Given the function below
Find the equation of the tangent line to the graph of the function at . Answer in form.

Use the tangent line to approximate .

Compute the actual value of . What is the error between the function value and the linear approximation?
Answer as a positive value only,
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Question Help:

To find the equation of the tangent line to the graph of the function at , you’ll need to follow a few steps:

So, if we denote the function and its derivative appropriately, we’d derive the slope and then obtain the equation for the tangent line.

For the approximation at :

Now, calculate the actual function value and find the absolute error .

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!