Understanding Derivatives and Tangent Lines

What is the primary purpose of finding a derivative in calculus?

• To solve differential equations

• To find the area under a curve

• To determine the slope of a tangent line at any point

• To calculate the volume of a solid

Which rule is used to find the derivative of the function f(x) = 2x^2 - 5x + 1?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

What is the slope of the tangent line to the function f(x) = 2x^2 - 5x + 1 at x = 2?

• 5

• 4

• 3

• 2

How do you find the y-value of a point on a function?

• By differentiating the function

• By integrating the function

• By plugging the x-value into the original function

• By setting the derivative to zero

What is the slope of the tangent line to the function f(x) = 4/x - x at x = 4?

• -5/4

• -1/4

• 1/4

• 5/4

What is the y-intercept of the tangent line to the function f(x) = 4/x - x at x = 4?

• 4

• 2

• 1

• 3

What does a horizontal tangent line indicate about the slope at that point?

• The slope is negative

• The slope is positive

• The slope is zero

• The slope is undefined

For the function f(x) = x^3 - 3x + 4, at which x-values does the function have a horizontal tangent line?

• x = 2 and x = -2

• x = 3 and x = -3

• x = 1 and x = -1

• x = 0

How do you find the x-values where a function has a horizontal tangent line?

• Set the derivative equal to zero

• Set the original function equal to zero

• Integrate the function

• Find the second derivative

What is the y-value of the point where the function f(x) = x^3 - 3x + 4 has a horizontal tangent line at x = 1?

• 5

• 4

• 3

• 2