Understanding Derivatives and the Fundamental Theorem of Calculus

The video tutorial explains how to find the derivative of a function defined by a definite integral using the fundamental theorem of calculus. The function g(x) is introduced, and the problem is to find g'(27). The tutorial demonstrates a shortcut using the fundamental theorem, which simplifies the process by directly evaluating the inner function at x, leading to the solution that g'(27) equals 3.

What is the function g(x) defined as in the problem?

• The derivative of a polynomial

• The definite integral from 19 to x of the cube root of t

• The sum of squares from 1 to x

• The product of x and a constant

What is the initial method suggested to find the derivative of g(x)?

• Using numerical approximation

• Applying the product rule

• Taking the anti-derivative and evaluating it

• Using the chain rule

What theorem provides a shortcut for finding the derivative of g(x)?

• The Pythagorean Theorem

• The Intermediate Value Theorem

• The Mean Value Theorem

• The Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, what is the derivative of g(x)?

• The logarithm of x

• The reciprocal of x

• The square of x

• The cube root of x

What is the value of g'(27) according to the solution?

• 27

• 3

• 9

• 1