Math Problem Statement

Bestem tangent for f.docx

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

Solution

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


Given function: f(x)=2x3+3xf(x) = 2x^3 + 3x Given point: x0=1x_0 = -1

1. Find f(1)f(-1):

f(1)=2(1)3+3(1)=23=5f(-1) = 2(-1)^3 + 3(-1) = -2 - 3 = -5 ✅ This part is correct! 😊

2. Find f(1)f'(-1):

First, we need to differentiate f(x)f(x): f(x)=6x2+3f'(x) = 6x^2 + 3 Now, substitute x0=1x_0 = -1: f(1)=6(1)2+3=6(1)+3=6+3=9f'(-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(x0)(xx0)+f(x0)y = f'(x_0)(x - x_0) + f(x_0) Substitute the values: y=9(x(1))+(5)y = 9(x - (-1)) + (-5) y=9(x+1)5y = 9(x + 1) - 5 y=9x+95y = 9x + 9 - 5 y=9x+4y = 9x + 4 ✅ This part is correct! 😊


So, the corrected solution is:

y=9x+4y = 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 x0=1x_0 = -1?
  4. If the point of tangency is x0=2x_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!

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Math Problem Analysis

Mathematical Concepts

Calculus
Differentiation
Tangent Line Calculation

Formulas

f(x) = 2x^3 + 3x
f'(x) = 6x^2 + 3
Tangent Line: y = f'(x_0)(x - x_0) + f(x_0)

Theorems

-

Suitable Grade Level

Grades 10-12