Curve Sketching Basics

When does a curve have a horizontal tangent line?

• When x = 0

• When y = 0

When does a curve have a vertical tangent line

• When x = 0

• When y = 0

What is the justification for a function to be decreasing?

• Function is increasing

• First Derivative is Positive

• First Derivative is Negative

• Second Derivative is Positive

What must be true for an x-value to be a critical number?

• A) The x-value is in the domain of the function

• B) The derivative at the x-value is 0 or undefined

• C) The derivative at the x-value is equal to the average value

• D) Both A and B

What is the justification for a function having a local minimum?

• The derivative changes from positive to negative

• The derivative changes from negative to positive

• The derivative changes from increasing to decreasing

• The derivative changes from decreasing to increasing?

For a point to be a local maximum according to the second derivative, f'(x)=0 and what else?

• The second derivative is negative

• The second derivative is positive

• The function crosses the x-axis

• The function crosses the y-axis

• cosx
• -cosx
• -sinx cosx
• sinx cosx

• sinx
• -sinx
• -cscx cotx
• sec²x

Find the derivative.

• x⁴ cosx - 4x³sinx
• x⁴ cosx + 4x³sinx
• 4x³cosx
• -4x³cosx

If a function has a derivative that is negative, what does that tell you?

• The function is increasing
• The function is decreasing
• The concavity of the function is up
• The concavity of the function is down