Estimating Function Values Using Tangent Lines

The video tutorial explains how to estimate function values using a tangent line. Given f(2) = 9 and f'(2) = -8, the tutorial demonstrates finding the tangent line's equation at the point (2, 9) using point-slope form. This tangent line, L(x) = -8x + 25, is then used to approximate values of f(3), f(2.5), and f(2.25). The closer the x-value is to 2, the more accurate the approximation. The tutorial emphasizes understanding the relationship between a function, its derivative, and the tangent line for estimation purposes.

What are the given values for f(2) and f'(2) in the problem?

• f(2) = -9 and f'(2) = 8

• f(2) = 9 and f'(2) = 8

• f(2) = 8 and f'(2) = -9

• f(2) = 9 and f'(2) = -8

What does the value f'(2) = -8 represent in the context of the problem?

• The x-intercept of the function

• The slope of the tangent line at x = 2

• The y-intercept of the function

• The maximum value of the function

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

• Quadratic form

• Point-slope form

• Standard form

• Slope-intercept form

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

• y = -8x + 25

• y = 8x - 25

• y = 8x + 25

• y = -8x - 25

How is f(3) approximated using the tangent line?

• By calculating l(3) = -8 * 3 - 25

• By calculating l(3) = 8 * 3 + 25

• By calculating l(3) = 8 * 3 - 25

• By calculating l(3) = -8 * 3 + 25

What is the estimated value of f(3) using the tangent line?

• 1

• 5

• 3

• 9

Why is the approximation of f(2.5) expected to be more accurate than f(3)?

• Because 2.5 is further from 2 than 3 is

• Because 2.5 is a whole number

• Because 2.5 is a prime number

• Because 2.5 is closer to 2 than 3 is

What is the estimated value of f(2.5) using the tangent line?

• 7

• 3

• 1

• 5

What is the final approximation for f(2.25) using the tangent line?

• 7

• 5

• 9

• 3

What is the main purpose of using the tangent line in this problem?

• To calculate the area under the curve

• To determine the maximum value of the function

• To approximate the function values near x = 2

• To find the exact values of the function