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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Concept explainers
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
![**Instruction: Calculating the Derivative Using First Principles**
To understand and master the concept of derivatives, we will go through the process of deriving \( f(x) = \sqrt{3x + 1} \) using its definition, also known as the first principles approach. Importantly, we will not make use of any pre-established calculus formulas.
**Instruction:**
- Use the definition to compute the derivative of \( f(x) = \sqrt{3x + 1} \).
- You may not use any calculus formulas.
**Detailed Explanation:**
The derivative of a function \( f(x) \) using the definition is given by the limit:
\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]
For this problem, we will proceed with the following steps:
1. **Substitute \( f(x) \):**
- Original function: \( f(x) = \sqrt{3x + 1} \)
2. **Determine \( f(x + h) \):**
\( f(x + h) = \sqrt{3(x + h) + 1} = \sqrt{3x + 3h + 1} \)
3. **Apply the definition of the derivative:**
\[ f'(x) = \lim_{{h \to 0}} \frac{{\sqrt{3x + 3h + 1} - \sqrt{3x + 1}}}{h} \]
By calculating this limit, students will gain a deeper understanding of how the derivative measures the rate of change of the function at a given point.
**Note:** To provide a complete solution, students will need to rationalize the numerator and then simplify the resulting limit. This process will include multiplying by the conjugate and potentially using algebraic manipulation to solve the limit.
The step-by-step procedure demonstrates the power of first principles in deriving fundamental results in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb1728920-6e41-44bf-9744-ac0a64f4d19a%2F20bed953-6d4e-4a6c-98e3-8f2dfc543ca9%2Ffbxu7ob_processed.png&w=3840&q=75)
Transcribed Image Text:**Instruction: Calculating the Derivative Using First Principles**
To understand and master the concept of derivatives, we will go through the process of deriving \( f(x) = \sqrt{3x + 1} \) using its definition, also known as the first principles approach. Importantly, we will not make use of any pre-established calculus formulas.
**Instruction:**
- Use the definition to compute the derivative of \( f(x) = \sqrt{3x + 1} \).
- You may not use any calculus formulas.
**Detailed Explanation:**
The derivative of a function \( f(x) \) using the definition is given by the limit:
\[ f'(x) = \lim_{{h \to 0}} \frac{{f(x+h) - f(x)}}{h} \]
For this problem, we will proceed with the following steps:
1. **Substitute \( f(x) \):**
- Original function: \( f(x) = \sqrt{3x + 1} \)
2. **Determine \( f(x + h) \):**
\( f(x + h) = \sqrt{3(x + h) + 1} = \sqrt{3x + 3h + 1} \)
3. **Apply the definition of the derivative:**
\[ f'(x) = \lim_{{h \to 0}} \frac{{\sqrt{3x + 3h + 1} - \sqrt{3x + 1}}}{h} \]
By calculating this limit, students will gain a deeper understanding of how the derivative measures the rate of change of the function at a given point.
**Note:** To provide a complete solution, students will need to rationalize the numerator and then simplify the resulting limit. This process will include multiplying by the conjugate and potentially using algebraic manipulation to solve the limit.
The step-by-step procedure demonstrates the power of first principles in deriving fundamental results in calculus.
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