Understanding the Unique Solution of a Function

The video tutorial explains how to find the unique solution for a function given an initial condition. It begins by determining the anti-derivative of the function, which involves integrating terms and adding a constant of integration. The initial condition, f(0) = 3, is used to solve for the constant C. The final unique solution is derived by substituting the value of C back into the function. The tutorial concludes with a summary of the steps taken to find the solution.

What is the initial condition given for the function f(x)?

• f(2) = 5

• f(3) = 6

• f(0) = 3

• f(1) = 4

What is the anti-derivative of 4^x?

• 4^x * ln(4)

• 4^x / ln(4)

• ln(4) / 4^x

• 4^x + ln(4)

How is the anti-derivative of x expressed?

• x^2

• x^2 / 2

• 2x

• x / 2

What is the value of 4^0?

• 1

• 2

• 4

• 0

What operation is performed to solve for c?

• Subtraction

• Division

• Addition

• Multiplication

What is the final expression for f(x) in terms of c?

• 4^x / ln(4) + 1/2 x^2 + c

• 4^x * ln(4) + 1/2 x^2 + c

• 4^x / ln(4) - 1/2 x^2 + c

• 4^x + ln(4) + 1/2 x^2 + c

What is the value of c after solving the equation?

• 3 * ln(4)

• 1/ln(4) - 3

• 3 - 1/ln(4)

• 3 + 1/ln(4)

What is the purpose of finding the constant c?

• To determine the slope of the function

• To find the unique solution for f(x)

• To solve for x

• To calculate the derivative

What is the role of the constant of integration in the function?

• It changes the variable

• It adjusts the y-intercept

• It determines the slope

• It provides the unique solution

What is the final form of the unique solution for f(x)?

• 4^x / ln(4) - 1/2 x^2 + 3 - 1/ln(4)

• 4^x * ln(4) + 1/2 x^2 + 3 - 1/ln(4)

• 4^x / ln(4) + 1/2 x^2 + 3 - 1/ln(4)

• 4^x + ln(4) + 1/2 x^2 + 3 - 1/ln(4)