Question
Given the function below
Find the equation of the tangent line to the graph of the function at
Use the tangent line to approximate
Compute the actual value of
Answer as a positive value only,
lerrorl
Question Help:
Find the equation of the tangent line to the graph of the function at
Use the tangent line to approximate
Compute the actual value of
Answer as a positive value only,
lerrorl
Question Help:
Ask by Sanders Barrett. in the United States
Mar 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Cannot determine the tangent line or approximation without the function
Solution
To find the equation of the tangent line to the graph of the function
- Find the derivative of the function
to determine the slope of the tangent line at . - Evaluate the function
at to find the point of tangency. - Use the point-slope form of the line to write the equation of the tangent line in
form. - Use the tangent line to approximate
. - Compute the actual value of
. - Calculate the error between the function value and the linear approximation.
Since the function
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Simplify this solution Beyond the Answer
To find the equation of the tangent line to the graph of the function at
- Find the derivative
at to get the slope of the tangent line. - Evaluate the function
to find the y-coordinate at that point. - Use the point-slope form
to write the equation of the tangent line.
So, if we denote the function and its derivative appropriately, we’d derive the slope
For the approximation at
- Use
to find .
Now, calculate the actual function value
If you plug in numbers, you should find:
The equation of the tangent line can provide us with valuable insights about the function’s behavior, especially near the point of tangency. For instance, tangent lines play a vital role in Newton’s method for finding roots of functions, where the line acts as an initial guess for the next iteration.
Understanding how tangent lines approximate functions beautifully captures the essence of calculus: using linear functions to understand and predict behavior of more complex shapes! It’s almost like being a detective, using simple clues (like slopes and points) to unveil the mysteries of intricate curves.
Whether you’re approximating values or solving for intersection points, having a grasp on how lines touch curves opens a world of mathematical adventures! Want to dive deeper? Explore calculus textbooks that focus on applications in physics and engineering, or even online resources that demonstrate real-life problems solved using tangent lines!
