Sect 4.9 #31, antiderivative of 1+cube root of x

TL;DR

Learn how to find the original function when given its derivative and a point on the function.

Transcript

here we are given the derivative some function we have f prime of x that's equal to 1 plus 3 square of x and then we also know a point on the function which is f of 4 is equal to 25. our goal is to figure out a formula for the original function because we're given the derivative we have to do the anti-derivative so that we can get the original but ... Read More

Key Insights

• ✊ The process of finding the original function from its derivative involves performing anti-derivative using the reverse power rule.
• 😀 The unknown constant 'c' is introduced in the anti-derivative to account for any constant term in the original function.
• 😥 Using a known point on the function helps in solving for the unknown constant 'c'.
• 👻 The reverse power rule allows for finding the anti-derivative of functions that include fractional exponents.

Questions & Answers

Q: How do you find the original function when only given its derivative?

To find the original function, you need to perform the reverse process of differentiation, known as anti-derivative or integration. This involves using the reverse power rule and solving for any unknown constants using given points on the function.

Q: What is the reverse power rule?

The reverse power rule states that when performing anti-derivative for x^n, you add one to the exponent (n+1) and then divide the new exponent by the new coefficient [(n+1)/(n+1)]. This allows you to determine the original function based on its derivative.

Q: What does the constant 'c' represent in the original function formula?

The constant 'c' represents an unknown constant that is added to the anti-derivative result. It is necessary to include 'c' because the derivative of a constant is always 0. By solving for 'c' using a given point on the function, the value can be determined.

Q: How does the known point on the function help in finding the original function?

A known point on the function, such as f(4) = 25, allows for solving the equation by substituting the x-value into the function formula. By solving for 'c', the value of the unknown constant can be determined, finalizing the original function.

Summary & Key Takeaways

• Given the derivative f'(x) = 1 + 3√x, the goal is to find the formula for the original function.

• By rewriting the derivative as 1 + 3x^(1/2), the reverse power rule can be used to find the anti-derivative of the function.

• Using the reverse power rule and the known point f(4) = 25, the original function is determined to be f(x) = x + 2x^(3/2) + 5.