Understanding Tangent Lines and Derivatives

The video tutorial explains how to find the equation of a tangent line to a given function at a specific point. It starts by identifying the function and the point of tangency, then proceeds to find the derivative of the function. The derivative is evaluated at the given point to determine the slope of the tangent line. With the slope and the y-intercept known, the equation of the tangent line is formulated. The process is verified by graphing the function and the tangent line, ensuring the line is tangent to the curve at the specified point.

What is the given function in the problem?

• f(x) = 6x + 2 - 3e^x

• f(x) = 6x^2 + 2 - 3e^x

• f(x) = 6x - 2 - 3e^x

• f(x) = 6x + 2 + 3e^x

What is the y-intercept of the tangent line?

• -1

• 0

• 1

• 2

What is the derivative of the constant term in the function?

• 1

• 0

• 6

• -3

What is the derivative of e^x?

• 1

• e^x

• 0

• x

What is the slope of the tangent line at x = 0?

• 4

• 3

• 2

• 5

What is the equation of the tangent line?

• y = 2x + 1

• y = 2x - 1

• y = 3x - 1

• y = 3x + 1

How do we verify the equation of the tangent line?

• By checking the second derivative

• By graphing the function and the tangent line

• By solving algebraically

• By using a calculator

What color is the graph of the given function?

• Red

• Yellow

• Green

• Blue

What is the point of tangency on the graph?

• (1, 0)

• (0, -1)

• (1, -1)

• (0, 0)

What is the slope of the tangent line as confirmed by the graph?

• 3

• 2

• 4

• 5