Tangent Line Approximations and Errors

The video tutorial explains how to use tangent line approximations to estimate function values near a point of tangency. It begins with given values of f(1) and f'(1) and demonstrates the process of approximating f(2), f(1.5), and f(1.25) using the tangent line method. The tutorial highlights the concept of differential y and delta y, and how they relate to the tangent line and the original function. Each approximation is calculated step-by-step, showing the small errors involved in the process.

What initial values are given for the function and its derivative?

• f(1) = 3 and f'(1) = -3

• f(1) = 2 and f'(1) = -2

• f(1) = 1 and f'(1) = -1

• f(1) = 0 and f'(1) = 0

What is the main idea behind using tangent lines for approximation?

• Tangent lines are used to determine the slope of a curve.

• Tangent lines can approximate function values near the point of tangency.

• Tangent lines are used to find the maximum value of a function.

• Tangent lines provide exact values for functions.

What is the estimated value of f(2) using the tangent line approximation?

• 1

• 2

• -1

• 0

What is the actual value of f(2) compared to the approximation?

• The actual value is -1.

• The actual value is 1.

• The actual value is 2.

• The actual value is 0.

How much does x increase when approximating f(1.5)?

• 0.25

• 1.5

• 1

• 0.5

What is the estimated value of f(1.5) using the tangent line approximation?

• 0

• 2

• 1

• 1.5

What is the error like when approximating f(1.5)?

• There is no error.

• The error is very small.

• The error is moderate.

• The error is very large.

How much does x increase when approximating f(1.25)?

• 1

• 0.25

• 1.25

• 0.5

What is the estimated value of f(1.25) using the tangent line approximation?

• 1.25

• 1.5

• 1

• 2

How does the error compare when approximating f(1.25)?

• The error is very small.

• There is no error.

• The error is very large.

• The error is moderate.