Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the y-intercept of a tangent line to the curve f(x) = 1/x at a point where x = k. It begins by visualizing the graph and the tangent line, then calculates the slope using the derivative. Finally, it solves for the y-intercept using the slope and a known point, concluding with the equation of the tangent line.

What is the main objective of the problem discussed in the video?

• To find the y-intercept of a tangent line

• To calculate the area under a curve

• To find the x-intercept of a curve

• To determine the maximum value of a function

What does the graph of f(x) = 1/x look like?

• A hyperbola

• A parabola

• A straight line

• A circle

At which point is the tangent line drawn on the curve f(x) = 1/x?

• x = 0

• x = 1

• x = k

• x = -1

What mathematical concept is used to find the slope of the tangent line?

• Matrix operations

• Algebraic manipulation

• Differentiation

• Integration

What is the derivative of f(x) = 1/x?

• -1/x^2

• x^2

• -x^2

• 1/x^2

In the slope-intercept form of a line, what does 'm' represent?

• The y-intercept

• The x-intercept

• The midpoint

• The slope

What is the equation of a line in slope-intercept form?

• y = ax + b

• y = mx + b

• y = x^2 + 2x + 1

• y = ax^2 + bx + c

How do you solve for the y-intercept 'b' in the equation y = mx + b?

• By setting y = 0

• By setting b = 0

• By setting x = 0

• By setting m = 0

What is the y-intercept of the tangent line at x = k?

• 1/k

• 2/k

• -1/k

• -2/k

What is the final equation of the tangent line?

• y = -1/k^2 x + 2/k

• y = 1/k^2 x - 2/k

• y = -1/k x + 1/k

• y = 1/k x - 1/k