Math Problem Statement
Solution
To solve the problem shown in the image, we need to find the equation of the tangent line to the curve at the point .
Steps:
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Find the derivative : To get the slope of the tangent line, we need the derivative of .
The function is:
Using the quotient rule:
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Evaluate the derivative at to find the slope: Substituting into the derivative:
So, the slope of the tangent line at is 0.8.
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Use the point-slope form of the equation of a line: The point-slope form is given by: where is the slope and is the point of tangency.
Plugging in , , and : Simplifying:
Thus, the equation of the tangent line is:
Let me know if you want further details, or if you have any questions!
Related Questions:
- What is the general formula for the quotient rule in calculus?
- How do you find the second derivative of the function ?
- What happens to the tangent line if the point of tangency changes?
- How can you check if a line is tangent to a curve using limits?
- What is the geometric interpretation of the slope of a tangent line?
Tip:
When using the quotient rule, make sure to correctly apply the formula to avoid calculation errors!
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Tangent Lines
Quotient Rule
Formulas
Quotient Rule: (f'(x)g(x) - f(x)g'(x)) / g(x)^2
Point-Slope Form: y - y1 = m(x - x1)
Theorems
Quotient Rule
Tangent Line Theorem
Suitable Grade Level
Grades 11-12 or Introductory College Level
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