Calculus Derivatives and Trigonometric Values

The video tutorial explains how to find the derivative of a function using the quotient rule. The function is given as f(x) = 3x / (2sin(x) + cos(x)), and the goal is to find f'(−π). The tutorial walks through applying the quotient rule, substituting −π into the derivative, and simplifying the expression using trigonometric values from the unit circle. The final result is presented as both an exact value and a decimal approximation, with an explanation of its significance as the slope of the tangent line at x = −π.

What is the main goal of the video tutorial?

• To find the integral of f(x)

• To graph the function f(x)

• To find the derivative of f(x) at x = -π

• To solve a quadratic equation

Which rule is used to find the derivative of a quotient of two functions?

• Chain Rule

• Quotient Rule

• Power Rule

• Product Rule

In the quotient rule, what is the expression for the derivative of the numerator?

• g times f prime

• f times g prime

• g prime times f

• f prime times g

What is the derivative of 3x?

• 1

• 0

• x

• 3

What is the derivative of 2sin(x) + cos(x)?

• 2cos(x) + sin(x)

• -2sin(x) + cos(x)

• 2sin(x) + cos(x)

• 2cos(x) - sin(x)

What is the value of cos(-π) using the unit circle?

• -1

• π

• 1

• 0

What is the value of sin(-π) using the unit circle?

• π

• -1

• 0

• 1

What is the simplified form of the denominator after substitution?

• 1

• 0

• -1

• 2

What is the final simplified expression for the derivative at x = -π?

• -3 + 6π

• 3 + 6π

• -3 - 6π

• 3 - 6π

What is the approximate numerical value of the derivative at x = -π rounded to four decimal places?

• -21.8496

• -20.8496

• -22.8496

• -23.8496