Linear Approximation and Tangent Lines

The video tutorial explains how to use linear approximation, specifically tangent lines, to approximate the value of a quotient, 1 divided by 0.504. It begins by defining the function f(x) = 1/x and selecting a nearby point, x = 0.5, for the tangent line. The tutorial then details the process of finding the equation of the tangent line, including calculating the point of tangency and the slope using derivatives. The tangent line equation is used to approximate the function value at x = 0.504, resulting in an approximation of 1.984. The video concludes by comparing this approximation to the exact value, highlighting the accuracy of the method.

What is the purpose of using a tangent line in linear approximation?

• To determine the maximum value of a function

• To find the exact value of a function

• To calculate the derivative of a function

• To approximate the value of a function near a point

Why is x = 0.5 chosen as a nearby value for approximation?

• It simplifies the calculation

• It is the only available value

• It is the exact value needed

• It is the midpoint of the function

What is the x-coordinate of the point of tangency?

• 0.504

• 1

• 0.5

• 2

How is the slope of the tangent line determined?

• By finding the second derivative

• By calculating the average rate of change

• By evaluating the derivative at the point of tangency

• By using the original function

What is the slope of the tangent line at x = 0.5?

• -2

• 2

• -4

• 4

Which form is used to derive the equation of the tangent line?

• Quadratic form

• Slope-intercept form

• Point-slope form

• Standard form

What is the equation of the tangent line derived in the video?

• y = 2x + 2

• y = -4x + 4

• y = 4x - 4

• y = -2x + 2

How is the approximation for 1/0.504 calculated using the tangent line?

• By calculating the average of nearby values

• By substituting x = 0.504 into the tangent line equation

• By using the midpoint formula

• By substituting x = 0.504 into the original function

What is the approximate value of 1/0.504 using the tangent line?

• 2.004

• 1.984

• 1.504

• 2.084

Why is the linear approximation close to the exact value?

• Because the function is linear

• Because the tangent line is close to the function near x = 0.504

• Because the function has no curvature

• Because the tangent line is far from the function