https://file.aiquickdraw.com/mathbot/file/1726053410460h0if4zk2.docx

ret min opgave. ret kun fejlene, og lav smiley ved dem der er rigtige

Here is the corrected solution for your task, along with smiley marks for correct parts:

---

**Given function:**
\[ f(x) = 2x^3 + 3x \]
**Given point:** \( x_0 = -1 \)

### 1. Find \( f(-1) \):
\[ f(-1) = 2(-1)^3 + 3(-1) = -2 - 3 = -5 \]
✅ This part is correct! 😊

### 2. Find \( f'(-1) \):
First, we need to differentiate \( f(x) \):
\[ f'(x) = 6x^2 + 3 \]
Now, substitute \( x_0 = -1 \):
\[ f'(-1) = 6(-1)^2 + 3 = 6(1) + 3 = 6 + 3 = 9 \]
❌ This part was incorrect (you had 8 instead of 9).

### 3. Equation of the tangent line:
Use the formula for the tangent line:
\[ y = f'(x_0)(x - x_0) + f(x_0) \]
Substitute the values:
\[ y = 9(x - (-1)) + (-5) \]
\[ y = 9(x + 1) - 5 \]
\[ y = 9x + 9 - 5 \]
\[ y = 9x + 4 \]
✅ This part is correct! 😊

---

So, the corrected solution is:

\[
y = 9x + 4
\]

---

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How do you find the derivative of a polynomial function?
2. What is the general formula for the tangent line of a function at a point?
3. How would you find the normal line to the function at \( x_0 = -1 \)?
4. If the point of tangency is \( x_0 = 2 \), what would the tangent line equation be?
5. What is the significance of the slope of the tangent line in the context of rates of change?

**Tip:** Always double-check the differentiation process, especially when calculating slopes for tangent lines!

Bestem tangent for f(x)=2x^3 +3x. X0 =-1. F(-1)=2*-1^3 +3*-1=-2-3=-5. F`(-1)=6*-1^2 +3 =8. Bruger ligningen Y=f`(x0)(x-x0)+f(x0). Y=9*(x—1)+(-5)=9x-9-5. Y=9x+4.

Learn how to find the tangent line for the function f(x) = 2x^3 + 3x at the point x = -1. This example covers the steps to calculate the derivative and apply the formula for the tangent line. Suitable for high school 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