -3 up to 4), plot them in the zy-plane, and sketch a line or curve through them to find the graph of y = g(x). Y₁ = g(x) -b - 3
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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This is the complete question 2. Please help me with this.
![### Understanding Derivatives at a Point
In mathematics, especially calculus, the concept of a derivative at a point helps us understand the behavior of functions. This section delves into how derivatives can be used to glean information from function graphs.
#### Key Concepts
- **Derivative as a Function**: The derivative of a function \( f(x) \) at a point can be represented as a new function, \( f'(x) \).
- **Two Pieces of Information**:
- \( f(a) \) denotes the height of the point \((a, f(a))\).
- \( f'(a) \) represents the slope of the tangent line to the curve at the point \((a, f(x))\).
#### Practical Application
(a) **Graph Analysis**
- You are given a graph of \( y = f(x) \).
- At various x-values (\(x = -4, -3, -2, -1, 0, 1, 2, 3, 4\)), draw tangent lines to estimate their slopes.
- For example, at \( x = 2 \), the tangent line’s slope is estimated as \( 1.0 \).
**Table**:
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
\text{Slope of tangent} & & & & & & & 1.0 & & \\
\hline
\end{array}
\]
(b) **Graphing the Derivative**
- The slopes listed in the table are \( f'(x) \) values.
- For instance, \( f'(2) = 1.0 \).
- Plot these values on the xy-plane and sketch a curve that represents \( y = f'(x) \).
(c) **Repeat with Another Function**
- Perform similar exercises with \( g(x) \), by calculating and plotting slopes for integers from \(-3\) to \(4\), and derive the \( y = g'(x) \).
(d) **Derivatives Interpretation**
- Evaluate two graphs:
- Solid line: \( y = f(x) \)
- Dashed line: \( y = g](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca69301f-c6c0-4122-aab4-f4b1ae58e0d6%2F6dd0f27d-45ff-4ce4-8f09-fe8793c2b0bb%2F8zccin_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Derivatives at a Point
In mathematics, especially calculus, the concept of a derivative at a point helps us understand the behavior of functions. This section delves into how derivatives can be used to glean information from function graphs.
#### Key Concepts
- **Derivative as a Function**: The derivative of a function \( f(x) \) at a point can be represented as a new function, \( f'(x) \).
- **Two Pieces of Information**:
- \( f(a) \) denotes the height of the point \((a, f(a))\).
- \( f'(a) \) represents the slope of the tangent line to the curve at the point \((a, f(x))\).
#### Practical Application
(a) **Graph Analysis**
- You are given a graph of \( y = f(x) \).
- At various x-values (\(x = -4, -3, -2, -1, 0, 1, 2, 3, 4\)), draw tangent lines to estimate their slopes.
- For example, at \( x = 2 \), the tangent line’s slope is estimated as \( 1.0 \).
**Table**:
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
x & -4 & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
\text{Slope of tangent} & & & & & & & 1.0 & & \\
\hline
\end{array}
\]
(b) **Graphing the Derivative**
- The slopes listed in the table are \( f'(x) \) values.
- For instance, \( f'(2) = 1.0 \).
- Plot these values on the xy-plane and sketch a curve that represents \( y = f'(x) \).
(c) **Repeat with Another Function**
- Perform similar exercises with \( g(x) \), by calculating and plotting slopes for integers from \(-3\) to \(4\), and derive the \( y = g'(x) \).
(d) **Derivatives Interpretation**
- Evaluate two graphs:
- Solid line: \( y = f(x) \)
- Dashed line: \( y = g
Expert Solution
Step 1: Given
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, to estimate its slope at different points and use the table of slope values to draw the graph of the derivative of the function.
Given is a graph of function, to estimate its slope at different points and use the table of slope values to draw the graph of the derivative of the function.
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