Arc Length and Derivatives Assessment

The video tutorial explains how to find the arc length of the function y = 4x^(3/2) on the interval from 2 to 4. It begins by introducing the concept of arc length and the formula used to calculate it. The tutorial then demonstrates how to derive the function's derivative and apply the arc length formula. Using u-substitution, the integral is simplified and evaluated to find the arc length. The result is verified using a graphing calculator, confirming the accuracy of the calculations. The arc length is approximately 20.7837 units.

What is the function for which we are finding the arc length?

• y = 2x^(3/2)

• y = 5x^(1/2)

• y = 4x^(3/2)

• y = 3x^(2/3)

What is the interval over which the arc length is calculated?

• [0, 5]

• [2, 4]

• [1, 3]

• [3, 6]

Which formula is used to calculate the arc length of a function?

• Integral of f(x) dx

• Integral of sqrt(1 + (f'(x))^2) dx

• Integral of f'(x) dx

• Integral of (f(x))^2 dx

What is the derivative of the function y = 4x^(3/2)?

• 5x^(1/2)

• 4x^(1/2)

• 3x^(1/2)

• 6x^(1/2)

What substitution is used to simplify the integral?

• u = 1 + 9x

• u = 1 + 4x

• u = 1 + 6x

• u = 1 + 36x

What are the new limits of integration after substitution?

• 70 to 140

• 60 to 120

• 80 to 160

• 73 to 145

What is the antiderivative of u^(1/2) with respect to u?

• u^(3/2) / (3/2)

• u^(2/3) / (2/3)

• u^(1/2) / (1/2)

• u^(3/2) / (2/3)

What is the approximate arc length calculated?

• 20.7837

• 19.8765

• 18.5678

• 22.3456

Which tool is used to verify the arc length calculation?

• Basic calculator

• Algebraic calculator

• Scientific calculator

• Graphing calculator

What is the purpose of using a graphing calculator in this context?

• To verify the accuracy of the arc length calculation

• To find the derivative of the function

• To plot the graph of the function

• To solve algebraic equations