https://file.aiquickdraw.com/mathbot/file/17261008229494y1fy1pc.jpg

Let's work through this step by step based on the provided image:

### Given function:
\[ f(x) = 3x^2 - 2x \]

---

### (a) **Find the derivative \( f'(x) \)**:
We will use the power rule to differentiate each term.

1. The derivative of \( 3x^2 \) is \( 6x \).
2. The derivative of \( -2x \) is \( -2 \).

Thus, the derivative is:
\[
f'(x) = 6x - 2
\]

---

### (b) **Find the instantaneous rate of change of \( f(x) \) at \( x = -1 \)**:
The instantaneous rate of change is simply the value of the derivative at \( x = -1 \). Substitute \( x = -1 \) into the derivative \( f'(x) = 6x - 2 \):

\[
f'(-1) = 6(-1) - 2 = -6 - 2 = -8
\]

The instantaneous rate of change at \( x = -1 \) is \( -8 \).

---

### (c) **Find the slope of the tangent to the graph of \( y = f(x) \) at \( x = -1 \)**:
The slope of the tangent line is the same as the instantaneous rate of change, which we found in part (b). Therefore, the slope of the tangent line at \( x = -1 \) is also \( -8 \).

---

### (d) **Find the point on the graph of \( y = f(x) \) at \( x = -1 \)**:
To find the point, we need to substitute \( x = -1 \) into the original function \( f(x) = 3x^2 - 2x \).

\[
f(-1) = 3(-1)^2 - 2(-1) = 3(1) + 2 = 3 + 2 = 5
\]

So, the point on the graph is \( (-1, 5) \).

---

### Summary of Answers:
(a) \( f'(x) = 6x - 2 \)  
(b) Instantaneous rate of change at \( x = -1 \) is \( -8 \)  
(c) Slope of the tangent at \( x = -1 \) is \( -8 \)  
(d) Point on the graph at \( x = -1 \) is \( (-1, 5) \)

---

Do you have any questions or need further details?

Here are 5 related questions:
1. How do you use the power rule for more complex functions?
2. What is the difference between instantaneous rate of change and average rate of change?
3. How do you find the equation of a tangent line given a function and a point?
4. How do higher-order derivatives affect the shape of a graph?
5. How can the second derivative test be used to determine concavity?

**Tip**: To find the derivative of a sum or difference of terms, you can differentiate each term individually and then combine them.

Let f(x) = 3x^2 - 2x. (a) Use the definition of derivative and the procedure given to find the derivative. (b) Find the instantaneous rate of change of f(x) at x = -1. (c) Find the slope of the tangent to the graph at x = -1. (d) Find the point on the graph at x = -1.

Learn how to find the derivative of f(x) = 3x^2 - 2x using the power rule and solve related problems including finding the instantaneous rate of change and the slope of the tangent line at x = -1. This guide is suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation