Understanding Antiderivatives and Integration Constants

The video tutorial explains how to find the unique solution for a function given f(1)=2. It involves determining the antiderivative of the function using the power rule, formulating the function with a constant of integration, and solving for this constant using the given condition. The final function is presented with the exact value of the constant.

What is the initial condition given in the problem?

• f(3) = 4

• f(2) = 3

• f(1) = 2

• f(0) = 1

Which rule is applied to find the antiderivative of the function?

• Chain Rule

• Power Rule

• Quotient Rule

• Product Rule

What is the antiderivative of x^4?

• x^2/2

• x^3/3

• x^4/4

• x^5/5

What is the next step after finding the antiderivative?

• Solve for x

• Differentiate the function

• Integrate again

• Substitute the initial condition

What equation is formed after substituting x = 1?

• 1/5 + 1/3 - 1 + c = 0

• 1/5 + 1/3 - 1 + c = 2

• 1/5 + 1/3 - 1 + c = 1

• 1/5 + 1/3 - 1 + c = 3

What is the value of 1/5 + 1/3 - 1?

• -1/15

• 7/15

• -7/15

• 1/15

How do you solve for the constant c?

• Subtract 7/15 from both sides

• Multiply both sides by 15

• Add 7/15 to both sides

• Divide both sides by 2

What is the final value of the constant c?

• 15/37

• 37/15

• 30/15

• 7/15

What is the final form of the function f(x)?

• 1/5 x^5 + 1/3 x^3 - x + 15/37

• 1/5 x^5 + 1/3 x^3 - x + 37/15

• 1/5 x^5 + 1/3 x^3 - x + 7/15

• 1/5 x^5 + 1/3 x^3 - x + 30/15

What is the purpose of finding the constant of integration?

• To determine the slope of the function

• To find the unique solution for the function

• To calculate the derivative

• To solve for x