Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the slope of a tangent line to the function f(x) = 25 - x^2 at the point (-5, 0) using the limit definition of the derivative. It demonstrates the process of simplifying the expression to find the derivative and then uses this to determine the slope of the tangent line. The tutorial also shows how to derive the equation of the tangent line in slope-intercept form and verifies the result graphically.

What is the initial function given in the problem?

• f(x) = x^2 - 25

• f(x) = 25 + x^2

• f(x) = 25 - x^2

• f(x) = 25x - x^2

What is the purpose of using the limit definition of the derivative in this problem?

• To calculate the area under the curve

• To find the maximum value of the function

• To find the y-intercept of the function

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

What is the expression for the limit definition of the derivative used in this lesson?

• lim (h -> 0) [f(x + h) + f(x)] / h

• lim (h -> 0) [f(x) + h - f(x)] / h

• lim (h -> 0) [f(x + h) - f(x)] / h

• lim (h -> 0) [f(x) - f(x - h)] / h

What is the result of squaring the binomial (negative five plus h)?

• 25 + 10h - h^2

• 25 - 10h - h^2

• 25 + 10h + h^2

• 25 - 10h + h^2

What is the common factor in the expression 10h - h^2?

• 1

• h^2

• 10

• h

What is the slope of the tangent line at the point (-5, 0)?

• -10

• 10

• 0

• 5

What is the equation of the tangent line at the point (-5, 0)?

• y = 10x + 50

• y = 10x - 50

• y = -10x - 50

• y = -10x + 50

How is the y-intercept of the tangent line determined?

• By setting x to zero in the original function

• By solving the equation f(x) = 0

• By using the point (-5, 0) in the tangent line equation

• By finding the derivative at x = 0

What does the graphical verification show?

• The tangent line is below the curve

• The tangent line is tangent to the curve at (-5, 0)

• The tangent line intersects the curve at multiple points

• The tangent line is parallel to the x-axis

What is the color of the tangent line in the graphical verification?

• Green

• Red

• Blue

• Yellow