Calc 2/13

Focus on linearization and its nuances.
Students will receive an additional assignment with eight problems (four canceled).
Emphasis on collaboration and understanding during problem-solving sessions.

Continue linearization for a couple of days before moving on to L'Hôpital's rule.
Quiz scheduled for Tuesday.
Two days to start the next unit before a four-day break.

Start from page 35 of the notes, specifically looking into the problem at the top.
Discuss and work through problems collaboratively for a better learning experience.

Understanding Zero:
- Recognize that zero, roots, and x-intercepts are the same concepts in calculations.
- Use x-intercepts to assist in solving problems involving linear approximation.
Linear Approximation Example:
- Approximate the cube root of 8.1 using the function ( f(x) = \sqrt[3]{x} ).
- Use the point ( x = 8 ) because it simplifies calculations (easy cube root).
- Understand overestimation or underestimation based on concavity of the function.
Calculating Slope and Tangent Line:
- Calculate the slope using the derivative: ( f'(x) = \frac{1}{3}x^{-\frac{2}{3}} ) at x=8.
- Build the tangent line equation from calculated slope and point coordinates: ( y - 2 = \frac{1}{12}(x - 8) ).
Evaluation of Approximation:
- Substitute ( 8.1 ) into the tangent line equation for approximation.
- Avoid reliance on calculators where possible; practice mental math with decimals and fractions for simplicity.

Approximating the Fourth Root of e:
- Use function ( f(x) = e^x ) to find the fourth root of e, e.g., ( e^{\frac{1}{4}} ).
- Work through tangent line approximation starting from easy points like ( x = 0 ).
- Calculate the approximation error by subtracting linear approximation from the actual value using a calculator.

Complete problems: 1, 2, 5, and 7 from the new worksheet.
Focus on understanding approximation accuracy (over or under estimation) based on second derivative analysis.
Encourage students to work collaboratively and ask questions as needed.