Math Problem Statement
find an equation of the tangent line to f(t) at a=-1
Solution
To find the equation of the tangent line to the function at , we can follow these steps:
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Find the point of tangency: This is .
So, the point of tangency is . -
Find the slope of the tangent line: This is given by , which is already calculated in the problem as .
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Use the point-slope form of the equation of a line: The point-slope form is: where is the slope and is the point of tangency. Substituting , , and , we get:
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Simplify the equation:
This is the equation of the tangent line in exact form.
Would you like a more detailed explanation of any part of the process?
Related Questions:
- How do we find the derivative using the definition of the derivative?
- Why is the point-slope form used for tangent lines?
- What is the difference between finding a secant line and a tangent line?
- How does the chain rule apply when differentiating functions like ?
- What does the slope of the tangent line represent geometrically?
Tip: The point-slope form is very useful for writing the equation of a tangent line because it directly uses the slope and a known point on the curve.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Tangent Lines
Formulas
f'(t) = lim (t -> a) [(f(t) - f(a)) / (t - a)]
Point-slope form: y - y1 = m(t - t1)
Theorems
Definition of the derivative
Point-slope formula for tangent lines
Suitable Grade Level
Grades 11-12 (Advanced Calculus)
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