2.3 Q4) Hey, I need help with the following calc problem, states to use the definition below. Thank you! ### Understanding the Derivative of a FunctionIn calculus, the derivative of a function at a point provides information about the rate at which the function's value changes at that point. The formal definition of the derivative is given by the following limit:\[ f'(a) = \lim_{{x \to a}} \frac{f(x) - f(a)}{x - a} \]#### Explanation:- **\(f'(a)\)**: Represents the derivative of the function \(f(x)\) at the point \(x = a\).- **\(\lim_{{x \to a}}\)**: Indicates that we are taking the limit as \(x\) approaches \(a\).- **\(\frac{f(x) - f(a)}{x - a}\)**: - The numerator \( f(x) - f(a) \) represents the change in the function's value as \( x \) changes from \( a \) to \( x \). - The denominator \( x - a \) represents the change in the input value from \( a \) to \( x \). - Together, the fraction \( \frac{f(x) - f(a)}{x - a} \) represents the average rate of change of the function over the interval from \( a \) to \( x \). As \( x \) gets arbitrarily close to \( a \), this average rate of change approaches the instantaneous rate of change at the point \( a \), which is the value of the derivative at that point.This formula is fundamental in calculus and is used to find the slopes of tangent lines to the graph of a function, velocities in physics, and rates of change in various scientific disciplines. ### Calculating Derivatives Using the DefinitionIn this exercise, we will use the definition of the derivative to compute the derivative of the given function at a specific value.**Function and Value:**\[ f(x) = 6x^2 - x \]\[ x = 2 \]**Derivative:**\[ f'(2) = \]**Steps to Compute:**1. **Definition of the Derivative:** The derivative of a function \( f(x) \) at a point \( x = a \) is given by: \[ f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h} \]2. **Substitute Given Values:** Substitute \( a = 2 \) into the definition of the derivative: \[ f'(2) = \lim_{{h \to 0}} \frac{f(2 + h) - f(2)}{h} \]3. **Function Evaluation:** - \( f(2 + h) \) can be found by substituting \( 2 + h \) into the function \( f(x) \): \[ f(2 + h) = 6(2 + h)^2 - (2 + h) \] - \( f(2) \) can be computed directly from the function: \[ f(2) = 6(2)^2 - 2 \]4. **Simplify Expression:** Simplify the expression obtained and apply the limit as \( h \) approaches 0.Place your solution in the blank provided:\[ f'(2) = \boxed{} \] Be sure to simplify your calculations and provide the final numerical value.

Answered: Image /qna-images/answer/e44afc70-d295-4c28-8a71-50c0b50aceb8.jpg

Answered: Use this definition to compute the…

Math

Calculus

Use this definition to compute the derivative of the function at the given value. f(x) = 6x² − x, x = 2 f'(2) =

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...

See similar textbooks

Related questions

Q: 3 4) 5.4 8 4 - 1 2

Q: The schematic map below shows a portion of a city area, including tunnels and bridges, Is it…

Q: (Type an integer or decimal rounded to four decimal places as needed.) (e) What is the probability…

A: It is given asμ = -0.05σ = 5.76n = 140

Q: What about part IX), is A similar to B, I think it is not similar, if I am correct or not please…

Q: If a coin is tossed 1000 times, we would expect approximately 500 of the outcomes to be heads. It…

A: Solution: From the given information, a coin is tossed 1000 times and the expected number of heads…

Q: 14 people try out for a basketball team, how many different ways are there to select 5 players if…

A: We hvae to find number of ways to arrange..

Q: 2. A C 2) 46.1 16.4 21 ស្ត Find x round to the nearest tenth. B D 26.9 4.9

Q: a) Is 0/5 defined or not? Write a word problem for 0/5, and use your problem to discuss whether or…

A: In this question ask that is 0/5 is defined or not. And second is is 5/0 is defined or not.

Q: A parent downloads an app that allows him to monitor the amount of screen time his child is using on…

A: Introduction :- Given :- A parent downloads an app that allows him to monitor the amount of screen…

Q: 1) If A = 3 5 (³2) and B=(4) -4 d) B-1

A: Given,

Q: Assume that we are distributing 25 indistinguishable ping-pong balls into 13 boxes. How many ways…

Q: c) 4(x+3)² + 16(y – 5)² = - -5 150 -5 5

A: To graph the equation 4(x+3)2+16(y−5)2=16, we can rewrite it in the standard form of an ellipse…

Q: 2. If 24x = 16, then x = (A) I 2 (B) (C) 4 (D) 8 (E) 12

Q: Evaluate the following. 10) 12! 100! 11) 94,109,400 96! 12) 12P5 13) 6C3 14) 10P8 95040 **BONUS**…

A: The formula for nPr is given by... nPr = n!/(n-r)!

Q: 5. Determine whether the given statement is true or false. (3} E{{3}, (6}} Choose the correct answer…

Q: Fill in the missing information in the table below. Sx Sy 4 9 14 9.6 0.6 14 12 X a) 30 b) 200 c) d)…

A: To find- Fill the given table: Concept Used- The linear regression equation is given by: y^ =…

Q: Calculate 3 2/5 + 1 2/3 in two different ways. a) In each case, express your answer as a mixed…

A: Do the addition as given in the figure. One way is by doing addition in fraction form and the other…

Question

100%

2.3 Q4) Hey, I need help with the following calc problem, states to use the definition below. Thank you!

### Understanding the Derivative of a Function

In calculus, the derivative of a function at a point provides information about the rate at which the function's value changes at that point. The formal definition of the derivative is given by the following limit:

\[ f'(a) = \lim_{{x \to a}} \frac{f(x) - f(a)}{x - a} \]

#### Explanation:

- **\(f'(a)\)**: Represents the derivative of the function \(f(x)\) at the point \(x = a\).
- **\(\lim_{{x \to a}}\)**: Indicates that we are taking the limit as \(x\) approaches \(a\).

- **\(\frac{f(x) - f(a)}{x - a}\)**:
- The numerator \( f(x) - f(a) \) represents the change in the function's value as \( x \) changes from \( a \) to \( x \).
- The denominator \( x - a \) represents the change in the input value from \( a \) to \( x \).
- Together, the fraction \( \frac{f(x) - f(a)}{x - a} \) represents the average rate of change of the function over the interval from \( a \) to \( x \).

As \( x \) gets arbitrarily close to \( a \), this average rate of change approaches the instantaneous rate of change at the point \( a \), which is the value of the derivative at that point.

This formula is fundamental in calculus and is used to find the slopes of tangent lines to the graph of a function, velocities in physics, and rates of change in various scientific disciplines.

### Calculating Derivatives Using the Definition

In this exercise, we will use the definition of the derivative to compute the derivative of the given function at a specific value.

**Function and Value:**
\[ f(x) = 6x^2 - x \]
\[ x = 2 \]

**Derivative:**
\[ f'(2) = \]

**Steps to Compute:**
1. **Definition of the Derivative:**
The derivative of a function \( f(x) \) at a point \( x = a \) is given by:
\[
f'(a) = \lim_{{h \to 0}} \frac{f(a + h) - f(a)}{h}
\]

2. **Substitute Given Values:**
Substitute \( a = 2 \) into the definition of the derivative:
\[
f'(2) = \lim_{{h \to 0}} \frac{f(2 + h) - f(2)}{h}
\]

3. **Function Evaluation:**
- \( f(2 + h) \) can be found by substituting \( 2 + h \) into the function \( f(x) \):
\[
f(2 + h) = 6(2 + h)^2 - (2 + h)
\]

- \( f(2) \) can be computed directly from the function:
\[
f(2) = 6(2)^2 - 2
\]

4. **Simplify Expression:**
Simplify the expression obtained and apply the limit as \( h \) approaches 0.

Place your solution in the blank provided:
\[ f'(2) = \boxed{} \]

Be sure to simplify your calculations and provide the final numerical value.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Given

VIEW

Step 2: Formula and calculation

VIEW

Solution

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

A parent downloads an app that allows him to monitor the amount of screen time his child is using on the phone. The child claims that she is only using 1 hour per day on the phone on average, but the app, over the course of 47 days, finds that the child used an average of 1.2 hours per day. The parent wants to know if the child is using an average of 1 hour per day for all days of use. a) The 1.2 is a/an: variable population parameter sample statistic observational unit b) The sample size is: 1 47 1.2 c) Each day is a/an: population parameter observational unit variable sample statistic
Will you show me step by step how to answer this
6.
Which is(are) true? Choose the best answer below, and show how you got your answer: i) 37nine = 35seventeen ii) 41six = 34seven iii) 100011two = 32eleven (i) and (ii) only (ii) and (iii) only (ii) only or None of these
Please type out the answer. I can't read most handwriting. (Plus, typing it out makes things easier for me) :D